Hearing Rhythm and Meter: Analyzing Metrical Consonance and Dissonance in Common-Practice Period Music 2019017506, 2019021212, 9781351204316, 9780815384472, 9780815384489

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Hearing Rhythm and Meter Hearing Rhythm and Meter: Analyzing Metrical Consonance and Dissonance in Common-Practice Period Music is the first book to present a comprehensive course text on advanced analysis of rhythm and meter. This book brings together the insights of recent scholarship on rhythm and meter in a clear and engaging presentation, enabling students to understand topics including hypermeter and metrical dissonance. From the Baroque to the Romantic era, Hearing Rhythm and Meter emphasizes listening, enabling students to recognize meters and metrical dissonances by type both with and without the score. The textbook includes exercises for each chapter and is supported by a full-score anthology. Matthew Santa is Professor of Music Theory and Chair of the Music Theory and Composition Area at the Texas Tech University School of Music.

Hearing Rhythm and Meter Analyzing Metrical Consonance and Dissonance in Common-Practice Period Music

Matthew Santa

First published 2020 by Routledge 52 Vanderbilt Avenue, New York, NY 10017 and by Routledge 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN Routledge is an imprint of the Taylor & Francis Group, an informa business © 2020 Matthew Santa The right of Matthew Santa to be identified as author of this work has been asserted by him in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Names: Santa, Matthew, author. Title: Hearing rhythm and meter : analyzing metrical consonance and dissonance in common-practice period music / Matthew Santa. Description: New York ; London : Routledge, 2019. | Includes bibliographical references and index. Identifiers: LCCN 2019017506 (print) | LCCN 2019021212 (ebook) | ISBN 9781351204316 | ISBN 9780815384472 (hardback) | ISBN 9780815384489 (pbk.) Subjects: LCSH: Musical meter and rhythm. | Music—19th century—Analysis, appreciation. | Music—18th century—Analysis, appreciation. Classification: LCC MT42 (ebook) | LCC MT42 .S26 2019 (print) | DDC 781.2/209—dc23 LC record available at https://lccn.loc.gov/2019017506 ISBN: 978-0-8153-8447-2 (hbk) ISBN: 978-0-8153-8448-9 (pbk) ISBN: 978-1-351-20431-6 (ebk) Typeset in Adobe Caslon Pro by Apex CoVantage, LLC Visit the eResources: www.routledge.com/9780815384489

Contents

Preface

ix

Chapter 1

Notated Meter and Sounding Meter Establishing Meter Conducting and Maintaining Meter Pulse Layers Steps to Identifying Meter Without the Score Factors in Metrical Perception Subjectivity in Metrical Interpretation Continuity and Conflicting Accents Polymeters Versus Multiple Interpretations Performing the Sounding Meter Versus the Notated Meter The History of Meter and Metrical Notation in Western Music Chapter Review Homework Assignments Further Reading

1 4 6 7 9 9 13 14 14 15 16 17 19 20

Chapter 2

Hypermeter and Phrase Rhythm Hypermeter Hypermeter at Fast and Slow Tempos Phrase Rhythm Levels of Metrical Structure and Grouping Structure Annotating a Metrical Interpretation Multiple Levels of Hypermeter Elisions and Hypermetrical Reinterpretations Conducting on and off the Podium Chapter Review

21 21 23 23 26 28 30 31 32 32

vi

Contents

Homework Assignments Further Reading

34 36

Chapter 3

Phrase Expansions and Hypermeter The Phrase Prefix Internal Repetitions Cadential Extensions Composed-Out Ritardandos and Fermatas Parenthetical Insertions Links Thematic Versus Non-Thematic Material Chapter Review Homework Assignments Further Reading

37 37 40 43 44 45 46 47 48 49 51

Chapter 4

Metrical Dissonances Metrical Consonance and Dissonance Grouping Dissonances Displacement Dissonances Broken Chords and Accompanimental Patterns Labeling Metrical Dissonances Identifying Grouping and Displacement Dissonances Subtactus-Level and Hypermetrical Dissonances Direct and Indirect Metrical Dissonances Asymmetrical Meters Subliminal Dissonances Comparing Metrical Dissonances Finding Metrical Dissonances in a Score A Sample Analysis: Bach, Partita, BWV 826, Sarabande Chapter Review Homework Assignments Further Reading

53 53 54 57 59 60 62 63 63 65 66 67 69 70 72 74 76

Chapter 5

Metrical Processes Augmentation and Diminution Families of Dissonances Tightening and Loosening Finding the Best Label for a Metrical Dissonance Metrical Upshifting and Downshifting Meter Changes and Tempo Modulation Fragmentation

77 77 79 81 82 83 86 86

Contents

Stretto Chapter Review Homework Assignments Further Reading

vii

88 89 90 92

Chapter 6

Metrical Maps Dissonance Logs Dissonance Logs for Slow Movements Conducting Plans Multiple Interpretations and Metrical Maps Chapter Review Homework Assignments Further Reading

93 93 100 104 108 109 110 111

Chapter 7

Meter in Music With Text Conventional English and German Text Setting Conventional Italian Text Setting Text Painting Speech Song and Lyric Song Methods of Textual Emphasis Metrical Dissonance as a Means of Textual Emphasis Metrical Maps of Vocal Music Chapter Review Homework Assignments Further Reading

113 113 115 116 119 121 122 124 126 128 130

Chapter 8

Form and Meter Using Metrical Changes to Delineate Formal Boundaries Metrical Changes and Formal Functions Case Study 1: Schumann’s Quartet, Op. 41, No. 1, First Movement Case Study 2: Brahms, Variations on a Theme by Haydn Chapter Review Homework Assignments Further Reading

131 131 132

Appendix

163

Bibliography

165

Index

173

136 143 157 158 161

Preface

This book owes its existence to many writers, and it was born from my struggles to synthesize their work. Its earliest seeds can be traced back to three articles by Carl Schachter on the topic of rhythm and linear analysis and to William Rothstein’s book Phrase Rhythm in Tonal Music. These writings greatly influenced my own approach to analysis, but the struggle to synthesize didn’t really come until I read Harald Krebs’ book Fantasy Pieces: Metrical Dissonance in the Music of Robert Schumann. Here was a book that I found equally valuable for its ability to provide insights into how music achieves its expressive power, but it was written from a fundamentally different perspective, and it was often hard to fully comprehend how the insights coming from these different perspectives fit together. As I was synthesizing the material for myself over a period of ten years while simultaneously pursuing a host of other projects, along came Danuta Mirka’s book Metric Manipulations in Haydn and Mozart, and it added many of the final pieces to the puzzle I was trying to solve: how do I present this research to students in a way that would benefit them as performers? Mirka’s book was an equally brilliant study, and it held new insights that I felt must be part of any course on rhythm and meter. It was at this point that I taught my first graduate course called Rhythm and Meter, and then decided to write the book you are reading now. Hearing Rhythm and Meter is a textbook designed for upper-division undergraduate courses as well as for first- or second-year graduate courses on rhythm and meter. It offers the following advantages over simply reading the sources already cited: • •

It is comprehensive, covering rhythm and meter in music from the Baroque to the Romantic (i.e. the common-practice period). It is appropriate for college readers, without lengthy discussions or references to musical works that are unfamiliar to college-age music students.

x

Preface

•

•

It is shaped by recent scholarship, but it does not read like a treatise. Research by Richard Cohn, Christopher Hasty, Ray Jackendoff, Harald Krebs, Fred Lerdahl, Justin London, Yonatan Malin, Danuta Mirka, William Rothstein, Carl Schachter, David Temperley, and others (much of which has been published in the last three decades) is taken into account, but the ideas are always presented in a fashion that is immediately accessible to college readers. It is supported by its own anthology, and, unlike many recent anthologies, the anthology for Hearing Rhythm and Meter is made up primarily of masterworks in full score.

Repertoire and the Anthology The choice of repertoire in this textbook and its accompanying anthology might seem odd to those who, like me, love twentieth-century music, whether it be concert music, rock, jazz, etc. Such music is often very rhythmically complicated, and it may seem as though any book on the topic of rhythm and meter should address it head on. There are many reasons this repertoire is not included, but the primary reason is pedagogical: to ensure student success, the basic concepts associated with rhythm and meter in music are presented first in their simplest forms, and in ways that are as accessible as possible to the student. Concepts are simultaneously taught aurally, visually, and kinesthetically whenever possible, and the most accessible and direct way to understand many of these concepts visually is through score study, a kind of study that is problematic in many ways when dealing with twentieth-century repertoires. When dealing with twentiethcentury popular music, its notation is often a transcription, or presented in the form of a lead-sheet, and in either case it is harder for a student to relate the visual representation of the music to its sounding realization. When dealing with twentieth-century concert music, much of it is far less accessible to the student in terms of harmonic and formal analysis than the music of the common-practice period, and thus less ideal as a context for presenting foundational concepts. The students taking the course for which this textbook is intended have usually had three or four semesters of courses focusing on the analysis of tonal music, with only one semester (or maybe less!) focusing on post-tonal music. Originally, I considered making the last chapter about applying the concepts taught in all the previous chapters to music written after 1900, but it seemed redundant, making the book longer without saying anything new, while driving the cost of the book up considerably as now permissions would be required for all the musical

Preface

xi

examples. I decided instead to write a book which I believe could conceivably be covered in the first two-thirds of a semester, leaving the last third of the semester for the application of the concepts to whatever repertoire the instructors wish to have their students study. All the works in the anthology are in full score. While some anthologies in the past have chosen to provide piano reductions for larger works, many instructors believe that students need as much practice reading full score as possible. With that in mind, the selections in the anthology are also printed as close to 100% of the original printing size as possible, so that the notes will be as easy to read as possible.

The Approach Instructors familiar with the recent research upon which this book is based will find many important debates in the field missing, sacrificed for the sake of concision and for a unified narrative. Three are worth mentioning in this preface, all of them related specifically to the understanding of hypermeter: duple versus quadruple groupings, beginning accents versus end accents (i.e. the “shadow meter” debate), and how phrases should be defined relative to grouping structure. In order to simplify things for the student, the book chooses one side of each debate, makes its position clear at the beginning, and does not bother rehashing the debate in full for the students. Such a review would only slow down the presentation of the concepts, and it would distract the students from the task of discovering how the concepts presented shape the repertoire. In this textbook, I have chosen to take four-bar groupings over two-bar grouping whenever both seem plausible, to assume that beginning accents should be preferred to end accents in interpreting phrase rhythm as a rule, and to insist that phrases can only inhabit one level of grouping structure at a time.

Acknowledgments I wish to first thank my friends and colleagues at Texas Tech University whose support, advice, editorial comments, and thoughtful suggestions helped to shape this book—Jeannie Barrick, David Forrest, and Peter Martens—as well as all of the students who took my Rhythm and Meter course and provided valuable feedback. I owe perhaps an even greater debt to Harald Krebs and William Rothstein, both of whom read earlier drafts of the text and were generous enough to

xii

Preface

provide me with detailed feedback and with enthusiastic encouragement. I hope that they will be pleased with the shape that this book has ultimately taken. The accompanying anthology was a team effort including Taylor Etheridge, Justin Houser, and Brian Spruill, and I thank them all for their contributions. The translations of German and Italian texts in the anthology were generously provided by Thomas Cimarusti, Sigrun Heinzelmann, and Yonatan Malin, and I sincerely appreciate the artful way that they retained not just the meaning, but the soul of each text. I also wish to thank Genevieve Aoki at Routledge, whose enthusiasm for the project was often just what I needed when the writing got difficult, and whose lightning-fast responses always kept the book on schedule and on the right track. Finally, I would like to thank my wife, Lisa Garner Santa, who helped me in so many different ways that I would need to write yet another book to describe them all.

chapter 1

Notated Meter and Sounding Meter

Like key signatures, time signatures serve to facilitate reading music. Together with bar lines, beaming, and ties, time signatures help us to group music into familiar rhythmic cells that are easy to recognize and thus read quickly at sight. While time signatures work best when they represent the sounding meter, it is important to recognize that the time signature does not always represent the meter, just as a key signature does not always represent the key of a particular musical passage. Accidentals are routinely added to certain passages to produce keys that are not reflected by the key signature, and, in the same way, grouping and accentuation patterns within a passage can create a sounding meter that is not reflected by the time signature. Listen to mm. 16–20 from the second movement of Schumann’s String Quartet Op. 41/2 and conduct along while following Example 1.1 in order to feel how well the music fits the notation.

EXAMPLE 1.1

Passage in 12/8.

2

Notated Meter and Sounding Meter

The notation of the music in Example 1.2 seems to reflect the music fairly well, but Schumann’s score in fact looks quite different, and it is given as Example 1.2.

EXAMPLE 1.2

Schumann, String Quartet, Op. 41/2, II, mm. 17–20.

No one can be certain as to why Schumann chose his notation over the alternative provided as Example 1.1 (though we will do much speculating about such choices later in the book), but the difference between these two examples draws attention to the fact that time signatures and bar lines do not always reflect the sounding meter. Sometimes the sound of a passage doesn’t clearly establish any meter at all, but is nevertheless written in traditional notation. Listen to the beginning of Beethoven’s String Quartet, Op. 127 and conduct along while following Example 1.3 in order to determine how well the music fits the notation.

EXAMPLE 1.3

Passage in 2/4.

Notated Meter and Sounding Meter

3

Again, the notation of the music in Example 1.3 seems to reflect the music fairly well. The three fermatas, however, prevent any strong sense of regular pulse from emerging, and thus any strong sense of meter from being established. Beethoven’s notation for this passage is given as Example 1.4.

EXAMPLE 1.4

Beethoven, String Quartet Op. 127, I, mm. 1–5.

Notice how easily one can accept either notation for the same performance. In Example 1.3, the notated measures are coordinated with the changes of harmony and with the sforzando markings, while in Example 1.4, the notated measures are marked by a steady quarter note pulse, albeit one that the performers must keep internally, since the only quarter notes in the passage actually cut against the quarter note pulse that defines the notated meter. Now listen to the beginning of the second movement from Beethoven’s Piano Sonata, Op. 7, and conduct along while following Example 1.5 in order to feel how well the music fits the notation.

EXAMPLE 1.5

Passage in 3/4.

4

Notated Meter and Sounding Meter

The notation of the music once again seems to reflect the music fairly well, and this time the attacks are more regularly spaced, and so a sense of meter is established. Beethoven’s notation of this passage is given as Example 1.6.

EXAMPLE 1.6

Beethoven, Piano Sonata, Op. 7, II, mm. 1–4.

Again, notice how easily one can accept either notation for the same performance. In Example 1.5, the notation places the changes of harmony on the downbeats, while in Example 1.6, the harmonic changes are on beat 2. If either notation seems to fit the performance, then which one of these is actually the sounding meter? In many such situations, there might not be a single correct answer, but in order to arrive at any answer with certainty, we should first address the question of what meter is and how it becomes established as a sounding phenomenon.

Establishing Meter A rhythm is a pattern of note durations and rests. While a rhythm may be regular or irregular (i.e. the notes may or may not be evenly spaced), meter depends upon regularity for its definition. Meter is a musical pattern of accentuation created by two coordinated layers of evenly spaced pulses. For meter to be established as a perceptual reality (not just a notational convenience in a score), we first must hear these two layers. The process of establishing a meter need not take long. We need only hear two pulses in a single layer to predict when the next pulse will occur, and if the third pulse in that layer arrives as predicted, it confirms that layer in our minds as part of a meter. Example 1.7 illustrates two models for establishing meter.

a)

b) ● ●

●

● ●

●

● ●

etc.

● ●

EXAMPLE 1.7 Two Models for Establishing Meter.

●

●

● ●

●

●

● ●

etc.

Notated Meter and Sounding Meter

5

By the third pulse in the bottom layer of Examples 1.7a and 1.7b, meter has been established. Dots are used in Example 1.7 rather than note values in order to generalize the process. Example 1.8 illustrates how one could substitute quarter notes for the dots in the top layer and half notes for the dots in the bottom layer in Example 1.7a and establish a 2/4 meter, or substitute quarters and dotted halves in Example 1.7b and establish a 3/4 meter. On the other hand, one could substitute eighths and quarters in Example 1.7a and establish 2/8, or substitute eighths and dotted quarters in Example 1.7b and establish 3/8, as shown in Example 1.9.

EXAMPLE 1.8

One Possible Realization of Example 1.7 Using Note Values.

EXAMPLE 1.9 Another Possible Realization of Example 1.7 Using Note Values.

One of the two layers establishing any meter is called the beat. The beat is the pulse layer one chooses to count or conduct in a sounding meter, but is often defined as the duration indicated by the lower number of the time signature in a notated meter, or three times that duration if the upper number is 6, 9, or 12, and the tempo is fast enough. While two layers is the minimum number necessary to establish meter, most of the music we hear every day has more than two. We typically categorize meter in terms of three layers. Simple meters divide each beat into two parts, while compound meters divide each beat into three parts, and so these two categories each indicate two different layers. The terms duple, triple, and quadruple meter provide the character of the third layer by indicating whether the beats themselves are grouped into twos, threes, or fours, respectively (larger beat groupings such as quintuple are typically heard as combinations of duple and triple). Example 1.10 illustrates two models for establishing simple duple and compound duple meters, respectively.

6

Notated Meter and Sounding Meter

a)

b)

division: ● ● ● ● ● ● ● ● ● etc. beat: ● ● ● ● ● grouping: ● ● ●

EXAMPLE 1.10

● ● ● ● ● ● ● ● ● ● ● ● ● etc. ● ● ● ● ● ● ● ●

Establishing Simple Duple and Compound Duple Meter.

By the third pulse in the middle layer of Example 1.10a, meter has been established, but it takes an additional two pulses in that layer to establish a meter we can securely identify as simple duple. One could substitute eighth notes for the dots in the top layer, quarter notes for the dots in the middle layer, and half notes for the dotes in the bottom layer in Example 1.10a and establish a 2/4 meter. Similarly, one could substitute eighths, dotted quarters, and dotted halves in Example 1.10b and establish a 6/8 meter. While it only takes two layers to establish a meter, three layers are required to categorize it in the traditional way. Look at Example 1.7 again. Example 1.7a could represent 6/8 just as easily as 2/4 if we take the top layer to represent beats and the bottom layer to represent downbeats because there would be no layer articulating how the beats are subdivided, and thus we wouldn’t be able to tell whether the meter was simple or compound. It could also represent 2/4, 3/4, or 4/4 if we take the top layer to represent divisions of the beat and the bottom layer to represent beats; because there would be no layer articulating downbeats and grouping the beats into measures, we wouldn’t be able to tell whether the meter was duple, triple, or quadruple. It only takes two layers to establish meter, but it takes three layers to establish a meter that may be unambiguously labeled both by its beat groupings and by how those beats are subdivided.

Conducting and Maintaining Meter There is one pulse layer that distinguishes itself by being the one we choose to conduct or count; we will call it the tactus. We could call it the beat (the terms “tactus” and “beat” are synonyms when referring to sounding meters), but not without risking confusion later on, since “beat” is often defined differently when referring to notated meters. Musicians typically have strongly ingrained ideas about what gets the beat based on various time signatures, but there are many cases in which what is heard and felt as the tactus does not correspond to what the time signature defines as the beat. The tactus level is also subjective, not universal: there are many cases where different listeners or performers will choose different pulse layers as the tactus for the same passage of music. It is not, however, completely unpredictable. There is a well-documented tendency for listeners and performers

Notated Meter and Sounding Meter

7

to choose the pulse layer moving closest to 100 bpm (beats per minute) as the tactus when the fastest three layers of a meter are all moving between 30 and 240 bpm; this speed is called the natural pace. The range of speeds found in the fastest three layers of a passage are important for the natural pace to hold its predictive power. When the fastest pulse layer in a passage is at or near 100 bpm, listeners more often choose a slower-moving tactus. It is not wrong (or “unnatural”) to choose a layer other than the one closest to 100 bpm as the tactus, even when the fastest three layers are all within the 30–240 bpm range; the natural pace is just useful in predicting what layer most people will hear as the tactus for a given passage (assuming you have a good idea of the tempo in which it will be performed). Once a meter is established, its tactus must be articulated almost constantly to remain a perceptual reality. The tactus need not be articulated continuously in any one part, and many musical textures will divide up the responsibility of articulating the tactus between multiple parts. However, if too many pulses in a row go unheard in any part—as in the case of a dramatic pause, for instance—the sense of meter will be interrupted and will once again need to be reestablished. This kind of interruption can occur after just two or three missing pulses, depending on the tempo. In addition to pulses that go missing, new accents that contradict an established meter will also lead to that meter’s replacement unless the layers of the established meter are constantly being rearticulated. Still, established meters are not such fragile things that they can’t withstand some syncopation. It is only when syncopated accents become regular enough to establish new pulse layers that a new meter might be perceived as replacing the old one.

Pulse Layers Pulse layers that establish or maintain meter may be divided between multiple parts or voices in a musical work, or they may be contained entirely within a single voice or part. Listen to the beginning of Mozart’s Piano Sonata, K. 332, given as Example 1.11, and conduct along as you listen:

EXAMPLE 1.11

Mozart, Piano Sonata in F Major, K. 332, Opening.

8

Notated Meter and Sounding Meter

The left-hand accompaniment in mm. 1–4 is enough to establish meter all by itself (in fact, so is the right hand all by itself ). By the downbeat of m. 3, a meter with three levels of pulse is firmly established (at that point we get the third pulse in the slowest moving layer, the one moving in dotted half notes). The steady stream of eighth notes obviously constitutes the fastest moving layer, but the pattern of alternating up and down motions that places the same note of the chord on beats 2 and 3 in each measure articulates the quarter note layer, while the changes in harmony and the low Fs in each measure articulate the dotted half note layer. Already by the third beat of the first measure we have a strong sense of meter because five eighth note pulses and three quarter note pulses have been articulated by that time, but it is too soon yet to know without looking at the score how the quarter notes themselves will be grouped together. The first beat of m. 2 is made to sound like the first beat in m. 1 both because of the low F in the left hand and the half note in the right hand. It is really at this point that most musicians would be able to predict the same three-layer meter that is most often represented in notation as 3/4, though we don’t get the confirmation of that meter until the downbeat of m. 3. Being able to identify the meter is not the same thing as being able to correctly identify the time signature without the score. In fact, there is no way to do this consistently, because the meter shown in Example 1.11 can be represented by a variety of time signatures. Example 1.12 illustrates another possible notation for the opening of Mozart’s K. 332:

EXAMPLE 1.12 Alternate Notation for the Opening to K. 332.

The alternative notation in Example 1.12 is an equally plausible notation for the sound of the music, and even if one might prefer Mozart’s original notation, one would not have to search long to find a 6/8 passage in the literature with a left-hand accompaniment similar to this one at this tempo. One could also imagine renotating this music in 3/8 simply by using bar lines in Example 1.12 to divide each measure in half. These last examples bring home the point that

Notated Meter and Sounding Meter

9

there is no consistent relationship between the sounding meter and the notated meter that applies to all musical scores. More specifically, musical notation represents the fastest layer of pulse in a variety of ways: as the beat value relative to the time signature, as a simple or compound subdivision, or even as a quadruple or sextuple subdivision.

Steps to Identifying Meter Without the Score 1 First tap along with the music, and by doing so, you will use your intuition to find the tactus. 2 After finding the tactus, continue to tap along, but begin to chant the divisions of the tactus on a neutral syllable like “ta.” Take note as to whether the tactus divides more naturally into two or into three, and use this to determine whether the meter is simple or compound. 3 After finding the natural division of the tactus, continue to tap along, but start counting the beats along with the divisions (e.g. “1-te, 2-te, 3-te, etc.” for a simple meter, or “1-la-li, 2-la-li, etc.” for a compound meter). If no particular grouping of beats seems obvious, try a duple meter first and see how well that fits with the music. If your counting doesn’t fit well, try triple next. If your duple counting does fit well, try quadruple next, and see if that fits just as well as duple. 4 Bear in mind that a single movement or work often does not maintain a single meter from start to finish, and, in many styles, one should actually expect the sounding meter to change. When you feel a change in meter, isolate it from the music that came before and begin the process again with Step 1.

Factors in Metrical Perception There are a variety of ways to create musical accents beyond simple accent marks, and the full spectrum of them must be considered to fully understand metrical perception. Any kind of musical change or musical grouping, whether it be melodic, harmonic, rhythmic, dynamic, textural, or registral, creates a kind of accent at the beginning of the change or group; such accents are called phenomenal accents. In addition, there are some general principles that apply to grouping regardless of the context. One of the most important of these is the primacy effect: given two or more evenly spaced and evenly accented pulses, a listener will naturally hear the first pulse as accented relative to the second pulse. Another is the principle of binary regularity: given a series of evenly spaced and evenly accented pulses, a listener will naturally group the pulses into pairs or into some factor of two. The accentuation patterns resulting from these two principles are much weaker than the

10

Notated Meter and Sounding Meter

patterns created by phenomenal accents. Whenever phenomenal accents contradict accents created by binary regularity or the primacy affect, the former accents easy overpower the latter. Nevertheless scientific evidence strongly suggests that the primacy effect and binary regularity are both factors in metrical perception, and composers sometimes exploit these principles musically. Figure 1.1 lists the most common factors in metrical perception. General Principles Primacy effect Binary regularity Continuity

FIGURE 1.1

Phenomenal Accents Harmonic rhythm Agogic accent Melodic grouping Dynamic or registral accents Changes in texture, register, or articulation Stressed syllables in texts to vocal music

Factors in Metrical Perception.

Harmonic rhythm, the rhythm at which harmonies change, is commonly considered the strongest factor in metrical perception. We can hear and feel it at work in Example 1.11, the opening to Beethoven’s Waldstein Sonata, Op. 53:

EXAMPLE 1.13

Beethoven, Waldstein Sonata, Op. 53, Opening.

The strongest harmonic changes at the beginning of this sonata occur every two bars. Listen to this opening while conducting the notated meter. Then listen again but this time conduct half as fast so that the downbeats of your pattern match the harmonic changes that happen on the downbeats of mm. 1, 3, 5, and

Notated Meter and Sounding Meter

11

7 (i.e. conduct the half note). You’ll notice that both metrical interpretations seem to match the music fairly well, and it is in large part the harmonic rhythm that makes the second interpretation feel just as satisfying as the notated meter. In addition, the repeated eighth note chords that fill the first eight measures of this sonata illustrate the principle of binary regularity: one can feel the quarter note pulse in the first two measures despite the fact that there is nothing in the music that articulates these quarter notes except for the weak harmonic change at the end of the second measure. An agogic accent is created by a relatively long note value. Return to Example 1.11 and remove its harmonic context by chanting the melody on a neutral syllable instead of singing it. You’ll notice that even without a sense of harmonic rhythm and without binary regularity, a sense of meter is projected by agogic accent: because the half notes in the first three measures are twice as long as the quarter notes in those measures, we naturally hear them as metrically stronger. If you conduct along while chanting, you’ll find yourself conducting either the notated meter or the four-bar phrase (i.e. one beat per measure), but in either case your beats will naturally mark the half notes as downbeats. You may experience the relative power of agogic accent directly by performing a simple experiment: chant a steady stream of alternating quarters and halves on a neutral syllable without conducting, taking care to chant each note with the same dynamic and at full value. As you are chanting, look at Example 1.14. Which of the two metrical interpretations in Example 1.14 seems to match your chanting best? Most people tend to prefer interpretation B. One can easily steer a listener onto interpretation A, however, by adding accents to the quarter notes, or make it almost impossible to hear interpretation A by instead placing accents on the half notes.

EXAMPLE 1.14

Experiment With Agogic Accents.

A melodic grouping is a segment of pitches that is heard as a single unit based on context, often because the segment is immediately repeated, either

12

Notated Meter and Sounding Meter

in the same part or in another (the repetition could be exact or varied). Melodic groupings can be very influential in the perception of meter if they occur in immediate succession and are all the same size. Example 1.15 shows how melodic groupings cut against the notated meter of 3/4 in the music of Brahms. The static harmony and constant stream of eighth notes in the passage mean that neither harmonic rhythm nor agogic accent are in play. Instead it is the melodic sequence that strongly projects a half note pulse throughout the passage because the transposed segments of the sequence are each four eighth notes long. The passage could be heard in 2/4 or 3/2, but to simply perform it as though it were just four more measures in 3/4 would be to miss something essential about the passage: the notated meter here is not the sounding meter.

EXAMPLE 1.15

Brahms, Violin Concerto, I, mm. 65–68 (Textural Reduction).

Other kinds of accents that are factors in metrical perception include changes in dynamics, register, texture, and articulation. Example 1.16 illustrates how textural changes and melodic grouping can work together to project a meter at odds with the notated meter. While the passage begins on beat 4 of the notated meter, the changes in texture and melodic grouping suggest that the beginnings of the melodic groups (the slurred sixteenths) are downbeats or at least strong beats. It is not only unclear whether the sounding meter is 6/8 or 12/8, but also which beats are strong and which are weak. While it is certainly possible to hear the notated meter as the sounding meter in Example 1.16, the texture and the melodic groupings in the example make it equally possible to hear the weak beats of the notated meter as strong beats, and vice versa. Conduct this passage while listening but without looking at the example, first in two, then in four, but in both cases conduct the first note you hear as a downbeat. You’ll notice that the first group of slurred sixteenths sounds like it falls on a downbeat, not a weak beat.

Notated Meter and Sounding Meter

EXAMPLE 1.16

13

Schumann, String Quartet, Op. 41/2, II, mm. 49–50 (Reduction).

Subjectivity in Metrical Interpretation Conducting Example 1.16 in both 6/8 and 12/8 highlights another important truth in the identification of meter: the process is to some degree subjective. It is of course not entirely subjective, as no one could reasonably argue that Example 1.16 is in a quintuple meter, but it is impossible here to definitively argue that this music is in a duple meter and not a quadruple meter or vice versa. Those who hear it as duple might point out that the slurred sixteenths are spread over two beats each time, and that the shift in texture also occurs every two beats. Those who would argue for quadruple might point out that the duple units themselves are paired to form larger four-beat units because the third group of slurred sixteenths is not simply a transposition of the first, but rather presents new melodic material, which is then transposed to form the fourth group of slurred sixteenths. Also, both of the two four-bar units have largely the same textural pattern: two beats played by the first violin are answered by two beats played by the second violin. In fact, there are three reasonable interpretations of this passage that begin with a strong beat: 1) the sounding meter could be in two, starting on beat 1; 2) the sounding meter could be in four, starting on beat 1; or 3) the sounding meter could be in four, starting on beat 3. This last interpretation is based on the fact that the first group of slurred sixteenths is presented without any accompaniment, almost like an anacrusis. When the second violin enters with the second group of sixteenths, so do the viola and cello, and because the cello’s material is in a register associated with a bass voice, this could easily be heard as a stronger candidate for a downbeat than the first beat of this passage. There is no single right answer among these three possible interpretations, and this is a common state of affairs in music that alternates between strong beats and weak beats. Because this situation is so common, this book will not discuss the choice between duple and quadruple each time it comes up in future examples, but you should assume that, for any music identified as being in quadruple meter, a duple interpretation is also possible.

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Notated Meter and Sounding Meter

Continuity and Conflicting Accents Another general principle of metrical perception is that of continuity: given two phenomenal accents of equal strength, one that reinforces an established sounding meter and another that contradicts that meter, a listener will more likely interpret the contradictory accent as an outlier and continue to interpret the music that follows it based on the established sounding meter. Many pieces of music begin with some factors suggesting one interpretation of the sounding meter, and other factors suggesting another. In such cases the listener will, unconsciously and automatically, weigh these factors and choose which interpretation to accept based on the relative strengths of each. After the listener unconsciously chooses a sounding meter, the principle of continuity then becomes a factor, and that sounding meter is not so easily replaced, provided its tactus is continually articulated. For a listener to abandon a sounding meter that is still being articulated by phenomenal accents, another pulse layer contradicting that sounding meter must be established by stronger phenomenal accents that simply overpower those articulating the previously established meter. As an example, listen and conduct along with mm. 1–12 from the second movement of Schumann’s String Quartet Op. 42, No. 2 (see Anthology, p. 252). Assuming that the notated meter is established as the sounding meter in mm. 1–10, the syncopations in m. 11 actually establish a pulse layer that contradicts the sounding meter: a series of evenly spaced pulses on the third part of every notated beat. This contradictory layer starts in the lower three parts, but is reinforced in the first violin part by the end of the second beat (the tied Bˉ). The sforzandi in the second violin and viola parts in the second half of the measure, however, ensure that the listener will still be able to hear the tactus established in mm. 1–10 even when confronted with the conflicting accents in m. 11. The principle of continuity is part of what tips the balance in favor of hearing the notated syncopations as syncopations against the sounding meter, and not as pulses articulating the tactus of a new sounding meter.

Polymeters Versus Multiple Interpretations Polymeter is a term referring to the simultaneous presentation of two meters, and numerous examples of polymeter are notated in scores from twentieth-century post-tonal composers (i.e. different parts read different time signatures simultaneously). While well established as a phenomenon in musical notation, experimental studies strongly suggest that listeners can only track one meter at a time, and

Notated Meter and Sounding Meter

15

so there is really no such thing as a sounding polymeter: in the case of notated polymeters, the listener chooses one meter as dominant and interprets the other as a series of conflicting accents against the dominant meter. While a listener may not be able to hear two meters simultaneously, it is definitely possible (and educational!) for a listener to hear the same passage multiple times and to focus on different metrical interpretations during different hearings.

Performing the Sounding Meter Versus the Notated Meter This book focuses, in part, on the factors that establish and maintain a sounding meter as a perceptual reality. This focus, however, does not mean that the composer’s choice of how to represent that meter in notation is unimportant or arbitrary, or that performers should feel free to ignore the composer’s choice of notation. Returning to the chapter’s opening pair of examples, Schumann’s choice to notate the passage as he did in Example 1.2 undoubtedly has a strong visceral impact on the performers and, to a lesser extent, on anyone following the score. A performer should therefore honor the composer’s choice of notation even while acknowledging the metrical conflict and do whatever is possible to bring that conflict out for the listener. Passages that completely suppress the downbeat, such as Example 1.2, present the greatest challenge to the performer, but in large ensemble music, the conductor can help to communicate the conflict by continuing to conduct the notated meter, not the sounding meter. In chamber music, performers of such passages can at least keep the notated meter in mind and resist the urge to make things easier by imagining the notation of the sounding meter, and the resulting tension of such a mental exercise may well be what the composer had in mind. Harald Krebs, one of the modern pioneers in the analysis of rhythm and meter, has suggested that pianists might press downward on the already depressed keys slightly to mark silent downbeats in such passages, and though it would have no audible effect, it might help the pianist to feel the notated downbeats. String players might coordinate their breathing to mark some of the notated downbeats in these passages, while wind players might make use of foot tapping or other bodily movements for the same effect. In passages without continually suppressed downbeats, performers are more empowered to communicate the tension between the sounding and notated meters to the listener. The first step is to analyze such passages carefully and determine what elements support the sounding meter, what elements support the notated meter, and the relative strengths of each. Then the performer must try to find a balance so that the conflict is still heard and felt as such by the

16

Notated Meter and Sounding Meter

listener, and not simply resolved in favor of the sounding meter by being careful not to weight the accents of that meter too heavily. In Example 1.16, for instance, the performers could lean a little more heavily on the strong beats of the notated meter than they normally would, and at the same time deemphasize the notated weak beats a little more than is customary, since the textural accents, registral accents, and accents created by the melodic groupings all contradict the notated meter. This book will regularly suggest that the student conduct the sounding meter first while listening, and the notated meter second (or not at all), but this practice is meant solely to promote a greater understanding of the music’s metrical complexity, and not meant to be interpreted directly as a performance suggestion. Once the sounding meter is identified, it is up to the performer to decide how best to negotiate any conflicts that may exist between that meter and the notated one.

The History of Meter and Metrical Notation in Western Music This textbook is for a one-semester course, and so it cannot possibly cover the entire history of meter and metrical notation, but some observations should be helpful when applying the concepts presented here to repertoire. The first is that the sense of how meter should be realized in notation came gradually, and some defaults used to notate the rhythms in musical scores today didn’t exist in earlier music. For example, using the quarter note to express the tactus in simple meters and the dotted quarter note to express it in compound meters only really became a deeply ingrained standard in the twentieth century; in earlier music, the half note or the eighth note were used to express the tactus almost as often. Also, the convention that the first beat with a clearly expressed harmony at the beginning of a work or movement should be notated as a downbeat didn’t begin until after the Baroque period; the Gavotte, to cite a well-known example of a Baroque dance in simple duple meter, would by convention begin strongly on the second beat of an incomplete measure. Second, one’s knowledge of musical styles can effectively speed the process of determining the sounding meter throughout a given work. For example, medium to fast pieces in triple meter during the Baroque period often used hemiolas (two-against-three cross rhythms) at important cadence points, and while the practice of changing time signatures to match changes in the sounding meter is a twentieth-century convention, temporary changes to the sounding meter were used to create musical tension by many famous nineteenth-century composers as well, including Brahms, Dvořák, Schumann, and Tchaikovsky.

Notated Meter and Sounding Meter

Chapter Review 1

Meter is a musical pattern of accentuation created by two coordinated layers of evenly spaced pulses. For it to be established as a perceptual reality (not just a notational convenience in a score), we first must hear these two layers.

2

Sounding meter is the meter a listener can perceive without reference to the score, whereas notated meter is the meter a performer sees represented in the score by a time signature and bar lines. The sounding meter may or may not match the notated meter.

3

While two layers is the minimum number necessary, most of the music we hear every day has more than two. We typically categorize meter in terms of three layers. Simple meters divide each beat into two parts, while compound meters divide each beat into three parts, and so these two categories each indicate two different layers. The terms duple, triple, and quadruple meter provide the character of the third layer by indicating whether the beats themselves are grouped into twos, threes, or fours, respectively.

4

The tactus is the pulse layer we choose to conduct or count. Though we have strongly ingrained ideas about what gets the beat based on various time signatures, there are many cases in which what is heard and felt as the tactus does not correspond to what gets the beat according to the time signature. The tactus is also personal, not universal: there are many cases where different listeners or performers will choose different pulse layers as the tactus for the same passage of music.

5

There is a well-documented tendency for listeners and performers to choose the pulse layer moving closest to 100 bpm (beats per minute) as the tactus when the fastest three layers of a meter are all moving between 30 and 240 bpm; this speed is called the natural pace.

6

Once a meter is established, its tactus must be articulated almost constantly to remain a perceptual reality. The tactus need not be articulated continuously in any one part, and many musical textures will divide up the responsibility of articulating the tactus between multiple parts. However, if too many pulses in a row go unheard—as in the case of a dramatic pause, for instance—the sense of meter will be interrupted.

17

18

Notated Meter and Sounding Meter

7

New accents that contradict an established meter will also lead to that meter’s replacement unless the layers of the established meter are constantly being rearticulated. It is only when syncopated accents become regular enough to establish new pulse layers that a new meter might be perceived as replacing the old one.

8

There are many factors that may contribute to establishing meter: the primacy effect, binary regularity, harmonic rhythm, agogic accent, melodic grouping, stressed syllables in vocal texts, and accents created by changes in dynamics, register, texture, and articulation.

9

Any kind of musical change or musical grouping, whether it be melodic, harmonic, rhythmic, dynamic, textural, or registral, creates a kind of accent at the beginning of the change or group; such accents are called phenomenal accents.

10

In addition to the phenomenal accents created by specific musical changes or musical groupings, there are some general principles that apply to grouping regardless of the context. One of the most important of these is the primacy effect: given two or more evenly spaced pulses without accents, a listener will naturally hear the first pulse as accented relative to the second pulse. Another is the principle of binary regularity: given a series of evenly spaced and accented pulses, a listener will naturally group the pulses into pairs or into some factor of two.

11

Whenever phenomenal accents contradict accents created by binary regularity or the primacy affect, the former accents easily overpower the latter. Nevertheless scientific evidence strongly suggests that the primacy effect and binary regularity are both factors in metrical perception, and composers sometimes exploit these principles musically.

12

Harmonic rhythm is the rhythm at which harmonies change, and it is commonly considered the strongest factor in metrical perception.

13

An agogic accent is an accent created by a relatively long note value.

14

A melodic grouping is a segment of pitches that are heard as a single unit based on context; this is often because the unit is immediately repeated, either in the same part or in another (the repetition could be exact or varied).

15

Another general principle of metrical perception is that of continuity: given two phenomenal accents of equal strength, one that reinforces an

Notated Meter and Sounding Meter

19

established sounding meter and another that contradicts that meter, a listener will more likely interpret the contradictory accent as an outlier and continue to interpret the music that follows it based on the established sounding meter. 16

Polymeter is a term referring to the simultaneous presentation of two meters, and numerous examples of polymeter are notated in scores from twentieth-century post-tonal composers (i.e. different parts read different time signatures simultaneously). While well established as a phenomenon in musical notation, experimental studies strongly suggest that listeners can only track one meter at a time, and so there is really no such thing as a sounding polymeter.

17

Although the sounding meter as a perceptual reality is important, that does not mean that the composer’s choice of how to represent that meter in notation is unimportant or arbitrary, or that performers should feel free to ignore the composer’s choice of notation. Once the sounding meter is identified, it is up to the performer to decide how best to negotiate any conflicts that may exist between that meter and the notated one.

18

The sense of how meter should be realized in notation came gradually, and some defaults used to notate the rhythms in musical scores today didn’t exist in earlier music.

Homework Assignment 1.1 Add time signatures and bar lines to the two melodies given below, then write one paragraph for each in which you label the meter by type and enumerate what elements of the melody contribute to establishing its meter (agogic accent, melodic grouping, etc.). Melody 1

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Notated Meter and Sounding Meter

Melody 2

Homework Assignment 1.2 Compose an eight-bar melody in simple quadruple meter in such a way that the meter is clearly established without any accompaniment, then write a paragraph that identifies what elements of the melody contribute to the establishment of its sounding meter.

Homework Assignment 1.3 Compose an eight-bar melody in compound duple meter in such a way that the meter is clearly established without any accompaniment, then write a paragraph that identifies what elements of the melody contribute to the establishment of its sounding meter.

Further Reading Hasty, Christopher. Meter as Rhythm. Oxford: Oxford University Press, 1997. Lerdahl, Fred, and Ray Jackendoff. A Generative Theory of Tonal Music. Cambridge, MA: MIT Press, 1983. London, Justin. Hearing in Time: Psychological Aspects of Musical Meter. Oxford: Oxford University Press, 2004. Mirka, Danuta. Metric Manipulations in Haydn and Mozart: Chamber Music for Strings, 1787–1791. Oxford: Oxford University Press, 2009. Temperley, David. The Cognition of Basic Musical Structures. Cambridge, MA: MIT Press, 2001.

chapter 2

Hypermeter and Phrase Rhythm

While time signatures usually identify the fastest three pulse layers in a notated meter, that is not always the case; sometimes the tactus corresponds to the downbeats of the notated meter, and the fastest pulse corresponds to its beats. When the tactus corresponds to the downbeats of the notated meter, the slower-moving layers are usually articulated by the work’s phrase organization. This chapter will focus on how rhythm and meter functions in these slower-moving pulse layers.

Hypermeter A hypermeter is a sounding meter in which the grouping of the tactus spans more music than the grouping of beats indicated by the notated time signature and bar lines. Examples 2.1 and 2.2 provide two passages from the second movement of Beethoven’s Ninth Symphony (see Anthology, pp. 51–83). In Example 2.1, every four-bar segment forms a hypermetrical group, or hypermeasure: mm. 1–4, 5–8, 9–12, etc. can each be conducted with a four-beat pattern. Later in the movement when this material returns, Beethoven specifies this grouping in the score: he writes above the music to mm. 234–237 “Ritmo di quattro battute,” which means “conduct this in four.” Conduct the hypermeter while listening to Example 2.1 (i.e. conduct dotted half notes in a four-beat pattern). Even without Beethoven’s instruction to conduct this passage in four, it is almost impossible to miss the four-bar hypermeter for at least three reasons. The most significant of these is the tempo: Beethoven’s tempo marking for this movement is “Molto vivace Ó. = 116.” It’s so fast that it maps the tactus onto the dotted half note of the hypermeter, and not the quarter note of the notated meter (a result of the natural pace discussed in Chapter 1). The second reason

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Hypermeter and Phrase Rhythm

is the formal structure: it is a fugal exposition, and the fugal subject is four bars long. With every four bars, a new voice presents either the subject or the answer (the subject in the key of the dominant). It is thus a perfect example of how accents created by the repetition of a melodic grouping can create meter. The third reason is that the form also entails an additive texture, growing by one voice every four measures and thus creating a textural accent there as well.

EXAMPLE 2.1

Beethoven, Symphony No. 9, II, mm. 9–28 (orchestral reduction).

Later in the movement, Beethoven specifies that a three-beat pattern be conducted (“Ritmo di tre battute”) because the four-bar segments from Example 2.1 have been shortened consistently to three-bar segments in this passage, as is shown in Example 2.2. Even though the notated meter of this passage is 3/4, the tempo is far too fast to conduct the notated meter in quarter notes: Beethoven is clearly intending for a three-beat hypermeter to be conducted. Conduct the hypermeter while listening to the passage (i.e. conduct dotted half notes again, but this time use a three-beat pattern).

EXAMPLE 2.2

Beethoven, Symphony No. 9, II, mm. 176–193.

Hypermeter and Phrase Rhythm

23

Hypermeter at Fast and Slow Tempos At fast tempos such as the ones in Examples 2.1 and 2.2, hypermeter is perceptually indistinguishable from regular meter, and thus functions purely as a notational concept, but at slower tempos, hypermeter functions very differently, and some might argue that it ceases to function as meter at all. If a given pulse layer moves too slowly, it ceases to be felt as meter, perceptually speaking. We might rationally understand a very slow layer as a kind of accentuation pattern tethered to the sounding reality by its hierarchical relationship to faster-moving pulse layers, but because we cannot experience it viscerally, it is less important to us as a way of understanding sounding meter, even while it might prove useful as a way of abstractly understanding the large-scale rhythmic organization of a composition.

Phrase Rhythm This chapter and the next would not have been possible without the groundbreaking research on phrase rhythm by William Rothstein. Rothstein considers phrase rhythm to be a general term that encompasses both phrase structure and hypermeter, but in an effort to simplify things, this book will define phrase rhythm somewhat differently. This book defines phrase rhythm as the rhythm created by successive phrase lengths. Phrase rhythm is directly related to hypermeter because phrases and their component parts (motives and subphrases) are melodic groupings, and the accents formed by these melodic groupings are often significant in articulating the different pulse layers of a given hypermeter. Before we can address the topic of phrase rhythm directly, a clear definition of the term “phrase” is needed. This book defines a phrase as an independent musical segment approximating what one could sing in a single breath, and encompassing a motion from one harmony to another. Phrases are typically four bars long, but two-bar phrases and eight-bar phrases are also common, and other lengths are also possible. It is very common for musicians to disagree about how much music constitutes a phrase, and because the definition is subjective, most musicians are at least tolerant of the notion that there are multiple interpretations of the phrase structure for a given piece of music. In Example 2.1 and 2.2, the phrase rhythm and the hypermeter both articulate the same groupings, but Example 2.3 illustrates a different relationship between phrase rhythm and hypermeter. In the example, the phrase rhythm begins in m. 1 with the first note of the first phrase, but the first hypermetrical downbeat doesn’t begin until m. 2 with the beginning of its accompaniment. The harmonic rhythm establishes a four-bar hypermeter, and the strong textural

24

Hypermeter and Phrase Rhythm

change creates the sense of a hypermetrical downbeat in m. 2. Listen to the passage and conduct the hypermeter (i.e. conduct dotted half notes in a four-beat pattern; note that the first measure is beat 4 of the hypermeter).

EXAMPLE 2.3

Strauss, Blue Danube Waltz, mm. 1–33.

Hypermeter and Phrase Rhythm

25

The phrase rhythm of the passage is subjective because what constitutes an independent musical idea and what can be called a cadence are both subjective questions, but most musicians will gravitate toward one of two interpretations: either they will hear the passage primarily in eight-bar phrases, or they will hear it primarily in sixteen-bar phrases. Though there is a strong sense of melodic closure in m. 4, most musicians would take this as a subphrase ending because there has been no change of harmony, and thus the harmonic motion we typically associate with a cadence is missing. By m. 8, a motion from tonic to dominant has occurred and thus one could read a half cadence ending the first phrase. Because this dominant has a seventh and is in second inversion, however, musicians who both demand that phrases end with cadences and that half cadences end on root-position triads will find the second harmony insufficient to provide harmonic closure. All musicians will agree that there is a phrase ending in m. 16, and that the cadence associated with this ending is authentic (even if only some of them will hear a phrase ending in m. 8). Regardless of which interpretation one chooses, phrases and subphrases begin on beat 4 of the hypermeter throughout the passage. Thus, the hypermeter and the phrase rhythm throughout are offset by one bar. While phrase rhythm can play an important role in establishing hypermeter, they are still to some degree independent concepts, and composers often exploit this independence, as can be seen in Examples 2.4 and 2.5. In Example 2.4, Mozart begins his primary theme on the second bar of a four-bar hypermeasure. That the hypermeter is in four (i.e. every four bars = one hypermeasure) seems confirmed by the harmonic changes in mm. 5 and 9, but the first two four-bar phrases both start on the second hyperbeat of each hypermeasure.

EXAMPLE 2.4

Mozart, Symphony No. 40, I, mm. 1–9.

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Hypermeter and Phrase Rhythm

In Example 2.5, the theme, now transposed to F˜ minor, begins the development section of this sonata form movement, but its relationship to its accompaniment has shifted. The accompaniment from mm. 1–2 now doesn’t begin until a measure after the theme has started, and so the theme now sounds as though it starts one bar before a hypermetrical downbeat, rather than one bar after, as in Example 2.4.

EXAMPLE 2.5

Mozart, Symphony No. 40, I, mm. 102–111.

Levels of Metrical Structure and Grouping Structure When speaking of meter, the hierarchy from large to small is as follows: hypermeasures are composed of measures, measures are composed of beats, and beats are composed of beat divisions. Figure 2.1 provides a hierarchy for simple duple meter, one that groups pairs of measures into hypermeasures, though a hierarchy with four-bar hypermeasures may be more common. Metrical hierarchies can be complicated somewhat by the fact that any single segment in the hierarchy may consist of four or more smaller segments. In such cases, one could argue for an additional level that divides segments of four or more into smaller segments of two or more. For example, a four-bar hypermeasure could be understood as a pair of two-bar segments (articulated musically by the alternation of strong and weak measures). Metrical hierarchies can also be complicated somewhat by the relationship of the sounding meter to the notated meter. In Example 2.1, for instance, the tempo is so fast that the beat according to the time signature is not subdivided, and so the beat sits at the bottom of the hierarchy in that context. While this book is about rhythm and meter, hypermeter in large part is dependent upon those melodic groupings created by musical form and phrase structure.

Hypermeter and Phrase Rhythm

27

hypermeasures: measures: beats: beat divisions: FIGURE 2.1 A Metrical Hierarchy for Simple Duple Meter.

When speaking of musical form and phrase rhythm, the grouping hierarchy from large to small is as follows: forms are composed of sections, sections are composed of phrase groups, phrase groups (including periods and sentences) are composed of phrases, phrases are composed of subphrases, and subphrases are composed of motives. Figure 2.2 provides a melodic grouping hierarchy that divides each unit neatly in two, though one might be hard pressed to find an example from the literature that divides in this way so consistently, especially at the lower levels. Not all musicians adhere to this approach to melodic grouping hierarchies: some prefer to conceptualize a hierarchy in which all of the aforementioned relationships are true, but in addition, larger phrases are composed of smaller phrases. For the sake of clarity, only one level of a work’s grouping structure will be designated as the phrase level in this book; larger units relative to this level will be called phrase groups, while smaller units will be called subphrases or motives. As with the grouping structures of meter, one or more of the levels within this hierarchy may be omitted: not all phrases can be divided into subphrases, and not all phrases can be grouped together with other phrases to form phrase groups.

sections: phrase groups: phrases: subphrases: motives: FIGURE 2.2 A Melodic Grouping Hierarchy.

EXAMPLE 2.6

Beethoven, Symphony No. 5, I, mm. 6–21.

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Hypermeter and Phrase Rhythm

Example 2.6 illustrates how the first theme in Beethoven’s Fifth Symphony might be understood in terms of grouping structures. The four-note motive that serves as the basis for this movement is bracketed in the example each time it appears (there are eleven iterations). The first three forms of the motive constitute the first subphrase, the second three constitute the second subphrase, and the last five begin a final subphrase (or phrase, depending on one’s interpretation) that ends with the half cadence at the fermata. Two interpretations of this theme’s phrase structure seem plausible: it could be heard as one long phrase composed of three subphrases, or as two phrases, with the first of those broken into two subphrases. Multiple interpretations of the theme’s hypermetrical structure exist as well. The tempo almost guarantees that the tactus will be felt as the half note (i.e. one beat per measure), and the entrance of the accompaniment on the downbeat of m. 7 ensures that m. 6 will be heard as a hypermetrical anacrusis to m. 7. The way that the measures group into larger units, however, is a matter of interpretation. Four-bar hypermeasures seem logical for the first two subphrases of the theme, as this matches both the melodic grouping and the harmonic progression (each subphrase is set to a single harmony). One could continue with the same fourbar hypermeter until the end of the theme, but at the beginning of its second half, the music seems to get twice as fast, when the melodic repetition defines two-bar units and the harmonic rhythm changes to one chord per bar. One could take this as a reason to argue for an interpretation that changes to two-bar hypermeasures for the second half of the theme, but this latter interpretation is not necessarily better or worse than the former. While a change in hypermeter would match changes in the phrase rhythm and the harmonic rhythm, the regular alternation of strong and weak beats that defines a four-beat group at any level already accounts for these changes, and an actual change of pattern may seem unnecessary. One could argue that two-bar units are already felt in any quadruple as a consequence of binary regularity, and so it would take a much more powerful accent on beat 3 of the four-beat unit to truly register a change of hypermeter. In short, both interpretations seem viable.

Annotating a Metrical Interpretation One can notate a metrical interpretation by placing numbers above each beat or hyperbeat in the top staff of each system in a musical score, and by adding a superscript to the downbeats to communicate the choice of beat pattern—12, 13, 14, etc.—one can facilitate conducting along with the interpretation (e.g. “14”

Hypermeter and Phrase Rhythm

29

over a note would indicate that one should start a four-beat pattern on that note). Examples 2.7 and 2.8 illustrate the two metrical interpretations of Example 2.6 discussed earlier.

EXAMPLE 2.7

Interpretation With Four Quadruple Hypermeasures.

EXAMPLE 2.8

Interpretation With Change in Hypermeter.

It is important when annotating a metrical interpretation to place the numbers precisely above the attack point that marks the beat, because, as we saw in Chapter 1, the sounding meter does not always correspond to the notated meter. Example 2.9 illustrates an annotated metrical interpretation of just such a passage.

EXAMPLE 2.9

Schumann, Op. 41/2, II, mm. 17–20 With Annotations.

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Hypermeter and Phrase Rhythm

In this passage, the sounding beats correspond to the third eighth note in each beat of the notated meter, as one can see from the careful placement of the numbers above the music. Numbers meant to reflect an interpretation should always reflect the sounding meter rather than the notated meter when the two are in conflict, but the numbers may be used to represent either the meter or the hypermeter. In Example 2.7 and 2.8, the numbers represent hyperbeats, while in Example 2.9, the numbers represent beats.

Multiple Levels of Hypermeter So far, we have only examined the lowest level of hypermeter, but as a single meter is maintained over time, more hypermetrical levels may potentially become established (though perceptually speaking, there is an upper limit defined by the listener’s short-term memory). Example 2.10 represents an interpretation of the first theme from Beethoven’s Fifth Symphony with three levels of hypermeter.

EXAMPLE 2.10 Three Levels of Hypermeter in Beethoven’s First Theme.

While multiple levels of hypermeter are common whenever the same meter persists for any length of time, annotating multiple levels in a score is impractical, both because there is seldom room on the page for all of the annotations, and, more importantly, because we only conduct one level at a time (using both hands to conduct two different metrical levels, while an interesting challenge, might result in confusion for anyone attempting to follow the conductor, and would also prevent the conductor from using one of the hands to cue entrances, to shape dynamics, etc.).

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31

Elisions and Hypermetrical Reinterpretations An elision is when the end of one melodic group and the beginning of the next share one or more notes. This rhythmic overlap between groups often creates a hypermetrical reinterpretation: what serves as the last beat of one hypermeasure simultaneously serves as the first beat of the next hypermeasure. Example 2.11 illustrates just such an elision in m. 12; the annotations reflect that m.12 is both beat 4 of a four-bar hypermeasure that starts in m. 9, and beat 1 of a new four-bar hypermeasure. Such reinterpretations often sound and feel like changes in hypermeter (e.g. a three-bar hypermeasure surrounded by four-bar hypermeasures), though these changes can vary widely in terms of how strongly the change is heard or felt, and some may be so subtle as to be almost imperceptible unless one is conducting along. In annotating a metrical interpretation, this textbook will always provide conducting patterns that mark the beginning of the new group as beat 1, and will reflect hypermetrical reinterpretations by giving a two-part superscript, x(y), for the hypermeasure that ends with a reinterpretation, where x is the beat pattern to be conducted and y is the actual number of hyperbeats when the reinterpreted measure is counted as the last beat in that hypermeasure. One can feel the reinterpretation by listening to the passage given as Example 2.11 twice: once while conducting mm. 9–12 as a four-bar phrase, and stopping the recording just before the high Bˉ in m. 12 that begins a cadential extension, and then once while conducting as indicated by the annotations and letting the recording play until the downbeat of m. 16. The act of conducting the annotations alone could be misleading because m. 9 is really the first beat of a four-bar hypermeasure (not a three-bar one), but it does reflect the fact that m. 12 serves as the first beat of the next four-bar unit.

EXAMPLE 2.11

Beethoven, Piano Sonata, Op. 13 (Pathétique), III, mm. 9–16.

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Conducting On and Off the Podium On the podium, a conductor is obliged to use a pattern suggested by the time signature, or if the music is too fast to conduct that pattern, then to use a pattern matching the hypermetrical level moving at or close to the natural pace. But using beat patterns on a conductor’s podium to keep an ensemble rhythmically in sync represents only a fraction of their true potential. An equally valuable function that beat patterns can serve is to rationalize the rhythms that we hear and feel when preparing music off of the podium. This book will suggest many exercises in which beat patterns will be used to teach us how we hear and feel the sounding meter, including the various layers of pulse that make up that meter. Some of these beat patterns will be unlike any a conductor would ever use on a podium: some will combine unequal beat sizes when they represent music that seems to either shorten or lengthen specific points within its metrical grid, and some will move so slowly that conducting them makes you feel as though you are in a slow-motion sequence of a movie (the highest levels of hypermeter often move this slowly). By taking such liberties with beat patterns off of the podium and conducting the different layers of a meter, one can make those layers visceral and bring them to conscious awareness in a way that merely marking them on a score cannot. In this chapter, we have already explored ways of conducting different layers of hypermeter for a single passage of music, some of them very slow. The next chapter suggests more exercises that will make use of more liberated beat patterns intended to promote the understanding of rhythm and meter.

Chapter Review 1

While time signatures usually identify the fastest three pulse layers in a notated meter, that is not always the case; sometimes the tactus actually corresponds to the downbeats of the notated meter, and the fastest pulse corresponds to its beats. When the tactus corresponds to the downbeats of the notated meter, the slower-moving layers are usually articulated by the work’s phrase organization or formal design.

2

A hypermeter is a sounding meter in which the grouping of the tactus spans more music than the grouping of beats indicated by the notated time signature and bar lines. A hypermeasure is a hypermetrical group.

3

A phrase is an independent musical segment approximating what one could sing in a single breath, and encompassing a motion from one

Hypermeter and Phrase Rhythm

harmony to another. Phrases are typically four bars long, but two-bar phrases and eight-bar phrases are also common, and other lengths are also possible. 4

Phrase rhythm is the rhythm created by successive phrase lengths. Hypermeter in large part is defined by a work’s musical form and phrase rhythm.

5

When speaking of meter, the hierarchy from large to small is as follows: hypermeasures are composed of measures, measures are composed of beats, and beats are composed of beat divisions. Metrical hierarchies can be complicated somewhat by the fact that any single segment in the hierarchy may consist of four or more smaller segments. In such cases, one could argue for an additional level that divides segments of four or more into smaller segments of two or more.

6

When speaking of musical form and phrase rhythm, the grouping hierarchy from large to small is as follows: forms are composed of sections, sections are composed of phrase groups, phrase groups (including periods and sentences) are composed of phrases, phrases are composed of subphrases, and subphrases are composed of motives. As with the grouping structures of meter, one or more of the levels within this hierarchy may be omitted: not all phrases can be divided up into subphrases, and not all phrases can be grouped together with other phrases to form phrase groups.

7

An elision is when the end of one melodic group and the beginning of the next share one or more notes. This rhythmic overlap between groups often creates a hypermetrical reinterpretation: what serves as the last beat of one hypermeasure simultaneously serves as the first beat of the next hypermeasure. Such reinterpretations often sound and feel like changes in hypermeter.

8

On the podium, a conductor is obliged to use a pattern suggested by the time signature, or if the music is too fast to conduct that pattern, then to use a pattern matching the hypermetrical level moving at or close to the natural pace, but an equally valuable function that beat patterns can serve is to rationalize the rhythms that we hear and feel when preparing music off of the podium. This latter function is served whenever we conduct the sounding meter, or when we conduct a beat that moves more slowly than the tactus.

33

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Homework Assignment 2.1 Listen to mm. 1–138 of the fourth movement from Haydn’s Symphony No. 101 while following the score (see Anthology, pp. 163–171) and conducting along in one (whole note = beat). While listening, focus your attention on how the measures group into larger units (often four-bar units). Then complete the following summary of its hypermeter. Listen again while conducting along with the completed summary and ensure that your continuation analyzes parallel music in a parallel way (e.g. if the music from mm. 1–8 returns unchanged, be sure to analyze it as two four-beat hypermeasures to match the analysis of mm. 1–8 given here). mm. 1–8, 9–28 (1st time), 9–24 (2nd time) = four-bar hypermeasures. The hypermetrical downbeats are marked by melodic groupings and harmonic changes throughout, and also by registral and textural changes in mm. 9–21. mm. 25–27 (2nd time) = four-bar hypermeasure conducted in three to account for the elision in m. 28 that forces a hypermetrical reinterpretation. mm. 28–59 = four-bar hypermeasures. The hypermetrical downbeat in m. 28 is marked by agogic, dynamic, textural, and registral accents. The downbeat in m. 32 is marked by the sudden change in register and texture. The downbeats in mm. 36–59 are marked by melodic groupings and the harmonic rhythm. mm. 60–61 = two-bar hypermeasure. The hypermetrical downbeat is marked by the sudden change in dynamics and texture. (to be continued by the student on a separate sheet of paper)

Homework Assignment 2.2 Listen to mm. 1–267 of the Scherzo from Beethoven’s Symphony No. 9 (see Anthology, pp. 51–69), while conducting along as indicated below (dotted half note = beat; after m. 150, conduct in one while listening for how the measures group together into hypermeasures). Then complete the following summary of its hypermeter. Be sure to listen again while conducting along with the completed summary and that your continuation analyzes parallel music in a parallel way. There are only three places where one is forced to either take a two-bar hypermeasure or a six-bar hypermeasure; the rest of the music up to the recapitulation easily groups into either four-bar or three-bar hypermeasures.

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35

mm. 1–116 = four-bar hypermeasures mm. 117–122 = a six-bar hypermeasure mm. 123–150 = four-bar hypermeasures (to be continued by the student on a separate sheet of paper)

Homework Assignment 2.3 Listen to the Scherzo from Beethoven’s Symphony No. 7 to m. 148 (see Anthology, pp. 42–50), first while conducting along in one (dotted half note = beat). Then complete the following summary of its hypermeter. Listen again while conducting along with the completed summary and ensure that your continuation analyzes parallel music in a parallel way (e.g. if the music from mm. 1–5 returns unchanged, be sure to analyze it as a two-bar hypermeasure followed by a three-bar hypermeasure to match the analysis of mm. 1–5 given here). mm. 1–2 = two-bar hypermeasure. mm. 3–5 = three-bar hypermeasure. The hypermetrical downbeat is marked by registral, textural, and dynamic accents. mm. 6–9 = four-bar hypermeasure. The hypermetrical downbeat is marked by the agogic accent and by the harmonic rhythm. m. 10 = one-bar hypermeasure. The hypermetrical downbeat is marked by the agogic accent and by the harmonic rhythm. mm. 11–14 = four-bar hypermeasure. The hypermetrical downbeat is marked by the harmonic rhythm and by the textural change. mm. 15–16 = two-bar hypermeasure. The hypermetrical downbeat is marked by the dynamics. mm. 17–24 = four-bar hypermeasures. The hypermetrical downbeats are marked by the change in texture and by the agogic accents. (to be continued by the student on a separate sheet of paper)

Homework Assignment 2.4 Find a short piece (one or two pages of music) at a medium or fast tempo in which the hypermeter is not perfectly regular from beginning to end. (If you have trouble finding one, try looking at Baroque- or Classical-period keyboard or vocal music.) Make a copy of the piece to turn in, and then add annotations that reflect a

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hypermetrical interpretation in which each measure is one beat of a hypermeasure. Finally, check your work by conducting along with a recording.

Further Reading Cohn, Richard. “The Dramatization of Hypermetric Conflicts in the Scherzo of Beethoven’s Ninth Symphony.” 19th-Century Music 15/3 (1992), 188–205. Mirka, Danuta. Metric Manipulations in Haydn and Mozart: Chamber Music for Strings, 1787–1791. Oxford: Oxford University Press, 2009. Rothstein, William N. Phrase Rhythm in Tonal Music. New York: Schirmer, 1989. Schachter, Carl. “Rhythm and Linear Analysis: A Preliminary Study.” In Music Forum 4, eds. William Mitchell and Carl Schachter, 281–334. New York: Columbia University Press, 1976. Reprinted in Unfoldings: Essays in Schenkerian Theory and Analysis, ed. Joseph N. Straus, 17–53. New York: Oxford University Press, 1999. Schachter, Carl. “Rhythm and Linear Analysis: Durational Reduction.” In Music Forum 5, eds. William Mitchell and Carl Schachter, 197–232. New York: Columbia University Press, 1980. Reprinted in Unfoldings: Essays in Schenkerian Theory and Analysis, ed. Joseph N. Straus, 54–78. New York: Oxford University Press, 1999. Schachter, Carl. “Rhythm and Linear Analysis: Aspects of Meter.” In Music Forum 6, eds. William Mitchell and Carl Schachter, 1–59. New York: Columbia University Press, 1987. Reprinted in Unfoldings: Essays in Schenkerian Theory and Analysis, ed. Joseph N. Straus, 79–117. New York: Oxford University Press, 1999.

chapter 3

Phrase Expansions and Hypermeter

As discussed in the last chapter, phrase rhythm and hypermeter are independent concepts, but they are closely related, and so departures from an established phrase length are often heard as changes to the hypermeter as well. The four-bar unit serves as the most common phrase length across all styles of tonal music, so much so that phrases of other lengths are often understood as alterations made to a four-bar model. Though four-bar phrases are most common, any phrase length can be established as a contextual norm as long as there are enough of them in a row to create an expectation. This chapter will examine the different ways that phrases may be expanded, and the consequences such expansions have for hypermeter.

The Phrase Prefix A phrase prefix is non-thematic material that is presented before a phrase begins, but that is clearly linked to the following phrase in some way, usually through a shared harmony. A phrase prefix by itself sounds too incomplete to be considered a phrase, but phrase prefixes are nevertheless part of the grouping structure that operates at the phrase level. Some phrases are preceded by a phrase prefix that is made up of the phrase’s own accompaniment: that is, the accompaniment that harmonizes the phrase is first presented on its own for a measure or two before the phrase begins. Examples 3.1 and 3.2 include

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phrase prefixes that do not function as an accompaniment to the first phrase, while Examples 3.3 and 3.4 include phrase prefixes that do. In three of the four cases, the prefix is a different number of measures than the phrase to which it leads, and when this happens, it creates what sounds like a change in hypermeter. In Example 3.1, the first two measures are identical and thus easy to group together, but the next twelve measures can just as easily be heard as three groups of four bars each because of the thematic nature of the cello part in mm. 3–6, the shift of attention to the first violin part in mm. 7–10, and the harmonic progression in mm. 11–14 that places a root-position tonic chord in m. 11 (embellished by a 4–3 suspension) and a cadential 6/4 chord resolving to a root-position dominant chord in mm. 13–14. In Example 3.2, the first five bars have the character of an introduction because of the two fermatas and the unaccompanied unison line, but its brevity leads one to hear it as a phrase prefix and not an introduction per se (assuming that introductions are thought of as inhabiting a higher level of grouping structure, and are generally composed of multiple phrases). In the sixth bar, the tempo becomes regular and Beethoven replaces the unison texture with a four-part texture, marking the beginning of the primary theme that was analyzed in the last chapter (see Examples 2.6–2.8). Regardless of what a conductor chooses to do with mm. 1–5 in performance, any sense of metrical regularity doesn’t begin until m. 6. In Example 3.3, mm. 9–20 from the first movement of Schubert’s Unfinished Symphony clearly divide up into a phrase prefix in mm. 9–12 and the first phrase

EXAMPLE 3.1

Beethoven, Symphony No. 3, I, mm. 1–15.

Phrase Expansions and Hypermeter

EXAMPLE 3.2

39

Beethoven, Symphony No. 5, I, mm. 1–14.

of the primary theme in mm. 13–20 because the accompanimental figuration in mm. 9–12 actually continues on to accompany the primary theme. Example 3.4, the secondary theme from the same movement, does the same thing. While the hypermeter in Example 3.3 can easily be taken in four despite the presence of a prefix, Example 3.4 forces one to take the prefix in two, and to begin a four-bar hypermeter with the start of the secondary theme in m. 44 (though one could conduct Example 3.4 in duple throughout, this would not fit the music as well, because the phrase is clearly a single four-bar unit, and doesn’t naturally divide up into subphrases). This chapter defines phrase prefixes as units that precede the phrase, but are not actually a part of it, but some other musicians have considered them to be an expansion of the phrase itself, which is why they are included in this chapter dedicated to considering phrase expansions and their impact on hypermeter.

EXAMPLE 3.3

Schubert, Symphony No. 8 (“Unfinished”), I, mm. 9–20.

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EXAMPLE 3.4

Phrase Expansions and Hypermeter

Schubert, Symphony No. 8 (“Unfinished”), I, mm. 42–47.

Internal Repetitions One common way of expanding a phrase is through internal repetition, both literal and varied, as is illustrated in Example 3.5. In Example 3.5a, one can see a parallel period consisting of two four-bar phrases, the first seven bars of which match mm. 80–86 in the first movement of Haydn’s Symphony No. 103, the beginning of the movement’s second theme. The eighth bar is a recomposition based on what one might expect given the style and the context provided by the first four-bar phrase. Example 3.5b is another recomposition, but this time the seventh bar is repeated twice, expanding the eight-bar parallel period given as Example 3.5a to ten bars. The hypermeter shown in Example 3.5a groups every two bars in a four-beat unit and aligns with the subphrases of the example. The hypermeter shown in Example 3.5b inserts two two-beat units in mm. 86–87, before returning to the four-beat units that end the period. While it would have been simpler to maintain the four-beat units throughout Example 3.5b, the switch to two-beat units in mm. 86–87 helps to underscore their function in expanding the phrase through repetition: m. 87 does not really serve as the second half of a four-beat unit, but is actually just the first two beats again. By conducting along with the hypermeter as notated in Example 3.5b, one is more likely to hear and feel the expansion by matching the repeated material with a repeated beat pattern. Conducting the passage in this way may lead to a deeper understanding of its form and meter, but does not come close to telling us everything about the passage’s meter. For one, a change in pattern alone cannot communicate why there is a change—it is up to us to keep the reason why in mind while conducting the passage. In this case, one must keep in mind that it is the varied repetition of m. 86 that motivates the change in conducting—in some sense, the beats in mm. 86–88 should be felt as a loop within the first half of a four-beat pattern that doesn’t conclude until m. 89. For another, because we only conduct one meter

Phrase Expansions and Hypermeter

41

at a time, the truth revealed by conducting in this way may hide another truth that would be revealed by conducting a deeper level of hypermeter.

3.5a

Parallel Period Consisting of Two Four-Bar Phrases.

3.5b

Parallel Period Expanded Through Internal Repetition.

EXAMPLE 3.5

Internal Expansion in a Melody Based on Haydn.

Example 3.6 provides a reduction of mm. 80–94, the expansion that Haydn actually wrote. In Example 3.6, the seventh bar (m. 86) is repeated once almost verbatim in m. 87, but then the phrase is expanded further by adding two new measures that are motivically linked to mm. 86–87: the motive in the top voice on beat 1 of m. 86 that is repeated three times over the next three beats, moves to the bass voice in mm. 88–89 and is transposed around to create a new chord progression. These new measures are then repeated in mm. 90–91 before the true phrase ending arrives with the tonic on the downbeat of m. 92. Again, it would be simplest to maintain the same hypermeter all the way through Example 3.6, but the change in conducting highlights the expansion of the consequent phrase through repetition.

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EXAMPLE 3.6

Phrase Expansions and Hypermeter

Haydn, Symphony No. 103, I, mm. 80–94.

Repetition alone does not necessarily indicate phrase expansion, as a careful consideration of Example 3.6 will show. In the top part of m. 82, the second half is a transposed repetition of the first half, and yet we do not consider the antecedent phrase expanded. What makes us hear an expansion beginning in m. 87 is the context: mm. 84–86 are a repetition of mm. 80–82, and so we expect an authentic cadence in m. 87 to answer the half cadence we had in m. 83. In recognizing phrase expansions of all types, the context is all important. Example 3.7 illustrates how phrase expansion through internal repetition might appear in a different context, this time against a triple hypermeter. The first phrase in mm. 1–6 divides neatly into two three-bar subphrases, but the second phrase in mm. 7–14 is expanded by two measures (the second phrase would make perfect sense even if the F5 on the downbeat of m. 9 skipped directly up to the second Bˉ in m. 11, and all the notes in between were omitted). The second phrase is also divided into two subphrases, but the first of these is expanded by a sequential repetition of m. 8 in m. 9 and again in m. 10, stretching the second phrase from six measures to eight

Phrase Expansions and Hypermeter

43

measures. If one focuses on the hypermetrical level that initially corresponds to the subphrases, one must switch to a pair of four-bar hypermeasures for the second phrase.

EXAMPLE 3.7

Mozart, Symphony No. 40, III, mm. 1–14.

Cadential Extensions A cadential extension is music that serves to bring a phrase to a more satisfying close after it has already reached a cadence. Cadential extensions are heard as a kind of suffix to the phrase itself, expanding its length and often turning what would have been a symmetrical phrase grouping into an asymmetrical one. Cadential extensions are probably the most common form of phrase expansion, and they typically take one of two forms: 1) they move from an initial cadence at its beginning to a different one at its end; or 2) they prolong the final chord of a single cadence by either repeating the final cadential motion, by repeating the final cadential chord, or by simply sustaining that chord. Example 3.8 illustrates the first type of cadential extension. The imperfect authentic cadence in the fourth measure of the example is too weak to serve as the ending of a theme, and so Mozart provides an additional two bars that lead to a perfect authentic cadence. The hypermetrical analysis provided in the example highlights the fact that cadential extensions, unlike internal expansions, often have a marked beginning. Example 3.9 illustrates the second type of cadential extension, one that prolongs the final chord. In Example 3.9, after the arrival of the Bˉ tonic chord in m. 23 that completes the perfect authentic cadence, the outer voices simply repeat the tonic pitch for seven measures. The hypermetrical change in m. 23 is heard because both the repeated countermelody in the inner voices and its attendant harmonic motion from V7/IV to IV span two bars.

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EXAMPLE 3.8

Cadential Extension in Mozart, Piano Sonata, K. 331, I, mm. 13–18.

EXAMPLE 3.9

Brahms, Variations on a Theme by Haydn, mm. 19–29.

Composed-Out Ritardandos and Fermatas Occasionally a composer will expand a phrase not by adding more material, as in the case of internal repetitions or cadential extensions, but simply by adding more time to the material already present. These additions are often indistinguishable from ritardandos and fermatas when listening without the score. Example 3.10 illustrates a written out ritardando in Haydn, while Example 3.11 illustrates a pair of written out fermatas in Beethoven. In the latter case, the fact that these are written out fermatas is made crystal clear by the context: when this material first occurs in mm. 23–28, the long chords last only a quarter note each.

EXAMPLE 3.10

Haydn, Piano Sonata No. 37, III, mm. 86–95.

Phrase Expansions and Hypermeter

EXAMPLE 3.11

45

Beethoven, Symphony No. 5, II, mm. 148–155.

Parenthetical Insertions Sometimes composers expand phrases by placing material in the middle of the phrase that sounds as though it were added parenthetically; that is, the material seems to interrupt an expected conclusion to the phrase, the material is markedly different than what came before it in the phrase, and, after the material is over, the phrase seems to pick up where it left off and conclude. Example 3.12 illustrates a parenthetical insertion in the third movement of Haydn’s String Quartet, Op. 74, No. 1.

EXAMPLE 3.12

Haydn’s String Quartet, Op. 74, No. 1, III, mm. 49–60.

This example also provides another illustration of phrase expansion through internal repetition in its last two measures: m. 59 sounds like a strong measure hypermetrically both because m. 60 simple continues the same tonic harmony, and because of the parenthetical insertion. The insertion can sound like a

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four-bar hypermeasure by itself despite the fact that it is only three bars long, because it begins with a hemiola that changes the sounding tactus from the dotted half to the half note. To confirm this both aurally and viscerally, listen and conduct the passage as follows: mm. 49–52: four-bar hypermeasure (dotted half = beat), mm. 53–55: four-bar hypermeasure conducted in three because of the insertion, mm. 56–58: four-bar hypermeasure with unequal beats (half-half-half-dotted half ), mm. 59–60: two-bar hypermeasure (dotted half = beat).

Links A link is music that serves as a bridge between two phrases, and is thus dependent upon what comes both before and after it for its definition. A link by itself sounds too incomplete to be considered a phrase, but links are nevertheless part of the grouping structure that operates at the phrase level, and in that way they are similar to phrase prefixes. A link is not a form of phrase expansion, but is discussed here because links share the same level of grouping structure. Example 3.13 illustrates a link in the first movement of Haydn’s Symphony No. 103. In this example, the link lasts four bars and thus it fits easily within the prevailing four-beat hypermeter, though this is by no means always the case. In terms of understanding hypermeter, recognizing the difference between a phrase and a link is important because the first harmonized downbeat of a phrase is typically heard and felt as a hypermetrical downbeat, while links have no such regular association with hypermeter.

EXAMPLE 3.13

Haydn, Symphony No. 103, I, mm. 138–145.

Phrase Expansions and Hypermeter

47

Thematic Versus Non-Thematic Material The distinction between phrases, links, and phrase prefixes, as well as the distinction between music that is an irreducible part of the phrase and music that is part of a cadential extension or internal repetition, is often based on whether or not the musical material is thematic. Thematic material is memorable and distinctive, while non-thematic material is repetitive and harder to remember. Deciding whether melodic material is thematic or non-thematic is usually very subjective, but one’s argument usually depends in large part on the amount of variety. Example 3.14 illustrates the difference in two passages from the first movement of Mozart’s Symphony No. 40. In mm. 2–5, the first phrase of the primary theme is heard as thematic because of the great variety it possesses: it includes seven different pitch classes, a memorable leap of a sixth, a rest that divides it neatly into two contrasting subphrases, and a transformation of the opening three-note motive in its second subphrase. In mm. 73–76, the same three-note motive is present, but it only includes two different pitch classes, there is no memorable leap, and there is no dividing rest or contrast between the two halves. While audience members might be caught whistling the music of mm. 2–5, it is unlikely that they would be whistling the music of mm. 73–76—the latter is clearly four iterations of the same motive, part of a phrase maybe, but not an irreducible part of a theme.

3.14a

Mozart, Symphony No. 40, I, mm. 2–5.

3.14b

Mozart, Symphony No. 40, I, mm. 73–76.

EXAMPLE 3.14 Thematic Versus Non-Thematic Material.

In terms of understanding hypermeter, recognizing the difference between thematic and non-thematic material is important because thematic beginnings are associated strongly with hypermetrical regularity (though phrase expansion often disrupts this regularity in some way later on). Non-thematic material, on

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the other hand, has no such association with hypermeter; non-thematic material is almost just as often hypermetrically irregular as it is regular. Non-thematic material is usually found in sections that are transitional or developmental in character, and while one should not necessarily expect such sections to be hypermetrically irregular, one should expect such sections to sound less stable. Hypermetrical changes are just one more way that a composer can give these sections their unstable, restless quality, which are often just as effective as devices such as modulation, chromatic alteration, dramatic shifts in dynamics and texture, fragmentation, etc.

Chapter Review 1

Phrase rhythm and hypermeter are independent concepts, but they are closely related, and so departures from an established phrase length are often heard as changes to the hypermeter as well.

2

The four-bar unit is the most common phrase length across all styles of tonal music, so much so that phrases of other lengths are often understood as alterations made to a four-bar model. Though four-bar phrases are most common, any phrase length can be established as a contextual norm as long as there are enough of them in a row to create an expectation.

3

A phrase prefix is non-thematic material that is presented before a phrase begins, but that is clearly linked to the following phrase in some way, usually through a shared harmony.

4

A cadential extension is music that serves to bring a phrase to a more satisfying close after it has already reached a cadence. Cadential extensions are heard as a kind of suffix to the phrase itself, expanding its length and often turning what would have been a symmetrical phrase grouping into an asymmetrical one.

5

Cadential extensions are probably the most common form of phrase expansion, but phrases may also be expanded by internal repetition, by composed-out ritardandos or fermatas, or by parenthetical insertions.

6

A link is music that serves as a bridge between two phrases, and is thus dependent upon what comes both before and after it for its definition. A link by itself sounds too incomplete to be considered a phrase, but

Phrase Expansions and Hypermeter

49

links are nevertheless part of the grouping structure that operates at the phrase level. 7

The distinction between phrases, links, and phrase prefixes, as well as the distinction between music that is an irreducible part of the phrase and music that is part of a cadential extension or internal repetition, is often based on whether or not the musical material is thematic. Thematic material is memorable and distinctive, while non-thematic material is repetitive and harder to remember. Thematic beginnings are associated strongly with hypermetrical regularity (though phrase expansion often disrupts this regularity in some way later on). Non-thematic material, on the other hand, has no such regular association with hypermeter.

Homework Assignment 3.1 Add annotations to mm. 1–100 from the first movement of Beethoven’s Piano Sonata, Op. 2, No. 1 (see Anthology, pp. 5–6) that will allow you to conduct the following hypermetrical interpretation (1 bar = 1 beat): mm. 1–12 = four-bar hypermeasures; mm. 13–16 = four-bar hypermeasure, conducted in 3 to account for an elision in m. 16; mm. 16–27 = two-bar hypermeasures; mm. 28–30 = three-bar hypermeasure; mm. 31–32 = twobar hypermeasure; mm. 33–48 = four-bar hypermeasures; mm. 49–54 = three-bar hypermeasures; mm. 55–80 = two-bar hypermeasures; mm. 81–92 = four-bar hypermeasures; mm. 93–100 = two-bar hypermeasures. Listen and conduct along with the above interpretation at least twice. Then, on a separate sheet of paper to be turned in, write a paragraph in which you address the following questions: 1) if one takes mm. 9–20 to be a single phrase expanded to stretch over three hypermeasures, what constitutes the basic phrase and how would you describe the expansion using terms from this chapter? 2) Why might someone describe the music of mm. 26–27 and mm. 31–32 as a link and not part of a phrase? 3) If one takes mm. 49–54 to be a single phrase expanded to stretch over two hypermeasures, what constitutes the basic phrase and how would you describe the expansion using terms from this chapter? 4) Why might someone describe the music of mm. 93–100 as a link and not a phrase?

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Phrase Expansions and Hypermeter

Homework Assignment 3.2 Add annotations to mm. 1–43 from the first movement of Mozart’s Symphony No. 40 (see Anthology, pp. 217–219) that will allow you to conduct the following hypermetrical interpretation (1 bar = 1 beat): mm. 1–12 = three hypermeasures in 4; mm. 13–15 = one hypermeasure in 3; mm. 16–27 = three hypermeasures in 4; mm. 28–33 = one hypermeasure in 6; mm. 34–41 = two hypermeasures in 4; mm. 42–43 = one hypermeasure in 2. Listen and conduct along with this interpretation at least twice. Then, on a separate sheet of paper to be turned in, write a paragraph in which you address the following questions: 1) what aspects of the music could justify taking mm. 13–15 as a three-bar hypermeasure and m. 16 as a hypermetrical downbeat in the above interpretation? 2) What aspects of the music could justify taking mm. 28–33 as a six-bar hypermeasure in the aforementioned interpretation when a four-bar hypermeasure is established as a norm in mm. 1–12? If not already addressed in your first two answers, could one relate any type of phrase expansion to the above changes in hypermeter? 3) What could be a possible justification for an alternative interpretation that takes m. 1 as a one-bar hypermeasure, and mm. 2–5 and 6–9 as four-bar hypermeasures? 4) What about the harmony in mm. 1–9 challenges this alternative interpretation?

Homework Assignment 3.3 Review your analysis of the hypermeter in mm. 1–138 of the fourth movement from Finale of Haydn’s Symphony No. 101 in Homework Assignment 2.1 while consulting the score (see Anthology, pp. 163–171), and for each hypermetrical unit that is greater or less than four bars, specify whether or not the odd length is related to the presence of an elision (see Chapter 2), phrase prefix, link, internal repetition, composed-out ritardando, composed-out fermata, or cadential extension. Write your analysis of each hypermetrical unit on a separate sheet of paper to be turned in to your instructor.

Homework Assignment 3.4 Review your analysis of the hypermeter in the piece you chose for Homework Assignment 2.4, and for each hypermetrical change, specify whether the change is related to the presence of a phrase prefix, link, internal repetition, composed-out

Phrase Expansions and Hypermeter

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ritardando, composed-out fermata, or cadential extension. Then, on a separate sheet of paper to be turned in, create a phrase diagram that labels all cadences by type and indicates the measure in which each cadence falls. In your diagram, use slurs to represent phrases, links, cadential extensions, etc.

Further Reading Hudson, Richard. Stolen Time: The History of Tempo Rubato. Oxford: Clarendon Press, 1994. Huron, David, and Matthew Royal. “What Is Melodic Accent? Converging Evidence From Musical Practice.” Music Perception 13/4 (1996), 489–516. Margulis, Elizabeth Hellmuth. “Silences in Music Are Musical Not Silent: An Exploratory Study of Context Effects on the Experience of Musical Pauses.” Music Perception 24/5 (2007): 485–506. Margulis, Elizabeth Hellmuth. On Repeat: How Music Plays the Mind. New York: Oxford University Press, 2014. Mirka, Danuta. Metric Manipulations in Haydn and Mozart: Chamber Music for Strings, 1787–1791. Oxford: Oxford University Press, 2009. Rothstein, William N. Phrase Rhythm in Tonal Music. New York: Schirmer, 1989. Temperley, David, and Christopher Bartlette. “Parallelism as a Factor in Metrical Analysis.” Music Perception 20/2 (2002), 117–149. Willner, Channan. “Sequential Expansion and Handelian Phrase Rhythm.” In Schenker Studies 2, eds. Carl Schachter and Heidi Siegel, 192–221. Cambridge: Cambridge University Press, 1999.

chapter 4

Metrical Dissonances

The last three chapters have explored how meters and hypermeters are established, maintained, and changed; this chapter will explore how they can be challenged, and how sometimes two different metrical interpretations of the same music can compete with one another for the listener’s attention. To do this, the chapter will begin by defining what it means for music to be metrically consonant or dissonant.

Metrical Consonance and Dissonance Simply put, metrical consonance is a state in which all pulse layers are aligned, and metrical dissonance is a state in which at least two pulse layers are not aligned. Example 4.1 provides three different dot diagrams to illustrate; Example 4.1a is metrically consonant, while Examples 4.1b and 4.1c are both metrically dissonant. a)

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States of Metrical Consonance or Dissonance.

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There are two kinds of metrical dissonances: displacement dissonances and grouping dissonances. A displacement dissonance is one in which two conflicting (i.e. non-aligned) pulse layers are moving at the same speed, but are out-ofphase; Example 4.1b is a displacement dissonance. A grouping dissonance is one in which two conflicting pulse layers are moving at different speeds, and the faster layer is not simply a multiple of the slower layer. Example 4.1c is a grouping dissonance. In grouping dissonances, the pulses within the two conflicting layers will continuously move in and out of phase with one another. In Example 4.1c, one can clearly see this phenomenon: after the initial alignment of the two slower layers, the second pulse of the slowest is not aligned with the conflicting middle layer, then the third pulse is, the fourth is not, and so on. A syncopation is a rhythm that conflicts with the prevailing meter. Some authors have defined syncopations as displacement dissonances, but this textbook will argue that any kind of metrical dissonance is a syncopation, since both grouping dissonances and displacement dissonances are defined by metrical conflict.

Grouping Dissonances One grouping dissonance is so commonly used that it has been given a special name: the hemiola. A hemiola is a 3:2 grouping dissonance, and our first example of a grouping dissonance in this chapter (Example 4.1c) is a hemiola. Hemiolas can be notated in a variety of ways: some of these place the aligned pulses on the downbeat of every notated measure, as shown in Example 4.2, while others place the aligned pulses on the downbeat of every other notated measure, as shown in Example 4.3. In the music leading up to the passage given as Example 4.2, the dotted quarter note tactus suggested by the time signature is firmly established, but in mm. 309–310 (shown in the example), a layer of quarter note pulses in the winds, brass, and low strings emerges to challenge the dotted quarter layer. In m. 309, there is nothing to articulate the dotted quarter layer save the melodic grouping in the high strings, which breaks the measure into two groups of three eighth notes each (the mid-measure change of direction in the strings articulates the 3+3 grouping). In m. 310, the winds and brass continue to articulate the quarter note layer, but the low strings abandon that layer and articulate the dotted quarter layer instead when they leap down to the low F in that measure, adding a registral accent to the melodic grouping accent that continues to articulated the dotted quarter layer in the high strings. To feel this dissonance, listen to this passage and conduct along as follows: conduct the first two bars in two as notated (beat = dotted quarter), but then conduct the next two bars in 3/4 (the beat will get faster).

Metrical Dissonances

EXAMPLE 4.2

Brahms, Variations on a Theme by Haydn, mm. 307–310 (Reduction).

EXAMPLE 4.3

Schumann, Symphony No. 3, I, mm. 7–14 (Reduction).

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In Example 4.3, the conflicting layers are the dotted half note that is the tactus at the beginning, and the half note pulses that span mm. 11–12 (where the hemiola is marked). Both the melodic and the harmonic rhythm clearly articulate the dotted half layer throughout the passage, but the bass line in mm. 11–12 is strong enough to project a half note layer that challenges the dotted half layer and creates a strong hemiola (the bass line rhythm in these measures is actually the primary rhythmic motive for the entire movement, and is stated three times in the melody just before the music given as Example 4.3 begins). To feel this dissonance, listen to the example and conduct the first four bars as a single four-beat hypermeasure (beat = dotted half ), but then conduct the next two bars as a single bar of 3/2 (the beat will get faster). This will be difficult if singing along with the melody, but if one shifts attention to the bass voice during the hemiola it will become much easier.

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While the hemiola is by far the most common kind of grouping dissonance, composers have explored many other possibilities. Examples 4.4 and 4.5 provide 4:3 grouping dissonances in the music of Chopin and Beethoven, respectively, while Example 4.6 provides a 5:3 grouping dissonance in the music of Brahms. The 4:3 dissonance is fleeting in Example 4.4, lasting only a single beat, but the 4:3 dissonance in Example 4.5 is much more substantial, lasting four measures. The 5:3 dissonance in Example 4.6 spans five measures. The effect there is striking because the notated meter of three beats per measure is strongly articulated leading into it, but then is not articulated at all when the five-note melodic grouping in the outer voices strongly articulates a five-beat layer (B-A-G-D-E in the violins against G-A-A˜-B-C in the low strings). To feel this dissonance, listen to the passage while conducting as follows: conduct the first two bars in three as notated, but then conduct the next five bars as three bars of five. (NB: in all examples with dot diagrams, non-articulated pulses in any layer are placed in parentheses.)

EXAMPLE 4.4

Chopin, Nocturne in Eˉ Major, Op. 9, No. 2, mm. 17–18.

EXAMPLE 4.5

Beethoven, Leonore Overture No. 3, Op. 72, mm. 343–351.

Metrical Dissonances

EXAMPLE 4.6

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Brahms, Violin Concerto, I, mm. 51–57.

While this textbook focuses on the music of the common-practice period for its examples, grouping dissonances are extremely common in jazz and world music, and they actually define the sound of certain world music genres: 3:2 and its close relative 6:4 are central to much of the indigenous music from subSaharan Africa, 3:2 is central to much Afro-Cuban and Latin American music, and both 3:2 and 4:3 have been explored extensively in jazz since the 1950s (for a striking example of 4:3 in jazz, compare the downbeats to the high-hat strikes in the Miles Davis Quintet’s recordings of “Oleo”). Other grouping dissonances are also very common in post-tonal music, with examples too numerous to mention here.

Displacement Dissonances Just as grouping dissonances can be either fleeting or strong enough to change the sounding meter, so too can displacement dissonances come in a wide variety of strengths. Example 4.7 provides a phrase by Mozart with a single displacement dissonance in its third measure. The half note in the third measure is on beat 2, a weak beat, and sustains through beat 3, a strong beat, so it is heard as a syncopation, and could be thought to belong to its own theoretical pulse layer moving at the speed of a half note from weak beat to weak beat. If the next three notes in the melody were also all on weak beats, this theoretical layer would be confirmed and would be enough to actually challenge the sounding meter. In Mozart’s actual score, however, the quarter note on beat 4 of that measure leads to another quarter note on beat 1 followed by a rest on beat 2, so the half note is not the beginning of

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a metrical pulse layer, but merely sounds like it belongs on beat 3 but was displaced to arrive one beat early.

EXAMPLE 4.7

Mozart, Symphony No. 40, I, mm. 48–51.

Example 4.8 illustrates a passage where the same syncopation occurs twice in succession. It would be an oversimplification to say that this displacement dissonance in this passage is twice as strong as the one in Example 4.7 because it has twice as many syncopations. As yet, there is no single widely accepted way to measure the relative strength of metrical dissonance in a given passage, but certainly we can say that the dissonance in Example 4.8 is stronger than that of Example 4.7.

EXAMPLE 4.8

Schumann, Novellette, Op. 21, No. 6, mm. 25–28.

Example 4.9 provides a passage where the displacement dissonances extend for eight bars, and consequently the sounding meter changes to one with downbeats that correspond to beat 2 in each measure of the notated meter. The dot diagram below the music illustrates how the perception of the meter shifts quickly over the course of the passage. The conducting annotations show one of many possible responses to interpreting the corresponding changes in hypermeter. To conduct

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the hypermeter of this passage, one is forced to conduct two asymmetrical beat patterns, one that shifts the hyperbeats to beat 2 of the notated meter, and another to shift the hyperbeats back to beat 1 of the notated meter. The interpretation annotated here makes the first shift by starting an asymmetrical three-beat pattern (dotted quarter-dotted half-dotted half ) in m. 129, and the shift back by starting an asymmetrical two-beat pattern (dotted half-dotted quarter) in m. 135.

EXAMPLE 4.9

Schumann, String Quartet No. 1, I, mm. 127–137.

Broken Chords and Accompanimental Patterns Example 4.10 provides the first prelude from Book I of Bach’s Well-Tempered Clavier. One only needs to listen to this prelude while following the score to realize that long notes beginning in metrically weak positions, such as the second note in each measure, are not always heard as metrical dissonances. In Example 4.10, the texture is immediately recognized as a series of broken chords, and so the different notes in each chord are not heard as part of rhythmically independent voices, but rather as part of a single rhythm. Broken chords, as well as accompanimental patterns such as Alberti basses (see Example 1.11) and oom-pah-pah accompaniments (see Example 4.11), are generally heard this way, and so they aren’t heard as syncopated unless the resultant rhythm—the rhythm resulting from the combination of all parts forming the broken chord or accompaniment—is itself syncopated. None of these examples would be considered metrically dissonant.

EXAMPLE 4.10

Bach, Well-Tempered Clavier, Book I, mm. 1–3.

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EXAMPLE 4.11

Schumann, Papillons, Op. 2, No. 1, 1–4.

Labeling Metrical Dissonances This textbook will spend a lot of time exploring the relationship between different states of metrical consonance and dissonance, and so for convenience it will adopt the labeling system established by Harald Krebs, one of the first to write extensively on metrical dissonance in the terms used throughout this book. To use his labeling system, two more of his terms must be defined. A metrical layer is a layer of pulses that groups the fastest pulse in a way that reflects the sounding meter, while an antimetrical layer is any layer of pulses that groups the fastest pulse in a way that contradicts the metrical layer. In Krebs’ system, displacement dissonance labels take the form of Dx+y, where x represents the distance between the pulses in the metrical layer AND between the pulses in the antimetrical layer (by definition, the metrical layer and the antimetrical layer will be moving at the same speed). distance between pulses within each layer (NOT between layers)

Dx+y offset (between layers) The second number in a displacement label, y, represents the offset between the metrical and antimetrical layers, always measured from a metrical pulse in the metrical layer to the next sounding pulse in the antimetrical layer. Distance and offset are both measured by a reference pulse, which is often, but not always, the fastest pulse. Thus, D2+1 can be represented in a dot diagram as follows: reference pulse layer: • metrical layer: • antimetrical layer:

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Grouping dissonance labels, on the other hand, take the form of Gx/y, where x represents the distance between pulses in the slower-moving conflicting layer (whether it be metrical or antimetrical), and y represents the distance between pulses in the faster-moving conflicting layer (with distance again measured by a reference pulse). distance between pulses in slower-moving layer

Gx /y distance between pulses in faster-moving layer Thus G6/4 can be represented in a dot diagram as follows: reference pulse layer: six-layer: four-layer:

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In labeling a displacement or a grouping dissonance, one must be clear about the note value that corresponds to the reference pulse: the hemiola in Example 4.2 would be labeled G3/2 (‰) to indicate that the pulses are three eighth notes apart in the stream articulated by the melody (the metrical layer) but are two eighth notes apart in the stream articulated by the accompaniment (the antimetrical layer). While it seems intuitive to match the reference pulse of the label to the fastest pulse in the music, it is not always so obvious given a particular musical context. For instance, the hemiola in Example 4.3 could be labeled G6/4 (‰) or G3/2 (Œ), the former label corresponding to the actual fastest pulse, but the latter label corresponding to a pulse layer that also moves faster than the tactus (the dotted half ), but is much closer to the natural pace. Some of the other metrically dissonant examples in this chapter so far would be labeled as follows: Example 4.6 includes G5/3 (Œ), Example 4.7 and 4.8 both include D2+1 (Œ), and Example 4.9 includes D2+1 (Œ.). One must be careful not to make the mistake of equating the number of dots in each layer of a dot diagram with the numbers in the label of the dissonance. In the dot diagram for G6/4 above, one sees three dots in the bottom layer underneath two dots in the middle layer and so the G3/2 label might be incorrectly applied to it. G6/4 is labeled as such because the pulses in one

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conflicting layer are six reference pulses apart, while the pulses in the other layer are four pulses apart; it is not labeled G6/4 because there are six pulses against four pulses. While G3/2 and G6/4 are equivalent in some sense, if we keep the reference pulse the same in both, then the labels capture the sense that the G6/4 dissonance unfolds at half speed compared to G3/2, and that ability to capture the relative speed of the dissonance is not a feature we would want to sacrifice.

Identifying Grouping and Displacement Dissonances It can be easy to confuse grouping and displacement dissonances because only some of the pulses in the antimetrical layer of a grouping dissonance actually conflict with the metrical layer, and because even though displacement dissonance labels include an offset, it is the relative speed of the antimetrical layer and not the offset that distinguishes the two kinds of dissonances. Consider Example 4.12a. There is no metrical dissonance in Example 4.12a, but by simply delaying the F by two eighth notes (Example 4.12b), a hemiola would be created. The hemiola is G3/2 (‰), and while the F might sound like it’s offset from the downbeat by two eighth notes, its duration doesn’t fit any of the pulse layers that define 6/8: the eighth, the dotted quarter, and the dotted half. One should also notice that even though the last two notes in the measure are the same as in Example 4.12a, by placing the first note on the third part of beat 1, the context of those last two notes in the measure has changed—they now sound like a two-part division of a quarter note. While the F in Example 4.12a is not part of an antimetrical layer, the F in Example 4.12b is, and because of its new context, it also defines the syncopation as a grouping dissonance. Compare Example 4.12b to Example 4.12c: by simply moving the F one eighth note earlier, we turn the grouping dissonance into a displacement dissonance. The length of the second note in Example 4.12c is now equal to a dotted quarter, which is the beat value of the notated meter. Thus it implies an antimetrical layer that is moving at the same speed as one of the metrical layers, but is simply displaced, forming D6+1 (‰).

EXAMPLE 4.12 A Metrically Consonant Motive and Two Dissonant Variations on it.

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To complicate matters further, grouping dissonances do not need to begin in alignment with the sounding meter. Consider mm. 25–26 from the first movement of Beethoven’s Symphony No. 3 (see Anthology, p. 10). The two half notes in a row suggest an antimetrical layer moving in half notes, a duration that doesn’t fit any of the pulse layers that define 3/4: the eighth, the quarter, and the dotted half. This is therefore a grouping dissonance, and should be labeled G6/4 (‰). This grouping dissonance is offset relative to the downbeat in m. 25, but the speed of the antimetrical layer defines it as a grouping dissonance, not a displacement dissonance.

Subtactus-Level and Hypermetrical Dissonances While the labeling system presented earlier is powerful, it can feel abstract when applied to metrical dissonances that are moving too quickly or too slowly to be felt when conducting the tactus. A subtactus-level dissonance is one in which the dissonance can be expressed entirely within one pulse of the tactus, while a hypermetrical dissonance is one in which the dissonance can only be expressed over multiple measures of the notated score. Examples 4.4 and 4.5 both illustrate subtactus-level dissonances, while Example 4.6 illustrates two different hypermetrical dissonances. Though the labeling system presented earlier works well for hypermetrical dissonances, it could be awkward when applied to subtactus-level dissonances like those in Example 4.4 and 4.5, because the reference pulse, as the common denominator of 3 and 4, would be 1/12 the size of the beat, and would thus be moving much faster than the fastest regular pulse stream of the passage. In the case of subtactus-level dissonances, this book will label them “Gx/y (sub)” to avoid using a reference pulse that is not part of the musical texture.

Direct and Indirect Metrical Dissonances Metrical dissonances need not have the conflicting layers presented simultaneously to be heard or felt. Direct metrical dissonances are those in which two conflicting layers are presented simultaneously, while indirect metrical dissonances are those in which two conflicting layers are presented in immediate succession. Indirect dissonances are just as easy for us to perceive as direct dissonances because once a meter is established, it takes more than one or two missing pulses for a listener to abandon that meter. Example 4.2, 4.3, 4.4, 4.5, 4.7, and 4.8 are all examples

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of direct dissonances because in all of these passages there is at least one stream articulating the metrical layer and one stream articulating the antimetrical layer at the same time. The 5:3 grouping in Example 4.6, however, is an indirect dissonance because the metrical layer ceases to be articulated when the antimetrical layer appears. So is the eight-measure displacement dissonance in Example 4.9. Usually when one perceives that the sounding meter has changed, an indirect dissonance is at work. To further illustrate this point, listen to the beginning of the finale to Brahms’ Symphony No. 4 while conducting along with Example 4.13. Do this a few times, then listen to it while conducting the actual notation given in Example 4.14. The alternate notation of Example 4.13 is based on the fact that the sounding meter is dictated by the harmonic rhythm, not the bar lines; the tympani rolls and horn pedal tones on beat 1 of mm. 9–16 sound like anacruses leading to the chord changes on beat 2 of those measures. The indirect displacement dissonance, D3+1 (Œ), requires one to add a beat to the eighth measure in order to put the changes in harmony on the downbeats, then requires one to remove a beat in order to keep those changes on the downbeats. While it is relatively easy to conduct the notated meter (Example 4.14) given the tempo and the steady stream of quarter notes; conducting the alternate notation (Example 4.13) feels more true to what’s going on musically because of the harmonic rhythm, and the meter suggested by the harmonic rhythm is further reinforced by the melodic groupings in mm. 16–24.

EXAMPLE 4.13 Alternate Notation for Brahms, Symphony No. 4, IV, mm. 1–26.

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EXAMPLE 4.14 Actual Notation for Brahms, Symphony No. 4, mm. 1–26.

Asymmetrical Meters Asymmetrical meters all incorporate one pulse layer with some kind of alternation between groups of two and groups of three, such as 3+2, 2+3, 3+2+2, 2+3+2, 2+2+3, etc. This alternating layer can serve as the tactus, in which case it creates a series of unevenly spaced beats, some beats lasting for only two pulses of an evenly spaced faster-moving layer, while other beats last for three (e.g. Dave Brubeck’s “Blue Rondo a la Turk”). On the other hand, the alternating layer can serve at a higher level, grouping the beats of an evenly spaced tactus layer in an uneven way (e.g. Dave Brubeck’s “Take Five,” Radiohead’s “Morning Bell,” or “All You Need Is Love” by The Beatles). In either case, there are at least three layers involved: two consisting of evenly spaced pulses, and one layer alternating between twos and threes. In common-practice period music asymmetrical meters are rare and can create the impression of an indirect grouping dissonance that is constantly being rearticulated, though they need not be heard as dissonant at all in other contexts. The family of asymmetrical meters includes all meters represented by time signatures with an upper number that is not divisible by two or three, such as meters in five or in seven, but it also includes other meters with groupings that alternate between two and three even though the sum of those groupings is evenly divisible by two or three (e.g. a 3+3+2 division of eight or a 2+2+2+3 division of nine; for a popular example of the latter grouping, listen to Dave Brubeck’s “Blue Rondo a la Turk”).

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Subliminal Dissonances There is one class of metrical dissonance that may only be perceptible to a listener following the notation or watching a conductor: the subliminal dissonance. A subliminal dissonance is one in which all musical features—harmonic rhythm, melodic groupings, dynamic accents, etc.—establish a meter other than the one notated, while none of the musical features establish or reinforce the notated meter. The accents reflecting the notated meter in a subliminal dissonance exist only in the mind of one looking at the notation, or in the gestures of a conductor; they are not realized in sound at all. Example 4.15 provides a subliminal D3+2 (Œ) dissonance in the music of Schumann (the offset is 2 because it is measured from the notated downbeats to the agogic accents on beat 3; the notated meter was established as the sounding meter in the music leading up to this passage). Example 1.2 from the first chapter is another example of a subliminal dissonance (for an alternate notation that represents the sounding meter of Example 1.2, see Example 1.1).

EXAMPLE 4.15

Schumann, Faschingsschwank aus Wien, op. 26, I, mm. 87–90.

It is easy to confuse the concepts of subliminal dissonance and indirect dissonance because subliminal dissonances that don’t begin a work or movement will usually start off as indirect dissonances as well. Neither subliminal dissonances nor indirect dissonances involve superimposed conflicting layers. But whereas indirect dissonances depend upon the immediate succession of conflicting layers, subliminal dissonances have no such requirement. Indirect dissonances require both conflicting layers to be sounded, and thus they only last as long as it takes for the listener to abandon the first layer and adopt the second. Subliminal dissonances, on the other hand, can be of any length.

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Comparing Metrical Dissonances There is no universal system for comparing the relative strength of a metrical dissonance, but one can identify the factors that contribute to its strength. First and foremost, how long a dissonance lasts must be taken into account. This does not necessarily mean that long-lasting direct dissonances and subliminal dissonances are more powerful than indirect dissonances, which by their nature are shortlived, because other factors certainly play a role. Among grouping dissonances, those with more unaligned pulses between the conflicting layers (measured from aligned pulse to aligned pulse) are more strongly dissonant than those with fewer unaligned pulses, at least when all other factors are equal. Example 4.16 illustrates a comparison of G5/3, with six unaligned pulses per cycle, and G3/2, with three unaligned pulse per cycle: G5/3 would be considered more dissonant than G3/2 because it has more unaligned pulses per cycle. ●

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Comparing the Relative Strengths of Dissonance in G5/3 and G3/2.

Among displacement dissonances, those that have conflicts between metrical and antimetrical layers more often (i.e. those with shorter distances between pulses in both the metrical and antimetrical layers represented by the label) are heard as more dissonant than those that have conflicts less often. Example 4.17 illustrates a comparison of D3+1 and D2+1: in the abstract, D2+1 would ●

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Comparing the Relative Strengths of Dissonance in D3+1 and D2+1.

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be considered more dissonant than D3+1 because it has more conflicts within the same span of time. While the neatness of simply counting unaligned pulses to measure the strength of metrical dissonance is appealing in the abstract, one merely needs to compare Examples 4.16 and 4.17 with the same dissonances in Examples 4.6, 4.7, and 4.14 to see the problem with it. In Example 4.6, the G3/2 dissonance that begins the passage is a direct dissonance, while the G5/3 that immediately follows it is indirect, and so the latter dissonance does not really feel any stronger than the former. Comparing the D2+1 in Example 4.7 with the D3+1 in Example 4.14 underscores the fact that how long a dissonance persists may well be a more important factor in how sharply a dissonance is felt than the number of unaligned pulses implied by its label: the direct D2+1 in Example 4.7 happens so fast that it is over before it has had time enough to make a lasting impression as a metrical dissonance, unlike the D3+1 in Example 4.14, which lasts for sixteen measures. Occasionally a passage will combine multiple dissonances at once, as in Example 4.18; such combinations will generally create a more powerful effect than either dissonance would have on its own. In the example, a G3/2 (Œ) is present: the downbeats of the notated meter are articulated by agogic accents in the right hand that are sometimes reinforced by sforzando markings, while the accompanimental pattern in the left hand starting in m. 29 groups the quarter notes in pairs against the notated meter. At the same time, a D3+1 is present: while the notated meter is articulated by the agogic accents and occasional sforzandi, the melodic groupings in the right hand place an accent on the second beat of each measure, one which is initially reinforced by a registral accent (the Eˉ5 on beat 2 in m. 29 is approached by a leap of more than an octave, and the Aˉ5 on beat 2 of m. 30 is approached by an octave leap).

EXAMPLE 4.18

Schumann, Carnaval, “Préambule,” mm. 26–31.

Rather than providing a dot diagram, Example 4.18 instead provides three different ways of conducting the hypermeter of this passage, each one tracing a different layer in one of the two conflicts. The tempo marking is piú moto, so

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the tactus is the dotted half note for the top two layers, and the half note for the bottom layer. Listen to this passage (and the music that immediately follows it) at least three times and conduct along with each of the different layers at least once. To do this successfully, you will need to shift your attention to different voices in the texture for different layers. To conduct the hypermeter aligned with the notated downbeats, you should focus on the bass in m. 28, and then shift your attention to the right hand in mm. 29–31. To conduct the hypermeter that aligns with beat 2 of the notated meter, you should focus on the right hand exclusively, and you should be ready to “float” on the second hyperbeat of the phrase prefix in m. 27, since the quarter rest on the downbeat of m. 28 will no longer be treated as a downbeat. To conduct the third layer, one should focus exclusively on the bass, and be ready to adjust the speed of the tactus from a dotted half note to a half note when moving from m. 28 to m. 29. Other factors that may well contribute to the relative strength of a metrical dissonance include those specific to its musical setting: changes in harmonic rhythm, melodic grouping, in dynamics, articulation, texture, and register, and also the number of voices involved in articulating the dissonance.

Finding Metrical Dissonances in a Score Labeling metrical dissonances can be complicated, but finding them in the score can be easy if one keeps in mind a few basic facts. One is that ties are often used when a single value would obscure the normal accentuation pattern associated with the notated meter—that is, in the case of syncopations. One should therefore examine every pair of tied notes carefully when looking for metrical dissonance. There are other uses for ties, so one must not assume that every pair of tied notes indicates a syncopation: tied dotted half notes in 3/4, a dotted quarter note on the downbeat tied to an eighth in 6/8, and a half note on the downbeat tied to an eighth in 4/4 are all examples of ties that do not necessarily involve syncopations. But when a pair of tied notes begins in a relatively weak metrical position, and the second note of the pair is in a stronger metrical position, the tie is being used to mark a syncopation. Another fact is that a longer note value beginning in a weak metrical position will mark a syncopation if that note value sustains through a stronger metrical position (e.g. in 4/4, a half note on beat 2). Also, notes beginning in a weak metrical position and followed by a rest in a stronger metrical position can often have the same effect (e.g. in 4/4, a quarter note on beat 2 followed by a quarter rest on beat 3).

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A Sample Analysis: Bach, Partita, BWV 826, Sarabande To illustrate how one might approach an analysis of metrical dissonances using the concepts taught in this chapter, the discussion that follows will apply these concepts to an analysis of the Sarabande from Bach’s Partita, BWV 826. Listen to a recording of it while conducting the notated meter (see Anthology, p. 4). In the very first measure there are two syncopations that are easy to find because they are both marked with pairs of tied notes, and in each case the first note of the pair is in a metrically weaker position than the second note of the pair. While finding these two syncopations is easy, identifying what kind of dissonance they represent is much more complicated. The first of these syncopations starts in the left-hand part halfway through beat 1, while the second starts in the right hand part halfway through beat two. A dot diagram of the measure looks like this: divisions: beats: tied notes: downbeats:

• •

• •

•

• •

• •

• •

•

• •

(•)

•

The diagram leaves out the sixteenths to simplify things, since the sixteenths are not necessary to understand the dissonance here (though if the tied notes began a sixteenth note or three sixteenths after the beat, we would have to include them in our diagram). Because both dissonances are off set from the notated beat by the same amount, it suggests that they are part of a layer moving at the same speed as the beats of the notated meter (the quarter note). The label would then be D2+1 (‰): the beats of the notated meter are two eighth notes apart, as are the first notes within each pair of tied notes, and these two layers are offset by one eighth note. This is only one way of understanding the metrical dissonances in the first measure, and one could argue that it is an incomplete understanding. D2+1 by itself does not account for where these two dissonances occur relative to the downbeat of the measure, nor does it address the fact that the left- and righthand parts sound contrapuntally independent. This contrapuntal independence suggests that the first syncopation in the left hand is more closely related to the syncopation in the left hand of m. 2 than it is to the syncopation happening a beat later. Something similar could be said about the syncopations in the right hand of mm. 1–2. To capture these aspects of the music, one could extend our

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dot diagram by a measure and consider only the syncopations in the left hand, then provide another diagram and consider only those in the right hand. They look like this: Right Hand divisions: beats: tied notes: downbeats: Left Hand divisions: beats: tied notes: downbeats:

• • • •

•

• •

•

• •

•

• •

•

•

• •

•

• •

•

• • • •

•

• •

•

• •

• •

•

• •

•

• • • •

• •

•

• •

•

• •

•

• •

•

The left-hand dissonance would be labeled D6+1(‰) because the antimetrical pulses are now six eighth notes apart, just like the downbeats of each measure, and they are offset from those downbeats by one eighth note. The right-hand diagram looks markedly different than the left-hand diagram in part because the tied notes in the right hand divide each measure exactly in half, and so when they are combined with the downbeats, they form an evenly spaced layer moving at the speed of the dotted quarter, not the same speed as any of our metrical layers, and thus not a displacement dissonance at all. The right-hand dissonance, because of its position relative to the downbeats, sounds like a grouping dissonance rather than a displacement dissonance. It would be labeled G3/2 (‰) because the distance from the downbeat to the tied note to the downbeat is a constant three eighth notes, while the beats of the notated meter are all two eighth notes apart. To confirm that this is a sounding reality and not just a notational phenomenon, listen to the first two measures again, but this time conduct them as four bars of 3/8 while singing along with the right-hand melody. D2+1, D6+1, and G3/2 dissonances, either in combination or alone, may also be found in mm. 5–6, 9–12, and in m. 20. But those are not the only ones in this sarabande. Another kind of dissonance is found first in m. 12 and then reappears in m. 16. In both of these measures, there is a tie in the right hand that connects beat 3 to the downbeat of the next measure, and thus indicates a syncopation. Because the pair of ties starts on a beat, it can only be understood as a displacement relative to a metrical layer moving more slowly than the beat,

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and that points the analyst to the layer of downbeats. The diagram for it looks like this: divisions: beats: tied notes: downbeats:

• •

•

• •

•

• • •

•

• •

•

•

• •

•

• • (•)

•

•

• • •

The dissonance would be labeled D6+4 (‰) because the downbeats of each measure are six eighth notes apart, and the pair of tied notes is offset from those downbeats by four eighth notes. Notice that in this case the antimetrical layer is hypothetical: there is no corresponding tied note at the end of m. 13 to match the one in m. 12. Any syncopation that cannot be grouped with other syncopations near it will necessarily be understood relative to a hypothetical antimetrical layer. This sample analysis will end with an examination of the dissonance in mm. 21–22. It is not the presence of the ties, but rather the melodic groupings and the harmonic rhythm in these two measures that marks this dissonance: the harmonic progression is C minor to Aˉ major to F minor, with each of those chords lasting two beats. The diagram looks like this: divisions: beats: harmonies: downbeats:

• • • •

•

• •

•

• • •

•

• •

•

• • •

•

• •

•

•

• • • •

The dissonance would be labeled G6/4 (‰) because the downbeats of each measure are six eighth notes apart, while the harmonic changes come every four eighth notes.

Chapter Review 1

Metrical consonance is a state in which all pulse layers are aligned, and metrical dissonance is a state in which at least two pulse layers are not aligned. A syncopation is a rhythm that conflicts with the prevailing meter: any kind of metrical dissonance is a syncopation.

2

A displacement dissonance is one in which two conflicting (i.e. nonaligned) pulse layers are moving at the same speed, but are out-ofphase.

Metrical Dissonances

3

A grouping dissonance is one in which two conflicting pulse layers are moving at different speeds, and the faster layer is not simply a multiple of the slower layer. A hemiola is a 3:2 grouping dissonance.

4

Broken chords, as well as accompanimental patterns such as Alberti basses and oom-pah-pah accompaniments aren’t heard as syncopated unless the resultant rhythm—the rhythm resulting from the combination of all parts forming the broken chord or accompaniment—is itself syncopated.

5

A metrical layer is a layer of pulses that groups the fastest pulse in a way that reflects the sounding meter, while an antimetrical layer is any layer of pulses that groups the fastest pulse in a way that contradicts the metrical layer.

6

Displacement dissonance labels take the form of Dx+y, where x represents the distance between the pulses in the metrical layer AND between the pulses in the antimetrical layer, and y represents the offset between the conflicting layers.

7

Grouping dissonance labels take the form of Gx/y, where x represents the distance between pulses in the slower-moving conflicting layer (whether it be metrical or antimetrical), and y represents the distance between pulses in the faster-moving conflicting layer.

8

Even though displacement dissonance labels include an offset, it is the relative speed of the antimetrical layer and not the offset that distinguishes the two kinds of dissonances. Grouping dissonances can also be offset relative to the sounding meter.

9

A subtactus-level dissonance is one in which the dissonance can be expressed entirely within one pulse of the tactus, while a hypermetrical dissonance is one in which the dissonance can only be expressed over multiple measures of the notated score.

10

Direct metrical dissonances are those in which two conflicting layers are presented simultaneously, while indirect metrical dissonances are those in which two conflicting layers are presented in immediate succession.

11

A subliminal dissonance is one in which all musical features—harmonic rhythm, melodic groupings, dynamic accents, etc.—establish a meter other than the one notated, while none of the musical features establish or reinforce the notated meter. The accents reflecting the notated meter in a subliminal dissonance exist only in the mind of one looking

73

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at the notation, or in the gestures of a conductor; they are not realized in sound at all. 12

Any subliminal dissonance that doesn’t begin a work or movement will start off as an indirect dissonance as well. Because indirect dissonances require both conflicting layers to be sounded, they only last as long as it takes for the listener to abandon the first layer and adopt the second. Subliminal dissonances, on the other hand, can be of any length.

13

Some of the factors that contribute to the strength of a metrical dissonance include: 1) how long a dissonance lasts; 2) how many unaligned pulses there are between conflicting layers; 3) the relative strength of changes in harmonic rhythm, melodic grouping, in dynamics, articulation, texture, and register; and 4) the number of voices involved in articulating the dissonance.

Homework Assignment 4.1 Listen to the Sarabande from Bach’s Partita in C minor, BWV 826 (see Anthology, p. 4) while conducting along with the notated meter, then review the discussion of it on pp. 70–72. Next, make a copy of the music to turn in and label all dissonances by adding brackets and labels to the score in a way that matches the metrical layer referenced by each label. For displacement dissonances, each bracket should begin at the first metrically strong position from which the antimetrical layer is offset, and each label should be of the form Dx+y. For grouping dissonances, each bracket should begin with the first note that can be understood as part of the antimetrical layer, and each label should be of the form Gx/y. When determining the x and y values for each label, use the eighth note as the reference pulse. Finally, check your work: you should have dissonances bracketed and labeled in mm. 1–2, 5–6, 9–12, 12–13, 16–17, 20, and 21–22 (NB: it is the harmonic rhythm, not the melodic rhythm, that defines the dissonance in m. 20).

Homework Assignment 4.2 Listen to the first movement of Beethoven’s Piano Sonata, Op. 2, No. 1 (see Anthology, pp. 5–8) while conducting along in two with the notated meter and paying special attention to any passages with syncopation. Then write a

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paragraph on a separate sheet of paper identifying the dissonances in the following four passages by type (Dx+y or Gx/y) and explaining in each case what articulates each layer of the dissonance: mm. 33–35, 41–46, 73–80, and 95–100 (note that there are at least two different dissonances sounding simultaneously in mm. 73–80).

Homework Assignment 4.3 Listen to the Menuetto that begins the third movement of Mozart’s Symphony No. 40 (see Anthology, pp. 235–236) while conducting along in one with the dotted half and paying special attention to any passages with syncopation. Then write a paragraph on a separate sheet of paper identifying the dissonance in mm. 1–2 by type (Dx+y or Gx/y), explaining what articulates each layer of the dissonance, and identifying every pair of measures after mm. 1–2 in which the same dissonance returns. Finally, review the discussion of its hypermeter that accompanies Example 3.7 (see pp. 42–43), and identify the hypermetrical dissonance that the phrase expansion in mm. 1–14 creates by type (Dx+y or Gx/y), again explaining what articulates each layer of the dissonance.

Homework Assignment 4.4 Listen to “Eusebius” from Schumann’s Carnaval Op. 9 (see Anthology, p. 244) while conducting along with the notated meter, and then answer the following seven questions in complete sentences on a separate sheet of paper. 1 What kind of dissonance occurs in m. 1? (Hint: one cannot hear a dissonance, but can certainly feel one while conducting along.) 2 What type of dissonance is in mm. 2–3? (Use the Dx+y or Gx/y format in your answer to this question and the next three.) 3 What type of indirect dissonance straddles the bar line between mm. 8 and 9? 4 What type of indirect dissonance is in mm. 9–10? 5 What type of direct dissonance emerges in mm. 11–12? 6 Schumann intended for this miniature to depict a specific side of his own personality. How would you describe its character? 7 Schumann also used metrical dissonance (not consonance) to depict a side of his personality that is the exact opposite. What kind of dissonances do you think he used in that miniature?

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Further Reading Frisch, Walter. “The Shifting Bar Line: Metrical Displacement in Brahms.” In Brahms Studies: Analytical and Historical Perspectives, ed. George S. Bozarth, 139–164. Oxford: Clarendon Press, 1990. Kaminsky, Peter. “Aspects of Harmony, Rhythm and Form in Schumann’s Papillons, Carnaval and Davidsbündlertänze.” Ph.D. diss., University of Rochester, 1989. Krebs, Harald. “Some Extensions of the Concepts of Metrical Consonance and Dissonance.” Journal of Music Theory 31/1 (1987), 99–120. Krebs, Harald. Fantasy Pieces: Metrical Dissonance in the Music of Robert Schumann. Oxford: Oxford University Press, 1999. Mirka, Danuta. Metric Manipulations in Haydn and Mozart: Chamber Music for Strings, 1787–1791. Oxford: Oxford University Press, 2009. Willner, Channan. “The Two-Length Bar Revisited: Handel and the Hemiola.” Göttinger Händel-Beiträge 4 (1991), 208–231.

chapter 5

Metrical Processes

Just as melodic material can be developed through transposition and inversion, rhythmic material can be developed through augmentation, diminution, and other kinds of transformations. In the same way, just as large-scale tonal processes can be understood in terms of key relationships that unfold over the course of a musical work, large-scale metrical processes can be understood in terms of the different metrical states a musical work moves through on the way to its conclusion. This chapter will explore how small- and large-scale rhythmic and metrical transformations may be understood.

Augmentation and Diminution Augmentation is a rhythmic variation in which the duration of each note value is increased by a common factor (usually doubled), and diminution is a rhythmic variation in which the duration of each note value is decreased by a common factor (usually halved). In addition to transforming melodic material, the techniques of augmentation and diminution can be used to transform metrical states. Example 5.1 illustrates how one can understand the G6/4 (‰) dissonance at the end of the Sarabande from Bach’s Partita in C Minor, BWV 826 as an augmentation of the G3/2 (‰) at its beginning. One can easily recognize dissonances that are related by augmentation or diminution because the numbers in their labels will form equivalent ratios. For example, G3/2, G6/4, G9/6, etc. are all related by augmentation or diminution because they can all be reduced to a 3:2 ratio; in fact, we can call them

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all hemiolas, because we can reduce each of the larger number labels to G3/2 simply by redefining the reference pulse (i.e. a G6/4 (‰) passage could just as easily be labeled a G3/2 (Œ) passage). This feature of the labeling system is one of its strengths; by keeping the same reference pulse throughout an entire movement or work, one can easily recognize augmentation and diminution. By the same token, it is not advisable to reduce dissonance labels down to their simplest terms in each case, as it might hide such relationships.

5.1a

G3/2 (‰) in mm. 1–2.

5.1b

G6/4 (‰) in mm. 21–22.

EXAMPLE 5.1

Bach, Partita in C Minor, MWV 826, Sarabande.

Displacement dissonances with labels forming equivalent ratios can likewise easily be recognized as related by augmentation or diminution. For example, D2+1 is a diminution of D4+2, as shown in Example 5.2; in Example 5.2a, D2+1 (Ù) is found in m. 22 (marked by a dot diagram below the music), while in Example 5.2b, D4+2 (Ù ) is found in mm. 41–43 (again marked by a dot diagram). In Example 5.2b, the D4+2 in mm. 41–43 begins as a direct dissonance, but the metrical layer is abandoned soon after the antimetrical layer is introduced in m. 41. By the end of the passage, one could argue that the dissonance has become subliminal, existing only in the mind of one looking at the notation, and thus is an excellent example of how abstract the concepts of augmentation and diminution can be when applied to metrical states rather than rhythmic motives.

Metrical Processes

5.2a

D2+1 (Ù) in mm. 20–23.

5.2b

D4+2 (Ù) in mm. 40–43.

EXAMPLE 5.2

79

Schumann, Papillons, Op. 2, No. 11

Families of Dissonances When one metrical dissonance is an augmentation or diminution of another, the two dissonances are said to belong to a family. These are the only kinds of families created among grouping dissonances, but Harald Krebs defines three other kinds of familial relationships that may exist among displacement dissonances. When two displacement dissonances share a common y value and have x values in which one is the multiple of the other, they can be considered relatives, such as D2+1 and D4+1, or D3+1 and D6+1. In a similar way, displacement dissonances that share a common x value can be considered relatives, such as D4+1, D4+2, and D4+3. Finally, when two displacement dissonances have x values in which one is the multiple of the other, and the sum of the x and y values of one equals the y value of the other, they can be considered relatives, such as D3+2 and D6+5, or D4+1 and D8+5. Within any given family, a member with a higher number of unaligned pulses per segment is considered a tight relative compared to a member with a lower number of unaligned pulses per segment (the segments in the comparison could be of any length, as long as they are the same and long enough for the dissonances to fully express themselves). Conversely, a member with a lower number

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of unaligned pulses per segment is considered a loose relative compared to a member with a higher number of unaligned pulses per segment. Examples 5.3–5.6 illustrate the four kinds of relationships that may exist among displacement dissonances. Example 5.3 compares the dissonances D2+1 and D4+2, which have equivalent ratios and are thus related by augmentation and diminution; D2+1 is a tight relative of D4+2, while D4+2 is a loose relative of D2+1. Example 5.4 compares the dissonances of D2+1 and D4+1, which have the same y value; D2+1 is a tight relative of D4+1, while D4+1 is a loose relative of D2+1. Example 5.5 compares the dissonances of D4+1 and D4+2, which have the same x value; because they have the same x value (i.e. the same number of unaligned pulses per segment), they cannot be characterized as tight or loose compared to one another. Finally, Example 5.6 compares the dissonances D3+2 and D6+5, in which the sum of x and y in the first equals the y value in the second; D3+2 is a tight relative of D6+5, while D6+5 is a loose relative of D3+2. Beneath the dot diagram in each example is a realization of the two dissonances being compared placed side by side; in each realization, the reference pulse is the eighth note.

EXAMPLE 5.3

D2+1:

• •

D4+2:

• •

D2+1 and D4+2 (Equivalent Ratios; Heard as Augmentation)

• • •

• • •

• • •

• • • •

• • •

•

EXAMPLE 5.4

D2+1: D4+1:

• • • •

• • •

• • •

• • • •

• • •

•

• • •

• • •

•

etc. tight relative of D4+2 (= more dissonant) etc. loose relative of D2+1 (= less dissonant)

D2+1 and D4+1 (Common Offset)

• • • •

• • •

• • •

• • • •

• • • •

• • •

• • •

• • • •

• • • •

• • •

• • •

etc. tight relative of D4+1 (= more dissonant) etc. loose relative of D2+1 (= less dissonant)

Metrical Processes

EXAMPLE 5.5

D4+1:

• •

D4+2:

• •

D4+1 and D4+2 (Common x Value)

•

•

•

• •

• •

•

•

• •

•

•

•

• •

• •

•

•

• •

•

EXAMPLE 5.6

D3+2:

D6+5:

• • • •

81

•

•

•

etc. relative of D4+2

• •

•

•

etc. relative of D4+1

•

•

D3+2 and D6+5 (x+y in the First Equals y in the Second)

•

•

•

• •

• • •

•

•

•

• • •

• • • •

•

•

•

• •

• • •

•

•

•

• • •

etc. tight relative of D6+5 (= more dissonant) etc. loose relative of D3+2 (= less dissonant)

Tightening and Loosening By defining the various possible relationships among different displacement dissonances, one may better observe two distinct metrical processes: tightening and loosening (these terms were also first defined by Krebs, though they have been broadened significantly here). Tightening is the process of moving from one displacement dissonance to a tight relative of that dissonance, and as such carries with it a sense of acceleration. Loosening is the process of moving from one displacement dissonance to a loose relative of that dissonance, and as such carries with it a sense of deceleration. Example 5.7 provides mm. 5–12 from the first movement of Beethoven’s Piano Sonata, Op 10, No. 2. In these measures, one can observe a tightening as the music moves from a D4+1 (‰) to a D2+1 (‰), and then a loosening, as the music moves back to D4+1 (‰). The process of tightening is very audible in the example because the first two dissonances actually share pulses, while it is harder to actually hear the subsequent loosening because the quarter rest in m. 8 separates the dissonances involved.

82

EXAMPLE 5.7

Metrical Processes

Beethoven, Piano Sonata, Op. 10, No. 2, I, mm. 5–12.

Finding the Best Label for a Metrical Dissonance Example 5.8 provides a passage from Beethoven’s Fifth Symphony that can be characterized by two different kinds of displacement relationships. A tied pair of quarter notes in the low register of mm. 182–183 forms a D4+2 (‰). But one could also view the tied quarter notes in mm. 182–183 as a D8+2 (‰). While the D4+2 label matches the label’s x value with the length of each measure in the notated meter, the alternative D8+2 reading of mm. 182–183 seems equally plausible because the two bars immediately preceding this dissonance have nothing but half notes on the notated downbeats, and so the listener perceives a deceleration in the movement’s momentum.

EXAMPLE 5.8

Beethoven, Symphony No. 5, I, mm. 174–185.

The difficulty in finding the best label for a metrical dissonance such as the one at the end of Example 5.8 is partly rooted in the subjective nature of metrical analysis: to label the dissonance, one must choose a reference pulse, and such decisions are subjective, just as choosing the tactus for a particular musical context is subjective. The difficulty is amplified when one is attempting to label a dissonance that is created by a single unaligned pulse, as at the end of Example 5.8, since the x value of the label is meant to capture the distance between pulses in both the metrical and the antimetrical layers, and with only one pulse representing the antimetrical layer, any such distance can only be theoretical. In such cases, recomposition is a useful tool that might help one decide between alternate labeling possibilities. Example 5.9

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83

provides two recompositions of the last dissonance in Example 5.8, one that continues the dissonance as a D8+2 (the tied quarters are eight pulses apart), and another that continues the dissonance as a D4+2 (the tied quarters are only four pulses apart). Through recomposition, one might better understand the character of the actual dissonance, as one can then hear and feel which of the two possibilities seems like a more natural continuation of the dissonance.

5.9a

Recomposed to More Strongly Articulate D8+2.

5.9b

Recomposed to More Strongly Articulate D4+2.

EXAMPLE 5.9

Recomposition of Beethoven, Symphony No. 5, I, mm. 180–185.

Metrical Upshifting and Downshifting This chapter so far has focused on relationships between metrical dissonances, but some metrical processes do not involve dissonance. Metrical upshifting is a transformation in which the fastest consistent pulse changes to a new, faster pulse layer within the same metrical consonance. Metrical downshifting is a transformation in which the fastest consistent pulse shifts to a slower pulse layer within the same metrical consonance. Example 5.10 illustrates metrical upshifting and downshifting in the first movement Haydn’s String Quartet, Op. 50, No. 2. The passage begins with the quarter note as the fastest consistent pulse (the sixteenths at the beginning are not regular enough to establish a pulse layer), then one hears the eighth note replace the quarter as the fastest consistent pulse in mm. 43–50. By conducting the hypermeter at the beginning of the passage and switching to the notated meter in m. 43, one can actually feel the upshift; the same holds true for the downshift if one switches from the notated meter back to the hypermeter in

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m. 51. The tactus in the passage doesn’t actually change, as one can feel by simply conducting the whole passage in four-bar hypermeasures, but by changing the speed of our conducting pattern, we highlight the process by making it visceral.

EXAMPLE 5.10

Haydn, Op. 50/2, I, mm. 39–57.

In Example 5.10, the sound of an upshift is reinforced by a change in the harmonic rhythm: in mm. 39–42, there are only two chords spread over four bars, but in mm. 43–52, the harmonic rhythm accelerates to one chord per bar. This is often the case, but not always. One can see a similar upshift in Example 5.11a: the fastest constant pulse changes from the quarter note to the eighth note in m. 95, and the harmonic rhythm accelerates to one chord per bar in m. 96 (the first hypermetrical downbeat with the eighth note as the constant pulse). In Example 5.11b, however, there is a downshift, but no sense of a deceleration in the harmonic rhythm, mainly because the G7 harmony that fills five bars in mm. 190–194 is enough to disrupt any sense of rhythmic regularity. When the half note is established as the new fastest pulse in mm. 196–199, the harmonic rhythm is one chord per bar. To experience the feeling of upshifting and downshifting in Example 5.11, conduct a four-beat pattern in whole notes instead of the tactus at the beginning of Example 5.11a, and

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85

switch to the tactus in m. 96; for Example 5.11b, conduct the tactus at the beginning and switch to conducting a four-beat pattern in whole notes at m. 196.

5.11a

mm. 88–101.

5.11b

mm. 188–207.

EXAMPLE 5.11

Beethoven, Symphony No. 5, I.

Remembering which term means an increase in activity and which one means a decrease in activity can be problematic, because we tend to equate the fastest pulse with the lowest level of the metrical hierarchy. The terms are rooted in the metaphor of a car’s gearshift: a higher gear is associated with more motion, and a lower gear with less motion. Upshifting is increasing the speed of the fastest pulse, while downshifting is decreasing the speed of the fastest pulse. Conducting along as suggested helps to

86

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make that process visceral, as we can feel our beat pattern speed up or slow down to match the music. Just as the objects outside the car window move by at a faster rate in a higher gear and at a slower one in a lower gear, so do the notes of the fastest pulse go by at a faster rate after an upshift and at a slower rate after a downshift.

Meter Changes and Tempo Modulation Changes in the sounding meter can involve grouping dissonances (as in Example 4.6), displacement dissonances (as in Example 4.9), upshifting, or downshifting. In the comparison between two meters, a change in the number of beats per measure indicates a grouping dissonance unless the larger number of beats is simply a multiple of the smaller number. Tempo Modulation (also known as “metric modulation”) is a simultaneous change to both the sounding meter and the tactus in which a pulse layer other than the tactus from the old meter remains constant and becomes a layer of the new meter. Though not a requirement, most musicians reserve the term for changes in which the new tactus is not simply a multiple or factor of the old tactus (i.e. 2:1, 3:1, etc.)—upshifting or downshifting seems more appropriate in such cases, even though these terms do not require a change in meter. Example 5.12 illustrates one of the most common forms of tempo modulation: a 3:2 relationship between the tactus of the new meter and the tactus of the old. In the example, the constant pulse between the two meters is the quarter note, while the tactus, indicated by the composer, changes from the dotted half note to the half note. This particular tempo modulation is an indirect grouping dissonance, like many if not most of those passages to which the term tempo modulation is regularly applied.

EXAMPLE 5.12

Prokofiev, Violin Sonata, Op. 94a, II, mm. 158–165.

Fragmentation The two final processes to be discussed in this chapter are not purely metrical processes in the same sense as those mentioned earlier, but they have a significant impact on how one perceives meter. Fragmentation is a developmental technique

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that divides melodic material into smaller segments and then repeats only one segment of the material multiple times. As a developmental technique, it affects our sense of hypermetrical rhythm, creating a sense of acceleration. Example 5.13 illustrates such fragmentation in the development section from the first movement of Mozart’s Symphony No. 40. Though a single hypermeter often persists throughout a passage in which fragmentation is used to develop material, changing one’s conducting pattern to reflect the acceleration helps to highlight the process. It is questionable whether the hypermeter of the three parts to Example 5.13 is actually different, but by conducting each part differently, one can actually feel the acceleration created by the fragmentation of the music in Example 5.13a. Because mm. 137–138 are an exact repetition of mm. 135–136, it sounds as though Example 5.13b is developing the music of Example 5.13a through fragmentation: specifically, the first half of Example 5.13a becomes the model for both halves of Example 5.13b (the first halves of each differ only by their transposition level and their last pitch). In a similar way, the first half of Example 5.13b is divided in half to become the model for Example 5.13c: while the repetition of the model is varied more in this case, it is quite easy to hear the four measures as four varied repetitions of the same idea because of the texture that pits the opening motive in the violins against that same motive in the bass four times, as well as the fact that the bass repeats the same pitches each time to create a pedal effect. Sing the example while conducting along as indicated.

5.13a

mm. 104–107 (Conduct in 4).

5.13b

mm. 135–138 (Conduct in 2).

5.13c

mm. 153–156 (Conduct in 1).

EXAMPLE 5.13

Fragmentation in Mozart’s Symphony No. 40, I.

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Stretto Another way to create a sense of acceleration is through the developmental technique of stretto. Stretto is the technique of writing overlapping imitative entries, and so stretto is possible in any imitative texture, and can commonly be found in inventions, in fugues, and in fugatos. Example 5.14 shows stretto in Bach’s Fugue in F Major from Book I of his Well-Tempered Clavier; Example 5.14a provides the first eight bars of the fugue where the subject and answer are first presented in succession with no overlap, while Example 5.14b provides mm. 36–44 where the subject, transposed to D minor, is presented in stretto. By conducting along as indicated by the annotations above the staff in the example, one can viscerally experience how stretto makes the music feel twice as fast: though the length of each subject entry presented in stretto is the same as it was in mm. 1–4 (four bars long), each successive entry overlaps the previous entry by two bars.

5.14a

mm. 1–8.

5.14b

mm. 36–44.

EXAMPLE 5.14

Bach, Well-Tempered Clavier, Book I, Fugue in F Major.

Metrical Processes

Chapter Review 1

Augmentation is a rhythmic variation in which the duration of each note value is increased by a common factor (usually doubled), and diminution is a rhythmic variation in which the duration of each note value is decreased by a common factor (usually halved). In addition to transforming a musical motive, the techniques of augmentation and diminution can be used to transform metrical states.

2

Those grouping dissonances that share equivalent ratios between x and y values are related by augmentation and diminution and thus considered part of a family.

3

There are four possible kinds of family relationships between displacement dissonances: 1) those that share an equivalent ratio between x and y values; 2) those that have x values in which one is the multiple of the other, and that share a common y value; 3) those that share a common x value; and 4) those that have x values in which one is the multiple of the other, and in which the sum of the x and y values of one equals the y value of the other.

4

Within any given family, a member with a higher number of unaligned pulses per segment is considered a tight relative compared to a member with a lower number of unaligned pulses per segment. Conversely, a member with a lower number of unaligned pulses per segment is considered a loose relative compared to a member with a higher number of unaligned pulses per segment.

5

Tightening is the process of moving from one displacement dissonance to a tight relative of that dissonance, and as such carries with it a sense of acceleration.

6

Loosening is the process of moving from one displacement dissonance to a loose relative of that dissonance, and as such carries with it a sense of deceleration.

7

It is difficult to find the best label for a metrical dissonance when it is created by a single unaligned pulse, since the x value of the label is meant to capture the distance between pulses in both the metrical and the antimetrical layers, and with only one pulse representing

89

90

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the antimetrical layer, any such distance can only be theoretical. In such cases, recomposition is a useful tool that might help one decide between alternate labeling possibilities. 8

Metrical upshifting is a transformation in which the fastest consistent pulse shifts to a new, faster pulse layer within the same metrical consonance. Metrical downshifting is a transformation in which the fastest consistent pulse shifts to a slower pulse layer within the same metrical consonance. Upshifting is increasing the speed of the fastest consistent pulse within a metrical consonance, while downshifting is decreasing the speed of the fastest consistent pulse.

9

Tempo modulation (also known as “metric modulation”) is a simultaneous change to both the sounding meter and the tactus in which a pulse layer other than the tactus from the old meter remains constant and becomes a layer of the new meter. Though not a requirement, most musicians reserve the term for changes in which the new tactus is not simply a multiple or factor of the old tactus.

10

Fragmentation is a developmental technique that divides melodic material into smaller segments and then repeats only one segment of the material multiple times. As a developmental technique, it creates a sense of acceleration.

11

Stretto is the technique of writing overlapping imitative entries, and so stretto is possible in any imitative texture, and can commonly be found in inventions, in fugues, and in fugatos. It also creates a sense of acceleration.

Homework Assignment 5.1 Listen to Bach’s Two-Part Invention in A Minor while conducting along with the notated meter (see Anthology, p. 1), then make a copy of it and add annotations that label each dissonance by type (use the eighth note as the reference pulse). Finally, write a sentence explaining why mm. 3–6 is an example of tightening.

Homework Assignment 5.2 Listen to the first movement of Haydn’s Piano Sonata in C Minor, Hob. XVI: 20 while conducting half notes in a two-beat pattern (see Anthology, pp. 157–162);

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when there is a notable increase in rhythmic activity, switch to conducting the notated meter, and when there is a notable decrease, switch back to conducting half notes in two. Then make a copy of the movement and annotate your conducting above each system. Finally, on a separate sheet of paper, cite all instances of metrical upshifting and downshifting using measure numbers and specify whether each one is an upshift or downshift.

Homework Assignment 5.3 One interpretation of the hypermeter in the first 108 measures of Beethoven’s Symphony No. 3 is provided; add annotations that represent this interpretation to your copy of the score so that you can conduct along with a recording, and then listen to it while conducting along (see Anthology, pp. 9–14). mm. 1–2: one hypermeasure in two (dotted half=beat) mm. 3–18: four hypermeasures in four (dotted half=beat) mm. 19–22: two hypermeasures in two (dotted half=beat) mm. 23–26: one hypermeasure in four (dotted half=beat) m. 27-m. 29, beat 1: one asymmetrical hypermeasure in three (dotted half-half-half ) m. 29, beat 2-m. 33, beat 1: two hypermeasures in three (half note=beat) m.33, beat 2-m. 36: one asymmetrical hypermeasure in four (dotted halfhalf-dotted half-dotted half ) mm. 37–64: seven hypermeasures in four (dotted half=beat) mm. 65–70: one hypermeasure in six (dotted half=beat) mm. 71–102: eight hypermeasures in four (dotted half=beat) mm. 103–108: one hypermeasure in six (dotted half=beat) Now answer the following questions on a separate sheet of paper (be as specific as possible, and use the eighth note as the reference pulse): 1 What type of dissonance occurs in mm. 7–8? 2 What two hypermetrical dissonances occur in mm. 7–10? (Hint: consider where the forzando markings fall within the hypermeasure.) Are either of these a tight or loose relative to the dissonance in mm. 7–8? 3 What type of dissonance occurs in m. 13? 4 What type of grouping dissonance occurs in mm. 25–26? 5 What type of grouping dissonance occurs in mm. 29–33?

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6 7 8 9

What type of dissonance pervades mm. 45–55? What type of dissonance occurs in mm. 71–72? What type of dissonance is articulated by the low strings in mm. 89–90? What type of dissonance is articulated by the winds and first violins in mm. 89–90? 10 What type of dissonance occurs in mm. 99–101? 11 What type of hypermetrical grouping dissonance is created by moving from four-bar hypermeasures to a six-bar hypermeasure, as in mm. 61–70 or 99–108? What is the relationship between these dissonances and the dissonance in mm. 29–33?

Homework Assignment 5.4 Find a short piece (one or two pages of music) that has at least three different types of metrical dissonances. (If you have trouble finding one, try looking at common-practice keyboard or vocal music.) Make a copy of the piece, and then add annotations in which each dissonance is labeled by type each time it occurs. Finally, check your work by conducting along with a recording.

Further Reading Kaminsky, Peter. “Aspects of Harmony, Rhythm and Form in Schumann’s Papillons, Carnaval and Davidsbündlertänze.” Ph.D. diss., University of Rochester, 1989. Krebs, Harald. “Some Extensions of the Concepts of Metrical Consonance and Dissonance.” Journal of Music Theory 31/1 (1987), 99–120. Krebs, Harald. Fantasy Pieces: Metrical Dissonance in the Music of Robert Schumann. Oxford: Oxford University Press, 1999. Mirka, Danuta. Metric Manipulations in Haydn and Mozart: Chamber Music for Strings, 1787–1791. Oxford: Oxford University Press, 2009. Rothstein, William N. Phrase Rhythm in Tonal Music. New York: Schirmer, 1989.

chapter 6

Metrical Maps

Now that we have established the difference between notated meters and sounding meters, the factors that contribute to establishing a sounding meter, the different kinds of consonant and dissonant metrical states possible, and the different processes that can transform one metrical state into another, we need a method of charting the progression of metrical states in a work or movement that is suitable for analysis. A metrical map is a summary of the metrical states in a musical work or movement from beginning to end. This chapter will introduce two kinds of metrical maps: dissonance logs and conducting plans.

Dissonance Logs A dissonance log is a summary of the metrical dissonances in a musical work or movement from beginning to end. Harald Krebs was the first to create metrical maps for analysis, and the ones he created were very similar to the dissonance logs discussed in this book. Example 6.1 provides a dissonance log for the exposition of the first movement from Schumann’s String Quartet, Op. 41, No. 1; compare this log to the score (see Anthology, pp. 246–247). A dissonance log is like a musical staff in that it represents time from left to right, and vertical slices of the log represent things that happen simultaneously. Time can be measured in either beats or measures of the notated score, depending upon the tempo, and groups of beats or measures are represented by boxes. Because the music of Example 6.1 is fast, the boxes with solid lines represent hypermeasures, and the boxes with dotted lines represent measures. The numbers inside the boxes represent the number of voices that articulate the antimetrical layer of each dissonance. In Example 6.1, for instance, the

4

145

42 4

50

105

1 1 1 1

54

109

(mm. 149150 = 3435)

4

99 101

3 1 3

46

113

2

3 4 4 4

2nd ending 151

2

58 2

62 4

4

70

72 4 4

4 1 1 1

4 1 1 1

4

S Theme (117137) 117 121 125

66

4

1

4

1

1

4 3 4 4 4 4 4

1st ending 129 133

1

EXAMPLE 6.1

Schumann, String Quartet, Op. 41, No. 1, I, Exposition.

1

2

Transion (76117) 76 80 84

reference pulse = eighth note boxes with solid lines = hypermeasures boxes with doed lines = measures NB: numbers within each box represent how many voices arculate the an-metrical layer

part: Closing (137148) mm.: 137 141 D12+3 2 1 3 D6+3 D6+3(2→1)

part: mm.: 88 92 95 D24+18 4 D24+12 D12+3 2 2 4 D6+3 4 4 4 D6+3(2→1) D3+2 2 2

part: P Theme (3475) mm.: 34 38 D24+12 4 D12+3 4 4 D6+3 D3+2 1

2 2

Metrical Maps

EXAMPLE 6.2

95

Schumann, String Quartet, Op. 41, No. 1, I, mm. 44–49.

dissonance log records the fact that all four voices of the quartet articulate D12+3 in m. 44, but only three voices articulate it in mm. 46 and 48, and only one voice articulates it in m. 47. In Example 6.2, one can confirm all these observations, but one can also see details that the metrical map leaves out: it is the upper three parts that articulate D12+3 in m. 46 and 48, while it is the cello that articulates it in m. 47. By recording the number of voices that articulate each dissonance, the dissonance log provides a rough measure of their relative strength, but this is only one of many factors that contribute to the relative strength of a dissonance. The dissonances are organized from top to bottom within each box, with those dissonances having the highest x numbers in their labels at the top, and those with the lowest at the bottom. This allows one to quickly see the processes of tightening and loosening, including augmentation and diminution. In Example 6.1, for instance, one can see that there is a tightening from D12+3 to D6+3 in mm. 36–52, and a further intensification from D6+3 to D3+2 in mm. 52–54. Consulting the score, one can see that the accented chords on beat 2 of every other measure in mm. 36–40 and 44–48 give way in mm. 52–54 to tied dotted quarter notes in the cello in every measure, and that, in m. 54, the first violin actually articulates D3+2 for six pulses before the music becomes metrically consonant once more. While most of the larger boxes indicating hypermeasures in Example 6.1 represent four-bar units, the large-scale dissonances of D24+12 and D24+18 that result from changes in phrase rhythm in mm. 72 and 95 have an impact on the hypermeter, creating a two-bar hypermeasure in mm. 70–71, and a three-bar hypermeasure in mm. 92–94. The metrical map in Example 6.1 reflects these changes by adjusting the placement of the solid lines. By comparing the eightbar phrase in mm. 34–41 that begins the primary theme to the last ten-bar phrase of that theme in mm. 66–75, it is clear that the former is the model for the latter: the first six bars of the violin I part are the same in both phrases. The latter is an internal expansion of the former through sequential repetition, as

96

Metrical Maps

shown in Example 6.3. The three-bar hypermeasure in mm. 92–94 sounds like a link because it is non-thematic and tonally unstable, so it stands on its own, with a hypermetrical downbeat clearly marked in m. 95 by the return of a rhythmic motive from the primary theme.

6.3a

mm. 34–41.

6.3b

mm. 66–75.

EXAMPLE 6.3

Schumann, String Quartet, Op. 41, No. 1, I, mm. 34–41 and 66–75.

One dissonance label in the map requires some explanation: D6+3 (2→1). The portion in parentheses is meant to reflect the fact that beat 2 starting in m. 131 now sounds like a downbeat, because the metrical layer stopped being articulated in m. 130, and the third chord change on beat 2 in a row after the last articulated downbeat comes in m. 131. While it is subjective exactly where beat 2 becomes the downbeat of the sounding meter, it is less subjective that it happens somewhere in mm. 131–135, as not a single notated downbeat is articulated again until m. 136 (refer back to the earlier discussion of this passage as Example 4.9 on pp. 58–59). This analysis provides the earliest possible place to take the change in the sounding meter, but most listeners would probably experience the change somewhere in that span, and once the sounding meter changes, D6+3 no longer sounds like a dissonance, it only looks like one; that is, it becomes a subliminal dissonance, dissonant only in the minds of those holding the score. Thus, “D6+3 (2→1)” represents the subliminal expression

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97

of that dissonance, while “D6+3” without the added portion in parentheses represents those direct and indirect expressions of it that actually sound like metrical conflicts. This change in the sounding meter occurs both in the first and the second endings, but it is only about half as long in the second ending. A metrical map such as the dissonance log in Example 6.1 can help a musician see the large-scale organization of metrical states in a work, facilitating observations that would be difficult to make otherwise, such as relationships between dissonances that appear on different pages of the score. In Example 6.1, for instance, the map highlights how the primary and secondary themes of the exposition are closely related. Metrically speaking, the first four bars of the second theme include the same dissonance found in the first four bars of the primary theme, but they also include three measures of D6+3, which is found in mm. 52–54 of the primary theme. The six consecutive bars of D12+3 in the primary theme (mm. 36–41) can likewise be found in the second theme (mm. 123–128), but starting with its second four-bar hypermeasure. On the whole, the second theme is more dissonant than the primary theme in part because it combines D12+3 and D6+3 in the same hypermeasures, as is shown in Example 6.4. While the primary theme includes both of these dissonances, they don’t occur within the same four-bar hypermetrical unit. The second theme also transforms beat 2 of the notation into the downbeat of the sounding meter in mm. 131–135 before its final cadence in m. 137 (see Example 4.9), while there is no similar challenge to the sounding meter in the primary theme. These differences run parallel to the tonal scheme: the primary theme, in addition to being less metrically dissonant, is in the primary key throughout, while the second theme begins in E minor and moves through D minor (mm. 121–123) before settling into the key of C major: that is, it is not just more metrically dissonant, it is also more tonally unstable.

EXAMPLE 6.4

Schumann, String Quartet, Op. 41, No. 1, I, mm. 117–124.

Examples 6.5 and 6.6 provide dissonance logs for the development and recapitulation of this movement. A quick comparison of Example 6.1 with Examples 6.5 and 6.6 reveals that the dissonances of the exposition are retained

EXAMPLE 6.5

Schumann, String Quartet, Op. 41, No. 1, I, Development.

EXAMPLE 6.6

Schumann, String Quartet, Op. 41, No. 1, I, Recapitulation.

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Metrical Maps

throughout the rest of the movement, and that the recapitulation is, metrically speaking, almost an exact duplicate of the exposition (the only difference between them is that when the music of m. 139 returns in m. 358, all four voices join in to articulate D12+3, whereas before only two voices articulated the dissonance). To fully appreciate what the maps have to offer, first use them to annotate the score with conducting patterns and then listen to the movement two times and conduct along, once while following the score, and then again while following the maps. (Because of the placement of the repeat signs, when following the map of the exposition, one should move directly from m. 148 back to m. 34, as mm. 149–150 are really just a repetition of mm. 34–35 and the repeats signs are not in the map.)

Dissonance Logs for Slow Movements Example 6.7 provides a dissonance log of the second movement of Schumann’s String Quartet Op. 41, No. 2. Because it’s a slow movement, the dissonance log tracks the dissonance from beat to beat rather than from measure to measure, and the solid boxes in the log represent measures from the score rather than hypermeasures. It is organized in the same way as the last dissonance log, but has an added feature: it marks where the downbeats of the sounding meter can be found when they are not the same as the first beat of the notated meter. A lighter shading is used when the sounding downbeat occurs on the last eighth note of the shaded beat, while a darker shading is used when the sounding downbeat occurs on the first eighth note of the shaded beat. This way of representing the sounding downbeats works well for this movement, but dissonance logs can and should be adjusted to fit the particular work that is the object of inquiry. Nevertheless, some strategy of representing the sounding downbeats when they conflict with the notated downbeats is a very useful feature in a dissonance log, as it allows one to better follow it while listening to the music that it models. Like the last dissonance log, this one also distinguishes between subliminal dissonances and sounding dissonances, but it uses the abbreviation “(sbl)” to do so, as it would be harder to capture that the third part of beat 4 in each measure becomes the sounding downbeat in an abbreviation. Example 6.8 provides the music of mm. 17–20 with annotations added that illustrate how one would conduct the sounding meter of the second theme in this double variations form. Example 6.9 provides the music of mm. 49–50 with

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annotations added that illustrate how one would conduct the sounding meter of the second variation. Example 6.10 provides the music of mm. 65–66 with annotations added that illustrate how one would conduct the sounding meter of the third variation. Listen and conduct along with each of these passages as indicated, and this should be enough of an introduction to feel comfortable conducting the sounding meter throughout the entire movement. Before doing so, take the time to transfer the conducting annotations from Example 6.8–6.10 to your own copy of the score (see Anthology, pp. 252–257), then use Example 6.7 to add similar annotations to the rest of the movement so that you have a guide to conducting it from start to finish. Listen to the movement while following the score and conducting along, then listen again and conduct along while following Example 6.7 (remember that the first note is actually a pick-up to beat 2 in m. 1, so the first sounding beat to be conducted is actually beat 2 of a four-beat pattern). There are at least two factors that contradict the interpretation of Variation II given in Example 6.7 and 6.9. The given interpretation depends upon taking the fourth beat of m. 48 as the downbeat of a shift in the sounding meter. The rationale behind the given interpretation depends upon the melodic groups and textural changes and was articulated in Chapter 1 (see pp. 12–13). The two factors that contradict the given interpretation are continuity and parallelism. If the string quartet chooses to keep a steady tempo throughout m. 48 and avoid any pause whatsoever between Variations I and II, then the momentum of the notated meter could well lead one to hear the notated meter as the sounding meter throughout Variation II (see the discussion of continuity on p. 14). This alternate interpretation is further supported by a parallelism: both Theme 2 and Variation I start with beat 4 of the sounding meter, and Variation II is based on the same theme. Because the previous two sections started with beat 4, listeners would be more likely to interpret the first beat of the new variation as beat 4. This alternate interpretation is not better than the interpretation given in Example 6.7 and 6.9, just different, but the alternate interpretation would change how we understand the dissonances in the Variation. If one takes the notated meter as the sounding meter in Variation II, now some of the melodic groupings and texture changes that were heard as reinforcing the meter in the earlier interpretation could be heard as dissonant. See p. 254 in the Anthology. The long note played by the viola in m. 49 would certainly be heard as expressing the antimetrical layer of D6+3 (‰), and this dissonance is also expressed by the viola in mm. 51, 60, 61, 63, and 64.

1

3 1

4 5 2222 11

6 1

1

7 1

8 2

9 2

10

11 22323

12 13 14 3222 11

4

1

44

1

1

(A') 45

2 1

1 1

46

1

47

1

2

2

1

1 1

48

11

1 2 11

1

1

37

1

1

38

Variaon II (on Theme 2) 49 50 51 52 1 2 1 2

1

1

Variaon I (on Theme 2) 33 34 35 36

2

53

1 111

1

39

EXAMPLE 6.7 Metrical Map of Schumann, String Quartet, Op. 41, No. 2, II.

40

2

54

111 11

reference pulse = eighth note sbl = subliminal NB: numbers in the map indicate the number of voices that arculate the anmetrical layer = sounding downbeat (on last eighth of notated beat) = sounding downbeat (on first eighth of notated beat)

secon: mm.: 43 D12+9(sbl) D12+3 D3+2 D3+1

secon: (A') mm.: 29 30 31 32 D12+9(sbl) D12+3 D3+2(sbl) 4 4 4 4 5 4 4 4 4 4 4 4 5 5 D3+2 D3+1

4

55

1

4 2 1

56

(B secon) 41 42 3 3

secon: Theme 2 (rounded binary) (B secon) mm.: 15 16 17 18 19 20 21 22 23 24 25 26 27 28 D3+2(sbl) 113333441133334411341144455455 1133225444444444 D3+2 4 3 3 2 4

secon: Theme 1 mm.: 1 2 D3+2 1 1 1 1

2

4

6

86

72

5

4

1

87

5

73

59

1

88

74

1122

5

4

60

6

1

4

6

5

2

4

64

1

1

1

Coda 106

1

1

107

1

1

2

108

80

1

2

1

1

109

1

81

Variaon III (free) 65 66 67 6 6 6 6 6

Variaon IV (free) 77 78 79

63

Theme 1 90-105 = 1-16

76

62

sbl = subliminal

89

6

75

(A') 61

4

1

EXAMPLE 6.7

(Continued)

NB: numbers in the map indicate the number of voices that arculate the anmetrical layer = sounding downbeat (on last eighth of notated beat) = sounding downbeat (on first eighth of notated beat)

reference pulse = eighth note

secon: mm.: 85 D12+6(sbl) D12+9 G3/2 D3+2 D4+1 1 1 D2+1 2

secon: mm.: 71 G6/4 G3/2(sbl) 6 D4+1 D2+1

secon: (B secon) mm.: 57 58 G6/4 G3/2(sbl) D12+9(sbl) 3 6

110

82

68

5

5

4

1

84

70 6

112 5

11132

5

4 112

111

1

83

5

69

104

Metrical Maps

EXAMPLE 6.8

Schumann, String Quartet, Op. 41, No. 2, II, mm. 17–20.

EXAMPLE 6.9

Schumann, String Quartet, Op. 41, No. 2, II, mm. 49–50.

EXAMPLE 6.10

Schumann, String Quartet, Op. 41, No. 2, II, mm. 65–66.

Conducting Plans A conducting plan is a metrical map that provides a strategy for conducting the sounding meter in a musical work or movement from beginning to end. Example 6.11 provides a conducting plan for the eighth variation from Brahms’ Variations on a Theme by Haydn. A conducting plan is particularly useful when the sounding meter is different than the notated meter much if not most of the time, as is the case in this variation.

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A section (mm. 322–341) four hypermeasures in 5, MM = c. 60 (dotted half ) B section (mm. 342–349), A′section (mm. 350-end) : B section: four hypermeasures in 3, MM = c. 90 (half note) A′ section: two hypermeasures in 4, MM = c. 60 (dotted half ) one hypermeasure in 3, dotted half note = beat : EXAMPLE 6.11

Conducting Plan for Brahms, Haydn Variations, Var. VIII.

The eighth variation represented by the map in Example 6.11 is in a rounded binary form, just like the theme upon which it is based. Repeat signs are not used in the A section, but when consulting the score (see Anthology, pp. 107–109), one can see that the second pair of five-bar phrases is a varied repetition of the first pair. Regarding the first five-bar phrase and its repetition, one can see that the line in the first three bars of the variation is repeated by the first violin and viola parts in mm. 332–334, only an octave higher. Regarding the second fivebar phrase and its repetition, one can see that the piccolo, clarinet, and bassoon parts in mm. 327–329 are repeated exactly in mm. 337–339 (the only difference between mm. 327–329 and 337–339 is the way those parts are accompanied in the strings). Thus, the A section is really ten bars long, just as long as the A section of the theme upon which it is based. Just as is typical in a rounded binary, the B and A′ sections are repeated as a block. Also in typical fashion, the return of the A section material is clearly marked by multiple factors: not only does the opening material return at pitch, but so does the opening dynamic, and the return is accompanied by a significant change in texture as well (even though the texture at the beginning of A′ is very different than the one at the beginning of A). What makes this variation very different than the theme and most rounded binaries, however, is the way that metrical dissonance is used to articulate the form. As the B section begins, the tactus accelerates, shifting from the dotted half note to the half note, as is shown in Example 6.12. The two four-bar phrases that constitute the B section are each heard as two hypermeasures in three, creating an indirect grouping dissonance with the A section that is articulated by the melodic grouping of the B section (starting with beat 3 in m. 342, the moving parts are paired in contrary motion, each articulating a contour pattern that divides the line into four-note groups). The shift of the tactus back to the dotted half in m. 350 links the music of that measure to the beginning of the variation and thus helps to articulate the form. Because of the elision in m. 354, however, the five-bar phrase from

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the A section is conducted with a four-bar hypermeasure, and the beginning of another: the strong perfect authentic cadence in m. 354, together with the entrance of the countermelody in the oboe and horn parts, is more than enough to create the strong sense of a hypermetrical downbeat. Listen to the variation while conducting along with the plan given as Example 6.11, then transfer the conducting patterns suggested by the plan to the score and listen to it again while conducting along with the score.

EXAMPLE 6.12

Brahms, Haydn Variations, Var. VIII, mm. 342–345.

The conducting plan is a simpler kind of metrical map, but one that nevertheless can sometimes more efficiently communicate the most important metrical changes to a musician in a visceral way. A conducting plan might be a good way to start an investigation, but it can only represent those metrical details that change the tactus or how the tactus is grouped. If one wanted a metrical map that could show the D2+1 (‰) in mm. 327–331 and 350–end, or the D6+4 (‰) in mm. 334–335 and 339–340, one would need a dissonance log. While always simpler than dissonance logs, conducting plans can be considerably more complicated than the one provided as Example 6.11. Example 6.13 provides a conducting plan for Schumann’s String Quartet, Op. 41, No. 2, II. Compare it both to the score (see Anthology, pp. 252–257), and to the dissonance log of the same movement given as Example 6.7. This conducting plan represents many of the displacement dissonances that define the character of this movement through pattern changes, but not all of them. A conducting plan, for example, can’t really do justice to such a small shift as the one shown in mm. 16–32 of Example 6.7 simply by changing patterns: one must actually change the length of individual beats within a pattern. This kind of shift is represented in the plan first by the instructions “one bar of 4 with a short beat 3 (= quarter note).” By shortening the third beat of Theme 1’s final measure from a dotted quarter to a quarter note, we are able to conduct the sounding meter of Theme 2 (see Anthology, p. 253). The uneven four-beat

Theme 1 (mm. 1–16), dotted quarter = 69 one incomplete bar of 4 starting on beat 2 fourteen bars in 4 one bar in 4 with a short beat 3 (= quarter note) (last beat is anacrusis to Theme 2) Theme 2 (mm. 16–32), rounded binary A section: seven bars of 4, one bar of 3 (beat 3 silent) B section: three bars of 4, one bar of 4 with long beat 4 (= half note) A′ section: one-beat anacrusis, one bar of 4 with short beat 1 (= quarter note), two bars in 4, one bar in 4 with long beat 3 (= half note) Variation I (mm. 33–48), rounded binary A section: seven bars of 4, one bar of 3 B section: three bars of 4, one bar of 5 A′ section: three bars of 4, one bar of 3 Variation II (mm. 49–64), rounded binary A section: eight bars of 4 B section: four bars of 4 A′ section: four bars of 4 pause Variation III (mm. 65–76), rounded binary, dotted quarter = 50 : A section: 2 bars of 3 followed by 1 bar of 6 : (tactus = quarter note) : B section: 2 bars of 3 followed by 1 bar of 6 A′ section: 2 bars of 3 followed by 1 bar of 6 : pause Variation IV (mm. 77–89), quarter note = 100 thirteen bars of 2 pause Theme 1 reprise (mm. 90–105), dotted quarter = 69 one-beat anacrusis, sixteen bars in 4 pause Coda (mm. 106-end) one-beat anacrusis, three bars of 4, one bar of 2 three bars of 4, with graduate ritardando throughout third bar fermata EXAMPLE 6.13

Conducting Plan for Schumann’s Quartet, Op. 41, No. 2, II.

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pattern that keeps one’s conducting aligned with the sounding meter consequently has eleven eighth-note pulses broken into four groups: 3 + 3 + 2 + 3. Listen to the movement while conducting along with the plan given as Example 6.13. The conducting plan does a good job of representing the displacements that offset the B sections of the second theme and of the first variation because they are both reflected by changes in the conducting pattern. The conducting plan also has the advantage of freeing one from the notation completely, thus reducing the subliminal influence that the notation can have on us and giving us a better chance to recognize antimetrical patterns that might be at work. For example, when conducting along with Example 6.13, one is less inclined to think of Variation II as very dissonant, but if one conducts while following the score, the bar lines and the placement of the music relative to the bar lines constantly reinforce the subliminal dissonance of D12+9 in the mind of the reader. On the other hand, there is a tension created between the sounding meter and the viola line in mm. 54–56 of that same variation (see Anthology, p. 254). When this line was first presented in mm. 22–24 (see Anthology, p. 253), it began with beat 4 of the sounding meter, but in Variation II, it now begins with beat 4 of the notated meter; this is beat 1 of the sounding meter according to the interpretation suggested by both the dissonance log given as Example 6.7 and the conducting plan given as Example 6.13. A conducting plan can only represent one interpretation of the sounding meter at a time, and so is unable to adequately represent such a conflict.

Multiple Interpretations and Metrical Maps Metrical maps present a single interpretation of what the sounding meter is at all times in a musical work, but this feature may at times be misleading, suggesting a certainty that is not part of the listening experience throughout. In reality, it is very possible that at some points within a musical work there may be passages where multiple interpretations of what is heard as the sounding meter seem equally plausible to the listener. This textbook suggests responding to such conflicts by providing alternative portions of the map, such as the one in Example 6.14, which provides an alternative conducting plan for Variation II of the movement discussed earlier. Listen to the variation again, but this time conduct along with Example 6.14 to hear and feel how well the music fits with this alternative interpretation (be sure to focus on the viola solo in mm. 54–56).

Metrical Maps

Variation II (mm. 49–64), rounded binary A section: five bars of 4, one bar of 5, one bar of 4, one bar 3 B section: four bars of 4 A′ section: four bars of 4 pause EXAMPLE 6.14 Alternative Plan for Schumann’s Op. 41, No. 2, II, Variation II.

Chapter Review 1

A metrical map is a summary of the metrical states in a musical work or movement from beginning to end. There are two kinds of metrical maps: dissonance logs and conducting plans.

2

A metrical map can help a musician see the large-scale organization of metrical states in a work, facilitating observations that would be difficult to make otherwise, such as relationships between dissonances that appear on different pages of the score.

3

A dissonance log is a summary of the metrical dissonances in a musical work or movement from beginning to end.

4

A conducting plan is a metrical map that provides a strategy for conducting the sounding meter in a musical work or movement from beginning to end.

5

The conducting plan is a simpler kind of metrical map, but one that nevertheless can sometimes more efficiently communicate the most important metrical changes to a musician in a visceral way. A conducting plan can only represent those metrical details that change the tactus or how the tactus is grouped.

6

Metrical maps present a single interpretation of what the sounding meter is at all times in a musical work, but it is very possible that at some points there may be passages where multiple interpretations of what is heard as the sounding meter seem equally plausible to the listener. This textbook suggests responding to such conflicts by providing alternative portions of the map.

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Homework Assignment 6.1 Review your analysis of the metrical dissonance in the first 108 measures of Beethoven’s Symphony No. 3 from Homework Assignment 5.3 (and refer to the music in the Anthology, pp. 9–13), then create a metrical map of those measures using a spreadsheet program like Microsoft Excel or on a sheet of graph paper (use those found in the chapter as models).

Homework Assignment 6.2 Listen to Bach’s Fugue in B Major from his Well-Tempered Clavier, Book I while conducting along with the notated meter (see Anthology, pp. 2–3) and pay special attention to the following dissonances: D8+2 (‰), D8+6 (‰), D4+1 (‰), and D4+3 (‰). Then create a metrical map that tracks those three dissonances throughout the fugue using a spreadsheet program like Microsoft Excel or on a sheet of graph paper (use those found in the chapter as models).

Homework Assignment 6.3 The Menuetto from Mozart’s String Quintet No. 3, K. 515 begins in a metrically ambiguous way uncharacteristic of most classical music (see Anthology, pp. 214–216). The first downbeat is not articulated any more strongly than the onebeat anacrusis that precedes it or the music that immediate follows it. An incomplete interpretation is provided here; complete the blanks in the interpretation by analyzing music in a parallel manner to what has already been given and make a copy of it to turn in to your instructor. Then add corresponding annotations in your copy of the score, and listen to a recording of it while you conduct along. anacrusis, mm. 1–2: one asymmetrical hypermeasure in 3 (half-half-dotted half; conduct the anacrusis as a downbeat) m. 3-m. 4, beat 2: one asymmetrical hypermeasure in 2 (dotted half-half ) m. 4, beat 3-m. 6: one asymmetrical hypermeasure in 3 (half-half-dotted half ) m. 7-m. 10, beat 2: one asymmetrical hypermeasure in 4 (3 dotted halves-1 half ) m. 10, beat 3–m. 12: ___________________________________________ m. 13–m. 14, beat 2: ___________________________________________

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m. 14, beat 3–m. 16: ___________________________________________ mm. 17–20: __________________________________________________ mm. 21–22: __________________________________________________ mm. 23–26: __________________________________________________ m. 27–m. 30, beat 2: ___________________________________________ m. 30, beat 3–m. 32: ___________________________________________ m. 33–m. 34, beat 2: ___________________________________________ m. 34, beat 3–m. 36: ___________________________________________ mm. 37–48: __________________________________________________ Make a copy of the conducting plan to turn in once you have completed it, and write two paragraphs to accompany it, one that explains what factors from Chapter 1 support interpreting the first seven quarter notes as three uneven beats, and another that identifies metrical dissonances in the following passages by type, and specifies which voice expresses the antimetrical layer: mm. 23–28, 52–55, 81–85, and 94–96.

Further Reading Krebs, Harald. “Some Extensions of the Concepts of Metrical Consonance and Dissonance.” Journal of Music Theory 31/1 (1987), 99–120. Krebs, Harald. Fantasy Pieces: Metrical Dissonance in the Music of Robert Schumann. Oxford: Oxford University Press, 1999. Mirka, Danuta. Metric Manipulations in Haydn and Mozart: Chamber Music for Strings, 1787–1791. Oxford: Oxford University Press, 2009.

chapter 7

Meter in Music With Text

One kind of music has been conspicuously absent in examples used up to this point: vocal music. Meter in vocal music adds a layer of complexity unknown in instrumental music, because the lyrics in vocal music have a rhythmic quality independent of the music that sets them: not only does each word carry its own pattern of accented and unaccented syllables, but the words of vocal music are most often organized in regular patterns of strong and weak syllables and rhyming lines that form poetic meters independent of the music. Another complication is that the words and the lines that they form carry meaning independent of the music, and thus one must always carefully consider the relationship of the music to the text: composers may use changes in meter or hypermeter or metrical dissonances to portray the general dramatic arc of the text, or to more specifically paint references to time and motion in that text. This chapter will concern itself with meter in vocal music, but also will address meter in programmatic music, as such music is also shaped by literary texts in some of the same ways.

Conventional English and German Text Setting To understand meter in vocal music, one must first understand the conventions of text setting. The most conventional kind of English or German text setting sets each line of text with the beginning of a new phrase or subphrase and aligns the stressed syllables of each word with the regular accents that define the meter. Thus the stressed syllables in the text and the accents of the meter in a conventional

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text setting mutually reinforce one another. Example 7.1 provides a conventional text setting by Hugo Wolf. These four measures set the first two lines of poetry, with each line beginning on a downbeat, and each stressed syllable beginning on a strong beat.

EXAMPLE 7.1

Hugo Wolf, “Verborgenheit,” mm. 3–6.

Example 7.2, on the other hand, provides an unconventional text setting by Schubert. (The notation in this example and in all other examples of vocal music in this book beams the notes in the vocal part according to the notated meter rather than reserving beams for melismas, as was the tradition in the nineteenth century. That is, in the original manuscript of the music in Example 7.2, Schubert had a separate flag for each note in the vocal part because each note carries a new syllable of text, but the notation here beams the notes in pairs according to the time signature, just as it would in an instrumental part.) As Wolf did in Example 7.1, Schubert also sets two lines of poetry, but he does so in just three bars, placing the first stressed syllable of the first line on a downbeat, but beginning the second line with the pick-up to beat 2 of the second measure, thus creating a G6/4 (‰) dissonance with the notated meter (the downbeats of the annotated two-beat pattern are four eighth notes apart, while the downbeats of the notated meter are six eighth notes apart). This hemiola is further reinforced by purely musical factors: the change of direction in m. 11, the sixteenth notes that seem to rush to mark the second beat of m. 12, and the open fifths in the piano accompaniment in those two spots all join the stressed syllables of the text to create the metrical dissonance that gives this unconventional text setting its character. Whenever one finds an unconventional text setting, one should explore the possibility that it may be related to a desire to interpret the meaning

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EXAMPLE 7.2

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Schubert, Die Winterreise, “Rückblick,” mm. 11–13.

of the  words, and that is definitely the case here. The first two stanzas of the text are about running away from a town where the singer is no longer welcome, and the text setting makes sense in that context: the singer is in too much of a hurry to wait for the next downbeat, but feels driven to get to the end of the second line as quickly as possible. Compare the eff ect of Schubert’s setting with the more conventional setting given as Example 7.3; Schubert’s setting certainly has a breathless, hurried quality to it that is lacking in Example 7.3.

EXAMPLE 7.3 Alternative setting of the first two lines from “Rückblick.”

Conventional Italian Text Setting The most conventional kind of Italian text setting also sets each line of text with the beginning of a new phrase or subphrase, but unlike text settings in English and German, the hypermetrical downbeats do not normally align with the first

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stressed syllable of each line. Instead, Italian settings are typically end-accented: they align the final stressed syllable of each line with the hypermetrical downbeats (e.g. each poetic line set to a four-bar hypermeasure is counted hypermetrically as 2-3-4-1). Example 7.4 provides a conventional text setting by Verdi. These measures set the first two lines of text, with each line beginning on the second hyperbeat of a two-bar hypermeasure (shown by brackets in the accompaniment), and the final stressed syllable of each line coming on a hypermetrical downbeat. The harmonic changes in the accompaniment define the hypermeter, while the two-bar groupings in the vocal part define the phrase rhythm. As is the case here, Italian settings will often begin with a measure of vamp (a phrase prefix) that gets the harmonic rhythm started before the voice comes in and sets up the end-accented alignment that is typical of Italian vocal music (see also Example 7.9). Such vamps that shift the hypermetrical downbeats to the end of each group can sometimes be found in instrumental music as well, and not just from Italy, as can be seen in Example 2.4.

EXAMPLE 7.4 Verdi, Rigoletto, “La donna è mobile,” Beginning (After Rothstein 2011).

Text Painting Musical text setting is not just about matching accents in the words to accents in the music, but is often equally concerned with either matching the meaning of that text with musical gestures that seem to depict that meaning, as we just saw in Schubert’s setting of “Rückblick,” or with music that provides a commentary on that text. In Example 7.5, Schubert’s setting of Goethe’s “Wandrers Nachtlied” instead depicts a protagonist in no hurry at all, but rather one who is waiting. He successfully depicts waiting through the obvious device of a fermata, but also by adding text repetition where the original poem had none: Schubert first repeats the penultimate line “Warte nur,” and then repeats the last two lines together with his

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own internal repetition included. The original form of the poem and its translation are given as Example 7.6. Without this text repetition, Schubert’s setting of the individual words would have taken only two measures, as is shown in Example 7.7. With his text repetition, the same two lines are spread over five measures.

EXAMPLE 7.5

Schubert, “Wandrers Nachtlied,” Op. 96, No. 3.

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Über allen Gipfeln Ist Ruh, In allen Wipfeln Spürest du Kaum einen Hauch; Die Vögelein schweigen im Walde. Warte nur, balde Ruhest du auch.

Over all hilltops Is peace, In all the treetops You sense Hardly a breath; The little birds are silent in the wood. Wait, soon You, too, shall rest.

EXAMPLE 7.6

Goethe, “Wandrers Nachtlied II” (“Wanderer’s Night Song II”).

EXAMPLE 7.7

Setting of the Last Two Lines With Text Repetition Removed.

Of equal importance is the fact that Schubert’s manner of text repetition interrupts the regular flow of four-beat units that characterizes the usual sound of the notated meter. Instead of the momentum that accompanies a steady and strongly articulated simple quadruple meter, Schubert prevents such a momentum from becoming established by setting some of the lines with groupings that are six quarter notes long: “Über allen Gipfeln ist Ruh,” “spürest du kaum einen Hauch,” and “Warte nur, warte nur, balde” all sound like measures of 6/4, and all are followed immediately by what sound like measures of 4/4 (see the conducting annotations in Example 7.5). What sounds like an unpredictable alternation of 4/4 and 6/4 helps to create the image of a hiker taking frequent pauses on his trek to admire the scenery, as opposed to a regular slow 4/4 setting which would better communicate a purposeful walk towards a destination. Text painting can involve rhythm and meter in many different ways, and can be used in both vocal music and in instrumental music that is programmatic. As seen in the two Schubert songs already discussed, it can paint references to time and to motion through unconventional text settings and text repetition

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in vocal music, but it can also paint such references through tempo and the stability of the sounding meter in both vocal and instrumental programmatic music. Composers can also color the general mood through the associations that certain rhythms, meters, and tempos have with well-known genres of music. Double-dotted rhythms at a slow tempo had an association with royal processions, and so Handel used them in the overture that begins his Messiah, a reference to Christ as king in the Christian tradition. Duple meters at a brisk walking tempo had an association with military marches when played by ensembles with military percussion instruments, and so Haydn chose one for the second movement of his Symphony No. 100, also known as the “Military” Symphony. And a slow triple meter when it sets a chromatically descending bass line is strongly associated with the topic of lament, and so one finds slow triple meters setting the “Crucifixus” text in Bach’s B-Minor Mass and the text of Dido’s lament, “When I Am Laid in Earth” in Purcell’s opera Dido and Aeneas. Some text painting can be very specific, as when Vivaldi uses specific pitch and rhythmic motives to represent specific birds such as the finch and the turtledove in his Four Seasons, or when Beethoven does the same thing in the second movement of his Symphony No. 6 (the “Pastorale” Symphony), a movement entitled “Scene by a Brook.” In vocal music, a composer might simply choose to set the word “long” with a long note value. In vocal music and in programmatic music, one should always explore the possibility that rhythms and meters were chosen by the composer to paint or comment on the text.

Speech Song and Lyric Song A melisma is a group of notes all sung on the same syllable, and melismatic writing is one way in which composers can extend the musical time spent on a word while maintaining a sense of forward momentum. One fascinating aspect of text setting is the ability to make some singing sound more like speaking. Speech song refers to singing that is mainly syllabic, includes many more repeated notes, and is rhythmically less predictable; it is commonly associated with recitatives in opera and oratorio, and with the verse in songs from classic Broadway musicals. It is used to get through more text more quickly, as when a character in an opera is setting the scene for an aria to follow, and communicating through song those elements of the plot that make its drama rich and detailed. Lyric song, on the other hand, refers to singing that is more melismatic, in which there are fewer repeated notes, and in which the rhythm is regular enough to establish a meter; it is associated with arias in opera and oratorio, and with the chorus in songs from classic Broadway musicals. It is meant to convey a mood rather than a story, and

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to be memorable—these are the tunes that the audience member might whistle on the way home from the performance. Example 7.8 is an example of speech song from the beginning of the finale to Act I from Verdi’s La Traviata, while Example 7.9 is an example of lyric song from later in the same scene.

EXAMPLE 7.8 Verdi, La Traviata, Act I, Scene 6, mm. 1–8.

EXAMPLE 7.9 Verdi, La Traviata, Act I, Scene 6, mm. 27–34.

The difference between speech song and lyric song is not just important to the dramatic structure of music theater genres such as opera and oratorio—it also serves to mark the roles of melody and accompaniment in other contexts. While the voice usually carries the main melody in vocal music, there are occasions

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when the voice serves to accompany a more important melody that is played by an instrumental part. Speech song, when paired with a lyrical instrumental melody, is often heard more as a commentary on that instrumental melody or as an accompaniment to it. A clear example of this relationship can be found in “Morgen” by Richard Strauss (see Anthology, pp. 258–259). The lyrical instrumental melody in mm. 1–15 repeats in mm. 16–30 almost verbatim, defining the one-part AA′ form of the song, while the singer plays a subordinate role musically, presenting the text in speech song as a counterpoint to that melody. This same relationship between an instrumental melody and a vocal part in speech song is everywhere in the music dramas of Wagner, so much so that one could consider it a defining characteristic of their sound. Listen to a recording of “Morgen” while conducting along with the score (see Anthology, pp. 258–259), and pay special attention to the relationship between the voice and the instrumental melody. There is not a single melisma in the vocal part, nor is the rhythm within any given measure ever repeated in the next measure. In fact, there are only two pairs of measures that share the same rhythm in the whole vocal part at all (mm. 18 and 22, and mm. 21 and 32). There are a total of thirty-five repeated notes in the vocal part altogether, compared to only four in the instrumental melody. Compared to the unpredictable rhythms in the vocal part, the rhythms in the instrumental melody are also much more memorable. If one takes the melody in four four-bar phrases with an elision in m. 16, then three of those phrases begin with the same rhythm in each of the first two measures (either two half notes, or four quarter notes). As one can see, instrumental melodies can communicate the sense of lyric song almost as effectively as vocal melodies through the careful manipulation of rhythm and note repetition, even though they cannot employ melismas to do so.

Methods of Textual Emphasis Just as gifted public speakers often drive to a single word within each sentence, and by doing so clarify their intended meaning, so too do gifted composers often abandon conventional text setting to emphasize a particular word within a stanza or couplet to clarify their own interpretation of the text. This emphasis in a text setting is often accomplished by giving the important word a particularly long duration or setting it to a long melisma. Fanny Mendelssohn Hensel’s setting of Heinrich Heine’s “Schwanenlied” (“Song of the Swan”) is a particularly good example of this device. Listen to a recording while following the score and conducting along (see Anthology, pp. 200–203).

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One thing that makes Hensel’s setting such a particularly clear example is the fact that she uses a conventional text setting almost exclusively. This makes three places in the song stand out rhythmically: her settings of the words “treiben” (“play”), “[Fluthen-] grab” (“[watery] grave”), and “verklungen” (“faded away”). The poem is four stanzas in length, and Hensel selected one word from the last line in each of the last three stanzas for special emphasis (for the translation, see Anthology, p. 207). The song is in a simple one-part AA′ form, with the first two stanzas set in the initial A section, and the last two stanzas set in an almost exact repetition of the music used for the first two. The first word to receive emphasis, “treiben” in mm. 20–22, is set to a three-bar melisma and consequently lasts almost six times as long as any other word in the first two stanzas. Because of the song’s one-part form, one would expect some word in the fourth stanza to receive a parallel treatment, and that word is “verklungen” in mm. 47–49. Even if one were to consider the meaning of the poem independently, this would be a natural choice for emphasis. Wolf ’s change to the A section in its varied repetition (the only change) highlights the compound word “Fluthengrab,” or more specifically, the second word of that compound: “grab” (“Grave”) in mm. 35–36. Comparing the setting of this word to the parallel place in the initial setting of the first two stanzas, m. 10, is telling. In m. 10, “seh” is set to a single dotted quarter note, and neither the consistent accompanimental pattern nor the consistent four-bar hypermeter is altered for its setting. In mm. 35–36, “grab” by contrast is set to a dotted half note tied to a quarter note with a fermata, the accompanimental pattern is replaced by a dramatic pianissimo arpeggio that climbs through five octaves, and the hypermeter is stretched to accommodate a five-bar hypermeasure. This is the climax of the poem, a poem about the death of a beautiful creature and the feeling of loss that defines such an event.

Metrical Dissonance as a Means of Textual Emphasis Another way in which composers may emphasize a particular word within a text setting is by placing it at the moment in which a metrical dissonance resolves. Example 7.10 shows this technique at work in Wolf ’s song “Ganymed.” The first two lines of text, “Wie im Morgenglanze du rings mich anglühst” (“As in the morning brilliance your glow surrounds me”), are set by two very syncopated measures followed by a measure of metrical resolution. In m. 2, a D4+1 (‰) is followed in the second half of the measure by D4+2 (‰), while in m. 3, D2+1 (‰) persists throughout, not resolving until the downbeat of m. 4 with the word “anglühst” (“glow”).

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EXAMPLE 7.10 Wolf, “Ganymed,” mm. 1–4.

Example 7.11 shows another instance of this same technique later in the same song. Here the text, “und deine Blumen, dein Gras drängen sich an mein Herz” (“And your flowers, your grass press on my heart”), is set by an anacrusis in m. 26 that leads to syncopations in mm. 27–28 (D4+1 and D4+2 in m. 27, and D4+1 in m. 28), all of which are resolved by the half note on the downbeat of m. 29 setting the word “Herz” (“heart”). In the second half of m. 28, the accents of the words conflict with the accents of the notated meter: the second, unstressed syllable of the word “drängen” (“press”) falls on a strong beat, while the stressed syllable “sich” falls in between beats 3 and 4. At first glance, one might consider this a mistake, but more than likely Wolf ’s intention here is for the singer to sing the words almost as if an accent mark were placed on the word “sich” (a German singer would probably find it unnatural to sing it any other way). Listen to the two examples while conducting along and pay special attention to the accents in the vocal part of m. 28. Note that while in the vocal part the syncopations of mm. 27–28 are resolved in m. 29, syncopations persist unabated in the piano part. One can feel the arrival in m. 29 of a more metrically consonant state, but that state is not consonant across the whole texture; it is only more consonant than what came before it.

EXAMPLE 7.11 Wolf, “Ganymed,” mm. 26–29.

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Metrical Maps of Vocal Music Metrical maps of vocal music are probably most effective when they incorporate the text directly, as can be seen in Example 7.12, a map of Hensel’s “Mayenlied.” A conducting plan for “Mayenlied” is given as Example 7.13. Before studying Example 7.12 in detail, first read the translation of the text (see Anthology, p. 207). Then transfer the conducting plan given as Example 7.13 to your copy of the score (see Anthology, pp. 204–206), and listen to a recording of it while conducting along with the score as indicated by the plan you just transcribed. The conducting in Example 7.13 marks beat 3 of the notated meter as a downbeat in mm. 4, 5, and 6. This interpretation is based both upon the tendency to hear the first accented syllable of the text as a downbeat and the strong arrival of tonic harmony on beat 3 after three beats of a dominant harmony over a tonic pedal. It also is based on the fact that the most important harmonic changes driven by the bass line happen on beat 3 in both mm. 5 and 6, which retrospectively adds support for hearing each of these as a downbeat. The C in the bass that begins on beat 3 of m. 6, however, lasts four beats rather than three, which realigns the sounding meter with the notated meter beginning with the downbeat of m. 8 (this same process happens at the beginning of the second verse in mm. 23–27). The shifted downbeats in this interpretation create an indirect G3/2 (Œ) as the music moves from a two-beat group to a threebeat group in m. 4, and then an indirect G4/3 (Œ) as the music moves from a three-beat group to a four-beat group in m. 6, and then back to a three-beat group in m. 8. These indirect grouping dissonances with a reference pulse of a quarter note are matched by direct grouping dissonances at the sub-tactus level, as is shown in Example 7.12. Each time a dotted eighth followed by a sixteenth in the vocal part appears, it appears against an eighth-note triplet, creating a direct G4/3 (sub), and each time the vocal part divides the beat evenly into two eighths in mm. 5–7 and 16–17, the accompaniment sets against it an eighth-note triplet, creating a direct G3/2 (sub). There are also indirect G3/2 dissonances at the subtactus level that relate to the one example of text painting in this song: as the boy shakes his fine blonde hair, the shaking motion is imitated by a wholesale abandonment of the triplet eighths that so far have defined the accompaniment in favor of a repeated pairs of eighth notes in mm. 13 and 15, after which time, the triplet eighths in the accompaniment immediately resume. The duple divisions represent the back-and-forth motion of the head shaking, and at the same time the abrupt departure from the triplets represents the boy’s abrupt shift from being part of the landscape to being an active figure upon it.

EXAMPLE 7.12

Metrical Map of Hensel’s “Mayenlied.”

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Introduction (m. 1–m. 4, beat 2) three bars in 3, one bar in 2 A section/First Stanza (m. 4, beat 3-m. 23, beat 2) two bars in 3, one bar in 4 fifteen bars in 3, one bar in 2 A′ section/Second Stanza (m. 23, beat 3-end) two bars in 3, one bar in 4 thirteen bars in 3 Coda (mm. 40-end) seven bars of 3 EXAMPLE 7.13

Conducting Plan for Hensel’s “Mayenlied.”

While the conducting plan given as Example 7.13 can give us a visceral appreciation of the largest grouping dissonances in this song, it is not able to represent sub-tactus grouping dissonances. It would therefore be unable to represent all of the metrical dissonances in Hensel’s setting, since G4/3 and G3/2 are each operating at two different levels throughout the song. While a conducting plan is often a good place to start, it just as often takes an additional layer of analysis to find and represent all of the nuances in a work’s metrical structure.

Chapter Review 1

The most conventional kind of English or German text setting sets each line of text with the beginning of a new phrase or subphrase and aligns the accented syllables of each word with the regular accents that define the meter. Thus the stressed syllables of the text and the accents of the meter in a conventional text setting mutually reinforce one another.

2

The most conventional kind of Italian text setting also sets each line of text with the beginning of a new phrase or subphrase, but unlike text settings in English and German, the hypermetrical downbeats do not normally align with the first stressed syllable of each line. Instead, Italian settings are typically end-accented: they align the final stressed syllable

Meter in Music With Text

of each line with the hypermetrical downbeats (e.g. each poetic line set to a four-bar hypermeasure is counted hypermetrically as 2-3-4-1). 3

Whenever one finds an unconventional text setting, one should explore the possibility that it may be related to a desire to interpret the meaning of the words.

4

Text painting can involve rhythm and meter in many different ways, and can be used in both vocal music and in instrumental music that is programmatic. It can paint references to time and to motion through unconventional text settings and text repetition in vocal music, and through tempo and the stability of the sounding meter in both vocal and instrumental programmatic music. Composers can also color the general mood through the associations that certain rhythms, meters, and tempos have with well-known genres of music.

5

A melisma is a group of notes all sung on the same syllable, and melismatic writing is one way in which composers can extend the musical time spent on a word while maintaining a sense of forward momentum.

6

Speech song refers to singing that is mainly syllabic, includes many more repeated notes, and is rhythmically less predictable; it is commonly associated with recitatives in opera and oratorio, and with the verse in songs from classic Broadway musicals. It is used to get through more text more quickly, as when a character in an opera is setting the scene for an aria to follow, and communicating through song those elements of the plot that make its drama rich and detailed.

7

Lyric song, on the other hand, refers to singing that is more melismatic, in which there are fewer repeated notes, and in which the rhythm is regular enough to establish a meter; it is associated with arias in opera and oratorio, and with the chorus in songs from classic Broadway musicals. It is meant to convey a mood rather than a story, and to be memorable—these are the tunes that the audience members might whistle on the way home from the performance.

8

Just as gifted public speakers often drive to a single word within each sentence, and by doing so clarify their intended meaning, so too do gifted composers often abandon conventional text setting to emphasize a particular word within a stanza or couplet to clarify their own interpretation of the text. This emphasis in a text setting is often accomplished by giving the important word a particularly long duration or setting it

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to a long melisma. Another way in which composers may emphasize a particular word within a text setting is by placing it at the moment in which a metrical dissonance resolves. 9

Metrical maps of vocal music are probably most effective when they incorporate the text directly.

Homework Assignment 7.1 Read the translation of Wolf ’s “Verborgenheit” (see Anthology, p. 260); then listen to a recording of it while following the score and conducting along (see Anthology, pp. 261–264). Write a paragraph on a separate sheet of paper in which you enumerate the ways in which “Wonne” in m. 9 and “wonniglich” in m. 26 are emphasized rhythmically. Which of these two words is emphasized by the resolution of a metrical dissonance? Then create a dissonance log for the whole song that tracks D2+1 (‰) and D8+4 (‰). Because the last A section is simply a repetition of the first, your log only needs to include the first 27 measures.

Homework Assignment 7.2 Read the translation of “L’empio, sleale, indegno” from Act I, Scene 6 of Handel’s Giulio Cesare (see Anthology, p. 156); then listen to a recording of it while following the score and conducting along (see Anthology, pp. 153–155). Note that hemiolas embellish most of the important cadences in this aria; many of these are not marked by ties in the upper parts, but instead by a strong harmonic change on beat 3, followed by an equally strong harmonic change on beat 2 of the next measure (e.g. mm. 10–11). Finish the incomplete conducting plan below, being careful to match your interpretation of mm. 16–end with what is already given for mm. 1–15. A Section (mm. 1–68) mm. 1–7: one hypermeasure of 4, one hypermeasure of 3 (Œ. = tactus) mm. 8–11: one hypermeasure of 2 (Œ. = tactus), one hypermeasure of 3 (Œ = tactus) mm. 12–15: one hypermeasure of 4 (Œ. = tactus) (. . . to be continued by the student on a separate sheet of paper)

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Homework Assignment 7.3 Read the translation of Schubert’s “Kennst du das Land?” (see Anthology, p. 243); then listen to a recording of it while following the score and conducting along (see Anthology, pp. 239–242). After listening, make a copy of the first two pages (mm. 1–40) and mark on it all instances of the direct D4+1 (‰) dissonance, as well as a conducting plan that accounts for both the indirect hypermetrical G4/3 (Ó) dissonance and the alternations between four-bar hypermeasures and two-bar hypermeasures in the B section (mm. 19–40). While these latter alternations do not create metrical dissonances, they contribute to the mood of the song. Turn your annotated score in to your instructor.

Homework Assignment 7.4 Read the translation of “Già il sole dal Gange” (see Anthology, p. 238); then listen to a recording of it while following the score and conducting along with the dotted half note in a one-beat pattern (see Anthology, pp. 237–238). Finally, review the discussion of Italian text setting on pp. 113–114, make a copy of the score to turn in, and add annotations that reflect a hypermetrical interpretation in which the dotted half note is the beat. It will be impossible to maintain a strictly duple interpretation from start to finish and at the same time respect the accents created by the text setting, so use those accents to find hypermetrical downbeats and let your analysis freely change between hypermeters based on the flow of the music and text. To start, mark the first eight bars as four two-beat hypermeasures. Note that each line of the translation is actually two lines of the original poetry. The final stressed syllable of each line in the first verse is marked in bold here: Già il sole dal Ganges—più chiaro sfavilla—e terge ogni stilla—dell’alba che piange. Finally, check your work by conducting along with a recording one more time.

Homework Assignment 7.5 Find a short song (one or two pages of music) that has at least two different types of metrical dissonances, and at least one example of text painting or of textual emphasis achieved through the manipulation of rhythm and meter. (If you have trouble finding one, try looking specifically at nineteenth-century music.) Make a copy of the piece, add annotations in which each dissonance is labeled by type each time it occurs, and write a paragraph explaining the example of text painting

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or textual emphasis that you found. Finally, check your work by conducting along with a recording, then turn in your annotated score and your paragraph to your instructor.

Further Reading Krebs, Harald. Fantasy Pieces: Metrical Dissonance in the Music of Robert Schumann. Oxford: Oxford University Press, 1999. Krebs, Harald. “Hypermeter and Hypermetric Irregularity in the Songs of Josephine Lang.” In Engaging Music: Essays in Music Analysis, ed. Deborah J. Stein, 13–29. New York: Oxford University Press, 2005. Krebs, Harald. “Text-Expressive Functions of Metrical Dissonance in the Songs of Hugo Wolf.” Musicologica Austriaca 26 (2007), 125–136. Krebs, Harald. “The Expressive Role of Rhythm and Meter in Schumann’s Late Lieder.” Gamut 2/1 (2009), 267–298. Krebs, Harald. “Fancy Footwork: Distortions of Poetic Rhythm in Robert Schumann’s Late Songs.” Indiana Theory Review 28/1–2 (2010), 67–68. Lewin, David. Studies in Music With Text. New York: Oxford University Press, 2006. Malin, Yonatan. “Metric Analysis and the Metaphor of Energy: A Way Into Selected Songs by Wolf and Schoenberg.” Music Theory Spectrum 30/1 (2008), 61–87. Malin, Yonatan. Songs in Motion: Rhythm and Meter in the German Lied. Oxford: Oxford University Press, 2010. Pau, Andrew. “‘Sous le rythme de la chanson’: Rhythm, Text, and Diegetic Performance in Nineteenth-Century French Opera.” Music Theory Online 21/3 (2015). www. mtosmt.org/issues/mto.15.21.3/mto.15.21.3.pau.html. Rothstein, William N. Phrase Rhythm in Tonal Music. New York: Schirmer, 1989. Rothstein, William N. “Metrical Theory and Verdi’s Midcentury Operas.” Dutch Journal of Music Theory 16/2 (2011), 93–111.

chapter 8

Form and Meter

Musical forms are shaped and articulated by composers in a variety of ways through the manipulation of every aspect of musical structure: melody, harmony, rhythm, meter, tempo, texture, dynamics, articulation, etc. Contrasts between different sections serve to mark the different parts of a musical form for the listener, and this chapter will focus in part on how contrasts between metrical states can serve that function. Various metrical processes can also serve musical forms by shaping their dramatic character in much the same way as crescendos and decrescendos can in the realm of dynamics, and accelerandos and ritardandos can in the realm of tempo. This chapter will explore how metrical processes and the controlled balance of metrical consonances and dissonances can serve to shape musical forms.

Using Metrical Changes to Delineate Formal Boundaries There are many ways of articulating formal boundaries, but moving decisively from one fairly stable metrical consonance to another is one of the more effective in terms of being easily identifiable by the listener. Example 8.1 illustrates such a boundary in the finale to Brahms’ Fourth Symphony. The change from 3/4 to 3/2 in m. 97 is accompanied by a notable change in texture as well, and such combinations are the norm in terms of marking a formal boundary. Though the movement is in a continuous variation form and this solo is based harmonically on the same eight-bar harmonic progression that has been the basis for every eight-bar segment of the movement since the first measure, the movement is also divided into the larger sections of a sonata form, and the melody in the flute part that begins in

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m. 97 sounds like it could be the second theme. It sounds more like the beginning of a second theme than any of the eight-bar segments that came before it largely because of the change in meter, and the simultaneous change in texture.

EXAMPLE 8.1

Brahms, Symphony No. 4, IV, mm. 94–98.

Another way in which a composer can use metrical changes to articulate a formal boundary is to insert a short passage featuring a change of metrical state in between two parts of the form; the passage serves either as a transition or introduction in itself, too short to be considered a part of the form in its own right. Example 8.2 shows such a change at work in Brahms; the hemiola in mm. 202–205 serves to highlight the structural divide between the transition that ends just before the example and the second theme that starts in m. 206.

EXAMPLE 8.2

Brahms, Violin Concerto, I, mm. 198–207.

Metrical Changes and Formal Functions In addition to marking formal boundaries, metrical changes can also help establish the character and formal function of a given section within a musical form, and metrical processes can manipulate the sense of momentum in moving

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from one section to another. Regarding the establishment of character, we have already discussed the difference between thematic and non-thematic material (see pp. 47–48), but it is worth reiterating here that thematic beginnings are strongly associated with hypermetrical regularity, while non-thematic material has no such regular association. One can divide sections that feature thematic material as primarily serving one of two formal functions: expository or recapitulatory. An expository section is one that presents thematic material for the first time, and is usually more stable both harmonically and rhythmically than those sections that surround it. A recapitulatory section is one that brings back thematic material that was presented earlier, though this return is often abbreviated or varied in some way; it is also usually more stable both harmonically and rhythmically than those sections that surround it. One first thinks of a sonata form when hearing these terms, but as formal functions they can be applied to sections of any form: binary, ternary, rondo, concerto, ritornello, fugue, etc. It is important to emphasize the relative nature of stability where formfunctional characteristics are concerned. It would be wrong to expect all expository and recapitulatory sections to be completely stable. While simplicity in the form of complete stability can be beautiful (e.g. “Twinkle, Twinkle, Little Star”), it is unlikely to hold a listener’s attention for long, and so composers writing concert-length works often incorporate some harmonic or metrical dissonances into their thematic material, even while establishing it as primarily stable. For example, the theme of Brahms’ Haydn Variations (see Anthology, pp. 84–85) would on the surface appear to be entirely consonant. The hypermeter at the beginning is asymmetrical, however, which makes it unstable at a deeper level than one could simply feel by conducting the notated meter. The first two phrases are five measures each, and each one is expanded internally to create a lopsided grouping of 3 bars + 2 bars. These two five-bar hypermeasures might be internally unstable, but because they form a parallel period and the second phrase answers the first, the first ten bars in a larger sense feel balanced. Compare the relative stability of the theme’s beginning to the relative instability at the beginning of its first variation in mm. 30–39 (see Anthology, pp. 86–87). The asymmetrical five-bar hypermeasures are retained in this variation, but now a direct sub-tactus level grouping dissonance of G3/2 is added throughout, as triplet eighths are pitted against pairs of eighths in every bar. This relative instability coupled with the multiple references to the theme mark this variation as having a developmental function, which is to be expected in any sectional variation form. A developmental section is one that is primarily non-thematic, relatively unstable, and is based on material presented in an expository section. One first thinks of the development section in a sonata form,

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but sections in a wide variety of other forms can serve a developmental function: the B section in a binary, the episodes in rondos and fugues, the variations in theme and variation sets, etc. While developmental sections are usually nonthematic, this is not always the case. Variation IV of Brahms’ Haydn Variations (see Anthology, pp. 94–97) has a melody that is arguably just as memorable as the original theme, but the theme is clearly based on the same melodic and harmonic structure as the main theme and thus is considered at the same time a development of it. One should not assume that the various sections of a given form always carry the same function from work to work, nor that they always communicate a single function, but by always making an effort to identify the formal function of a given section, one is more likely to discover its role in the dramatic structure of the work, and how its stability or instability plays a part. There are three other formal functions that are primarily non-thematic: sections can also be introductory, transitional, or cadential. An introductory section is one that is primarily non-thematic, and either comes at the very beginning of a movement or work, or is easily identifiable as an introduction to a later section; as, for example, when the same introductory material returns to introduce each major section of a sonata form movement, as it does in the first movement of Beethoven’s Piano Sonata, Op. 13 (“Pathétique”). The first thirtythree measures of Schumann’s String Quartet, Op. 41, No. 1 (see Anthology, p. 245) provides a clear example of how an introductory section distinguishes itself from sections with other functions. Besides the fact that it comes at the beginning, it is decidedly non-thematic compared to the theme that follows it in mm. 34–75. The two-bar subject of this fugue-like introduction has no leaps, nor does it have a very memorable rhythmic profile, especially compared to the theme that begins in m. 34. A transitional section is one that is primarily non-thematic, is unstable, and connects two contrasting themes or tonalities. The transition in mm. 76–116 from the first movement of Schumann’s String Quartet, Op. 41, No. 1 (see Anthology, pp. 246–247) provides a clear example of how transitions typically function in a sonata form—they lead from the primary theme to the second theme, and at the same time, from the primary key to the key of the dominant. What makes Schumann’s transition unstable is not really its metrical dissonance, as it primarily develops the dissonance first presented in mm. 36–37 of the primary theme, but its imitative texture and the way it wanders from key to key: first to Bˉ major in mm. 88–93, then to C major in mm. 94–100, to D minor in mm. 105–108, to G in mm. 109–112, to E minor in mm. 113–116, before giving way to a second theme whose beginning is marked by the return of the homophonic texture (the second theme actually begins in E minor and

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passes through D minor before settling into C major). This transition is also a good example of how sections can often have the characteristics of more than one function, as it begins by developing the primary theme and is thus also serving a developmental function at that point. A cadential section is one that is primarily non-thematic, is primarily stable, and serves to end a work or a group of smaller formal units (e.g. the parts of a sonata form exposition or recapitulation). The closing in mm. 137–148 of Schumann’s String Quartet, Op. 41, No. 1 (see Anthology, p. 247) provides a clear example of the cadential function closing sections typically have in a sonata form—they serve to both confirm the secondary key and to provide a definitive ending to the movement’s first large section, the sonata form’s exposition. Like the transition discussed already, it also shares the characteristics of more than one function: it begins by developing the opening four bars of the primary theme, and in that way also serves a developmental function at that point. It is, however, undeniably more metrically stable than what preceded it, and in fact is the most stable part of the exposition so far, as one can partly see by consulting the metrical map given as Example 6.1 (see p. 94). Example 6.1 makes it clear that only one span of twelve measures is less metrically dissonant than this twelve-bar closing: mm. 105–116, the end of the transition. One must bear in mind that Example 6.1 only shows metrical dissonance, however, and by consulting the score, one can confirm that the end of the transition is very unstable tonally, while the closing is very stable. Composers can create instability and the concomitant drive toward resolution in many different ways, and metrical dissonance is only one of them. This section of the chapter has identified six types of formal functions: expository, recapitulatory, introductory, developmental, transitional, and cadential. It has also identified the expository, recapitulatory, and cadential functions as being relatively stable (or at least starting that way) compared to the developmental and transitional functions, which are relatively unstable (it makes no claims regarding the stability of introductory sections). Finally, it has pointed out that, just because a section is metrically more consonant or dissonant than other sections, that does not mean that it is necessarily more stable or unstable overall, because tonal stability is just as important to our sense of overall stability as metrical stability, and other factors contribute to that sense as well (e.g. an imitative texture might be heard as more unstable than a homophonic one, since there is not a clear hierarchy among the voices). The rest of the chapter will explore the role that metrical stability and instability plays in the formal structures of two specific works: in the first movement of Schumann’s String Quartet, Op. 41, No. 1, and throughout Brahms’ Variations on a Theme by Haydn.

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Case Study 1: Schumann’s Quartet, Op. 41, No. 1, First Movement This textbook has already taken a fairly close look at this movement in Chapter 6, and has added further observations regarding its formal structure in the previous section, so what follows will consist of providing what further analysis is necessary to provide a comprehensive view of the movement’s form and how metrical stability and instability play a role in shaping it. First, Example 8.3 completes the picture provided in Examples 6.1–6.3 by adding a metrical map of the movement’s introduction. This map tracks one element that has nothing to do with meter: the tonal distance (abbreviated “T. Dist.”). The tonal distance is the distance between the primary key and a secondary key, as measured by the number of accidentals that are different between their key signatures, with the raised leading tone counted as part of the key signature in minor keys for counting purposes (so relative keys differ consistently by one accidental, and parallel keys differ consistently by two). The Appendix in the back of the book provides a matrix that calculates the distance between any two keys in the tonal system. The tonal distance does not account for all accidentals, but only those that are the product of tonicization or modulation (i.e. chromatic notes that are not diatonic to the current key will be ignored). The addition of this element affords the analyst a slightly more comprehensive view of stability, though it is by no means complete. Turning to the score (see Anthology, pp. 245–251), one can quickly see why many would consider this introduction to be a separate movement and take the sonata form starting in m. 34 to be the beginning of a second movement, though we will continue to consider it all part of the same movement here. Those in favor of taking it as a separate movement would point to the fact that it begins and ends in A minor (if you take the double bar at the end of m. 29 to be its end), and the whole quartet ends in A minor. It is also in a different meter, and has a very different character. Those in favor of interpreting it as the first part of the first movement would point to the fact that there is no end bar in m. 29, that there is a long tradition of slow introductions to fast sonata form movements, and that changing the time signature and character along with the tempo is part of that tradition (in fact, the first movement of Haydn’s Symphony No. 103, found on pp. 182–199 of the Anthology, is a good example of this tradition). Nevertheless, it is not common for any single movement to begin in one key and end in another, and so the debate is perhaps irresolvable. Schumann’s introduction is a fugato. It begins in a stable metrical state, but in the first beats of mm. 4–6, the first violin introduces an antimetrical pulse suggesting D4+2, and in the second beats of mm. 5–6, the second violin joins

EXAMPLE 8.3

Schumann, String Quartet Op. 41, No. 1, Introduction.

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in (throughout this discussion, the sixteenth is the reference pulse for all Dx+y labels). These are not simply passing, as D4+2 is present in every bar from m. 5 until m. 11, when D2+1 (the tight relative of D4+2) first appears in the viola and second violin (one can hear the syncopations in m. 11 as a diminution of those in mm. 5–6). In mm. 12–19, the music reverts to a consonant metrical state while it tonicizes D minor, but in the second half of m. 20, D4+3 in the first violin and viola is accompanied by D4+1 in the second violin, and D4+2 in the cello, all of which seem to coalesce into the tight relative they all have in common, D2+1 in mm. 21–22, and then pull back through augmentation to D4+2 in mm. 22–23. The subject of the fugue returns in m. 24 played by the second violin at its original pitch level (albeit down an octave), and with it comes a noticeable decrease in the amount of metrical dissonance. This restatement is accompanied by cadential gestures in the outer parts that create a full measure of overlap between a phrase that spans mm. 17–25 and a phrase that spans mm. 24–29. The cadence on the downbeat of m. 25 essentially marks the end of the introduction, but in the brief codetta that follows, D4+2 returns to end it, with the ties first starting in the cello part of m. 27, but then spreading to the first violin in m. 28, and finally articulated by all four voices in m. 29. Mm. 30–33 serve as a brief transition between the introduction and the beginning of the exposition, and are metrically consonant. While the fugato in mm. 1–29 serves an introductory function for the movement as a whole, it can also be seen as having a self-contained form of its own. The first six measures serve as the fugato’s exposition, their expository function clearly marked by the staggered entrances of the subject in all the voices. This expository function matches up well with our expectations, even though we identified this subject as non-thematic relative to the primary theme in m. 34. In the context of the first twenty-nine measures, it is clearly treated as a subject and is thus heard as thematic in that context. As we would expect, it begins in a very metrically stable way, though metrical dissonances begin to creep in after it reaches its halfway mark. What follows in mm. 6–25 functions as a fugal episode with a restatement elided to its ending in mm. 24–25. As we would expect, the episode serves a developmental function and is less stable, at first both metrically and tonally (tonally, because the subject starts to be transposed to other pitch levels), but in the middle, it is only the tonicization of D minor in mm. 13–19 that marks it as unstable. As the episode nears the restatement in mm. 24–25, the metrical dissonances return. The restatement has a recapitulatory function, and though it happens very quickly, the metrical and tonal stability of the opening returns with the opening material.

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By comparing this interpretation of the introduction to the analysis of the exposition that follows it, one can see a number of similarities that are probably masked by the number of obvious differences when listening. A new map of the exposition is provided as Example 8.4, identical to Example 6.1 save for the fact that it now also tracks the tonal distance from the primary key. The comparison first reveals that both begin with an expository section (the fugal exposition and the P Theme, respectively) and with a metrical consonance that gives way to a few metrical dissonances before moving to the next section. One should note that the most active dissonance in the P Theme, D12+3, is related by augmentation (a loose relative) to an active dissonance in the fugal exposition, D8+2, though the tempo difference between their executions is such that, while they are certain to be heard as related, they are unlikely to be heard as related by augmentation. The following section in each case (the episode of the fugato and the transition of the sonata exposition) is less stable through a combination of tonal instability (reflected by the amount and variety of tonal distances in the maps) and metrical instability. In each case, the second section begins with very little metrical dissonance but overall contains a greater variety of metrical dissonances and tonal centers than the opening section. In both cases, there are measures that tonicize secondary keys but are metrically consonant, as well as measures that are stable both in terms of tonal distance and meter but might still be heard as less stable because of their imitative texture. Finally, the dissonance D6+3 expressed by all four voices in mm. 92–94 is related through augmentation to the D2+1 and D4+2 dissonances in all four voices of mm. 21–23, and so there is one final way in which the structure of the introduction foreshadows the structure of the sonata form exposition that follows it. The rest of the introduction has little in common with the second theme or the closing of the sonata form, but we should still address the larger question of how the balance between stability and instability in both meter and tonality shapes the rest of the form. The second theme defies convention in that it is unstable from its very beginning, though its beginning is easily distinguishable from part of the transition for several reasons: the texture changes from imitative to homophonic, the marking dolce appears (a very conventional way of marking themes starting in the classical era), the rhythm changes to something very similar to the rhythm of the primary theme (compare the first, third, and fourth bars of each), and the dynamic shifts suddenly to piano. Though all of these factors mark it as a beginning, its thematic character, and thus its expository function, depends upon the variety displayed by the melody in the violin I part compared to what immediately precedes it. The instability of the second theme is far beyond what one might expect of a classical sonata form, in that

EXAMPLE 8.4

Schumann, String Quartet Op. 41, No. 1, Exposition.

EXAMPLE 8.5

Schumann, String Quartet Op. 41, No. 1, Development.

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it starts in E minor and moves through D minor in its first eight bars before settling in C major, the key of the dominant, where one would have expected it to start. The end of the second theme (see the first and second endings to the exposition) is arguably the most dissonant part of the whole exposition because of the shift in the sounding meter that offsets downbeats by a dotted quarter note (represented as “2→1” in the map). This forceful displacement seems to mark the downbeat of m. 137, its point of resolution, as the climax of the entire exposition. Though the closing might appear on the map to be more unstable that one would expect, if one ignores the consistent tonal distance reflecting the key of the dominant, then it is actually more consonant than the primary theme, and far more consonant than the secondary theme. The development and recapitulation of the movement have already been discussed in detail in Chapter 6 (see pp. 97–100), but have not yet been discussed in relationship to formal function. The metrical map of the development given as Example 6.2 only includes metrical dissonances, and so it cannot hope to do justice to the total sense of instability in that section. Simply adding the tonal distances traversed to the map as was done in Example 8.3 is not as effective when applied to the development, as there are at least three places where the tonal center is uncertain for two measures: this lack of certainty is the direct result of the diminished seventh chords in mm. 165–166, 185–186, and 193–194. Because any given diminished seventh chord can resolve to four different tonics, it is impossible to identify for certain the keys to which these chords belong, and thus to chart their tonal distance from the primary key. Example 8.5 is a metrical map of the development that provides a solution to this problem by tracking another aspect of tonal instability: the presence of harmonically presented tritones. Major-minor seventh chords contain a single tritone, while fully diminished seventh chords provide two, and so one can get a rough sense of how harmonically dissonant each measure of the development is even when the tonal distance from the primary key is uncertain (in these three places, the map includes a question mark in place of a tonal distance). As the example illustrates, the development is more tonally and harmonically dissonant than it is metrically dissonant, but what is perhaps more interesting is how Schumann balances the metrical dissonance with the tonal and harmonic dissonance. In those places where the tonal distance to the primary key is relatively close, such as in mm. 187–190, 223–226, and 245–252, he uses more metrically dissonant material to maintain the dramatic tension. Conversely, when he uses more metrically consonant material, such as in mm. 199–214 or 229–244, he keeps his tonal distance from the primary key to maintain the tension. The recapitulation is so similar to the exposition that a new map of it will not be offered, but instead only a few general observations regarding function and stability.

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The first is that mm. 253–310 are an exact duplicate of mm. 34–91. It is only in m. 311 that Schumann alters the material so that it circles back around to the primary key instead of moving to the dominant. This alteration is what any student should expect of sonata form, and what follows is a wholesale transposition of the rest of the exposition up a fourth so that it can end in the tonic key. This transposition affects not only the material from the exposition that was in the key of the dominant, but also every other key that was tonicized along the way, and so the tonal tension in the transition and at the beginning of the second theme is retained in the recapitulation though the actual distances change. The only significant change in terms of stability is that the closing ends in the primary key and Schumann adds a brief codetta to it with no metrical dissonances at all, so that the last twelve bars are relieved of virtually all tonal, harmonic, and metrical tension.

Case Study 2: Brahms, Variations on a Theme by Haydn In Schumann’s First String Quartet in the previous section, we saw formal functions operating at multiple levels: even while the introduction served an introductory function for the movement as a whole, it was broken down into expository, developmental, and recapitulatory functions when analyzed as a fugato in and of itself. One can see formal functions operating at multiple levels in Brahms’ Variations on a Theme by Haydn as well. The theme is in a rounded binary form (see Anthology, pp. 84–85), with an expository A section in mm. 1–10, a developmental B section in mm. 11–18, and a recapitulatory A’ section in mm. 19–23, the final cadence of which is extended until m. 29, with the extension serving a cadential function. In the form of the entire work, however, the theme as a whole serves an expository function, while each subsequent variation serves a developmental function, until mm. 448–457, when the A section returns only slightly altered to serve a recapitulatory function, after which the cadential extension of the theme returns to serve a cadential function and end the work. In what follows, we will examine how metrical changes and dissonances are used to mark formal boundaries and communicate formal functions. A map of the theme is given as Example 8.6. The first two five-bar phrases of the theme each subdivide into 3+2 and are thus metrically dissonant in the same way that all asymmetrical groupings are, but otherwise the theme is entirely consonant, with only two exceptions. First, an indirect grouping dissonance is created when the five-bar phrases of the A section are replaced by the four-bar phrases of the B section, represented by G20/16 (‰) in the map. Second, there are two displacement dissonances in which the bass articulates the antimetrical layer: D2+1 (‰) in m. 5 at the end of the first phrase, and the same dissonance in m.17 at the end of the B section.

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The metrical dissonance in each case articulates a formal boundary: the D2+1 in m. 5 reinforces the character of the opening antecedent phrase as one that demands a consequent, and the D2+1 in m. 17 strengthens the cadence at the end of the B section through its resolution in m. 18, and in that way reinforces the boundary between the B and A’ sections. While there is a hypermetrical change in m. 27 when the four-bar phrasing leads to a final grouping of three bars, this map will not bother to show it, as the sense of dissonance is washed out by the fact that it occurs at the end.

EXAMPLE 8.6

Brahms, Variations on a Theme by Haydn, Theme.

All of the variations except for the finale retain both the rounded binary structure of the theme and its relative proportions and phrase structure, and so the opposition of an A section with five-bar hypermeasures and a B section with four-bar hypermeasures helps to articulate the rounded binary form in all of them. On the other hand, each variation is metrically more dissonant than the theme, and those dissonances serve to establish the primary formal function of each as developmental when looking at the work as a whole. Compare the map of Variation I, given as Example 8.7, to the score (see Anthology, pp. 86–87). Two sets of measure numbers are provided for this and all subsequent variations: one that gives the measure numbers of the variation, and another that shows the corresponding measure numbers in the theme. In Variation I, there is a constant G3/2 subtactus-level dissonance as pairs of eighths are pitted against eighth-note triplets throughout. This G3/2 is only absent in a few places within the first variation: in mm. 44–46 and in its last three measures. In mm. 44–45, the feeling of metrical dissonance doesn’t decrease because a new dissonance is added in these measures: D4+2 (‰) in the flutes, trumpets, and lower strings. This is an augmentation of the D2+1 that embellishes the end of the B section in the theme, and one could view it as a specific reference to that earlier dissonance. It actually plays a similar role: together with the temporary cessation of G3/2, it helps to articulate the formal boundary between the two phrases of the B section. In the last three measures, the absence of any metrical dissonance adds to the ending’s stability and thus underscores its cadential function within the variation itself.

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EXAMPLE 8.7

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Brahms, Variations on a Theme by Haydn, Variation I.

Compare the map of Variation II, given as Example 8.8, to the score (see Anthology, pp. 88–90). Variation II, while less dissonant than Variation I, is still more dissonant than the theme, incorporating the same G3/2 subtactus-level dissonance from Variation I at the end of both phrases in its A section, and in m. 80 of its A’ section as well. In the last phrase of its B section (mm. 73–74), it also includes D4+2 in the parallel place to where it was found in Variation I. While dissonances are not as pervasive in Variation II as they were in Variation I, the placement of each one serves to reinforce a phrase boundary.

EXAMPLE 8.8

Brahms, Variations on a Theme by Haydn, Variation II.

Compare the map of Variation III, given as Example 8.9, to the score (see Anthology, pp. 91–94). This is the first variation in which Brahms writes varied repeats of the A section and of the material after the initial A section as well, and so there are now two different measure numbers in the variation that correspond to each measure of the theme (e.g. mm. 88 and 98 both correspond to m. 1 of the theme). To avoid an unnecessarily complicated map, Example 8.9 will include a dissonance even if it doesn’t occur in both iterations of the given section. Variation III is the first to challenge the 5+5 division of the A section by not only slurring its first ten bars together in the oboe, bassoon, cello, and contrabass parts, but also by sewing up the usual seam between the cadence in

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the fifth bar and the start of the second phrase in the sixth in three other ways: with a sequential repetition in the melody that binds the fifth and sixth bars (mm. 92–93) together, by an uninterrupted descent through a scale in the bass at the same time, and by a crescendo marking that begins in m. 92 and ends in m. 93. Nevertheless, one can still see the seam in the viola part’s 5+5 slurring and hear it clearly in the horn part that seems to cadence on the downbeat of the fifth bar. The varied repetition of this variation’s first ten bars in mm. 98–107 retains this same conflict, with the added voices doing nothing to resolve it.

EXAMPLE 8.9

Brahms, Variations on a Theme by Haydn, Variation III.

By weakening the division of this variation’s first ten bars into 5+5, Brahms allows another possible grouping to emerge: 3+3+4. This latter grouping is supported by sequential repetitions in the melody (mm. 89–90, 92–93, and 94–95), and by a registral and dynamic accent in m. 94 (one has to assume that the low F in the contrabass is really meant to go down a whole step based on the doubling of that line in the viola and cello parts, and that the Eˉ an octave higher in m. 94 is just an octave displacement necessitated by the range of the instrument). It is also supported by the presence of D2+1 in mm. 88, 91, and 94) which, though in an accompanimental role, still marks each of those measures as a beginning. In the map, the 3+3+4 grouping is represented by the indirect G20/12, with the “20” referring to the expectation that the first phrase will be five bars long (i.e. twenty eighth notes), but there are two three-bar groupings (i.e. twelve eighth notes) articulated instead. To test this interpretation’s validity, listen to a recording of mm. 88–107, and conduct along with the hypermetrical 3+3+4 groupings (i.e. with the tactus as the half note, conduct two three patterns and one four pattern for mm. 88–97, then do the same thing again for mm. 98–107). The B and A’ sections in mm. 108–126 and their varied repetition in mm. 127–143 echo some of the same dissonances heard earlier. D2+1 is used once

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again to strengthen the cadence ending the B section each time (see mm. 114 and 133), and its use at the beginning of the A section returns when the A’ section returns each time (see mm. 116–117 and 135–136). D4+2 appears in the bassoon for three bars at the end of the first time through the cadential extension, but only for two bars the second time through, a fitting alteration that supports its cadential function as the true end of the variation. One more change highlights the developmental role of the B section in the rounded binary form of the variation itself: Brahms rewrites the two four-bar phrases of the B section to create a sentence. The sentence as a pattern of phrase organization typically consists of a two-bar basic idea, a repetition of that idea, followed by a fragmentation where a one-bar motive from the basic idea is then itself repeated, and ending with a two-bar cadential idea, resulting in the proportions 2+2+1+1+2. In this case, the horn motive in m. 108 answered by the clarinet in m. 109 constitutes the two-bar basic idea; the horn motive in m. 110 answered by the first violin in m. 111 constitutes the repetition of the basic idea; and the oboe in mm. 111–114, with its 1+1+2 slurring, constitutes the remainder of the sentence structure. Of all the phrase organization patterns, the sentence with fragmentation is the one that most clearly communicates an internal development since the fragmentation in its second half clearly develops the basic idea. Thus, mm. 112–113 carry a developmental function across three levels of grouping structure: they are developmental within the sentence structure that constitutes the B section, they are developmental as part of the B section within the rounded binary of this variation, and they are developmental as part of the variation within this theme and variations set. There is a natural sense of acceleration within a sentence when the fragmentation occurs, since the internal repetitions make it clear that two-bar groups are replaced by one-bar groups. This can be felt by listening and conducting along with a hypermeter that reflects this grouping: conduct two two-bar hypermeasures for the basic idea and its repetition, followed by two one-bar hypermeasures for the fragmentation, and ending with one two-bar hypermeasure for the cadential idea. A musician can use this same strategy to confirm viscerally the presence of a sentence with fragmentation whenever one is suspected, but in a context that makes identifying it through score study difficult. Compare the map of Variation IV, given as Example 8.10, to the score (see Anthology, pp. 94–97). In this variation, Brahms writes varied repeats of the A section and of the material after the initial A section as well, and so once again two different measures in the score will correspond to each measure number given in Example 8.10. This is the first variation in triple meter, and a new dissonance that wasn’t possible in duple appears for the first time and shapes this

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variation: D6+4 (G30/24 in this variation is really just an augmentation of the G20/16 dissonance of the theme and doesn’t sound new). Though it doesn’t show up in our metrical map, the B section also makes use of a sentence structure to reinforce its developmental function. Note that some dissonances in the map are in parentheses because they only occur in a section’s first pass, but not in its second, or vice versa: D6+4 is in the first pass through the A section, but not the second (i.e. in mm. 154–155 but not in mm. 164–165), and it is in the second pass through A’, but not the first (i.e. in m. 195 but not in m. 175). Many instances of D2+1 in this variation are articulated through the use of bowing indications or slurs (e.g. mm. 154–155 and 164–165), which is another first in the expression of metrical dissonances in this set of variations.

EXAMPLE 8.10

Brahms, Variations on a Theme by Haydn, Variation IV.

Compare the map of Variation V, given as Example 8.11, to the score (see Anthology, pp. 97–100). In this variation, Brahms once again writes varied repeats of the A section and of the B and A’ sections, though in this case, the rewrite of the A section introduces so many new metrical dissonances that it has been mapped separately. This variation employs the greatest variety of dissonances so far, and they contradict the general expectations one has of the relative stability associated with its formal functions. While the general level of instability is very much in keeping with the variation’s placement within the work as a whole, the amount of dissonance at the beginning undercuts the expository function of the initial A section, and the amount at the end undercuts the cadential function of that music within the variation’s rounded binary form. The map of the A section’s varied repetition doesn’t really do justice to the relatively high degree of instability: in mm. 220–221 (corresponding to mm. 5–6 of the theme), G3/2 is now an indirect dissonance, with all three voices having abandoned the notated meter of 6/8, and the strings and the winds articulate measures of 3/4 that are offset by one eighth note. While the number of voices in general can give us a very rough

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measure of a dissonance’s strength, it arguably fails to do so here; although there are only three independent voices, all are doubled at the octave, and so it almost sounds like there are six out of a possible six voices in a dissonant relationship with the variation’s primary metrical consonance of 6/8. The map also does not clearly reflect the extent to which multiple interpretations of this variation are not only possible, but probable. The opening neighbor-note motive, Bˉ-A-Bˉ often appears in stretto with itself at some level of transposition, as in mm. 211, 226, 230–232, etc. The interpretation mapped in Example 8.11 takes the motive when starting on the second eighth of the measure as a kind of lead-in to the second beat, and is thus not metrically dissonant, but one can easily see how another musician might interpret the beginning of the motive as carrying an accent based on its association with the downbeat forged in the very first measure and reinforced by the bass line in many other places. If the motive carries an accent on its first note regardless of its placement, then this variation includes many more instances of D3+1 than those included in Example 8.11. Compare the map of Variation VI, given as Example 8.12, to the score (see Anthology, pp. 101–104). This variation seems much more straightforward than the previous two, and seems to conform to expectations of stability based on formal function as well. The expository function of the A section is expressed by music

EXAMPLE 8.11

Brahms, Variations on a Theme by Haydn, Variation V.

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that is metrically almost as stable as the A section of the theme. The developmental function of the B section is expressed by metrically dissonant music throughout. The recapitulatory function of the A’ section is reflected by music that is only metrically dissonant in its first measure, but is otherwise completely stable. The metrical state at the ending of this variation is the only way in which Brahms defies convention in that it is almost as unstable as the B section even though its function is clearly cadential, and one would therefore expect more stability. In this way, it has more in common with Variation V and with Variation VII, both of which have very metrically dissonant endings, than it does with the theme, which does not.

EXAMPLE 8.12

Brahms, Variations on a Theme by Haydn, Variation VI.

Compare the map of Variation VII, given as Example 8.13, to the score (see Anthology, pp. 105–106). While this variation sounds very pastoral on the surface, that sound belies a texture filled with metrical dissonance, especially at the end. Once again, Brahms’ use of metrical dissonance reinforces three of the four formal functions by matching conventionally stable functions with stable metrical contexts. The one exception is the cadential function at the end, but in this way, Brahms actually creates another link between this variation and the two that came before.

EXAMPLE 8.13

Brahms, Variations on a Theme by Haydn, Variation VII.

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This map does not capture the way in which the sounding downbeats shift in m. 314 (corresponding to m. 22 of the theme). Only the bass line reinforces the hypermetrical division between the A’ section and the cadential extension (i.e. between the final two hypermeasures corresponding to mm. 23–26 and 27–29 of the theme, respectively). Example 8.14 gives a conducting plan for the variation; take a moment and transfer the notes from the plan to your score, then listen to it while conducting along with your annotations. Notice how this conducting plan reflects the notation that Brahms chose: by consistently beaming across the bar line in mm. 314–319, Brahms is essentially pointing out the metrical conflict that the plan is designed to highlight. The conducting plan is intended as an off-the-podium exercise, not meant for performance, and yet a conductor approaching a performance of this work without the kind of insight into metrical conflict that the plan reflects should probably be considered underprepared. The experience of following the conducting plan given as Example 8.14 while listening helps to clarify the meaning of the labels in Example 8.13. The last six iterations of D6+3 are subliminal dissonances that relate the notated downbeats to the sounding downbeats according to the interpretation given in the conducting plan. The G3/2 in the last eight bars begins with the shift of the sounding downbeat in m. 314, also reflected by the beginning of beams that cut across the bar lines. The G6/4 at the end reflects the shift from conducting measures in three to conducting measures in two. This shift corresponds to the point where the beaming across the bar lines stops, but is really articulated by the repetition of the rising half step C˜ to D in the flute, horn I, and violin I parts. Finally, the

A section (mm. 293–302) : ten measures in 2, dotted quarter = tactus (as notated) : B section (mm. 303–310), A’ section (mm. 311-end) : six measures in 2, dotted quarter = tactus (as notated) two measures in 3, quarter note = tactus three measures in 2, dotted quarter = tactus (as notated) one measure in 1, dotted quarter = tactus five measures in 3, quarter note = tactus three measure in 2, quarter note = tactus one measure in 1, dotted quarter = tactus : EXAMPLE 8.14

Conducting Plan for Variation VII.

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D2+1 is not measured against the notated meter, but is a dissonance between the sounding simple duple meter established by those rising half steps and the F Horn part that becomes offset by one eighth note against it: specifically, the note on the downbeat of the penultimate measure in that part is actually syncopated against the sounding duple meter, as are its next three notes. Compare the map of Variation VIII, given as Example 8.15, to the score (see Anthology, pp. 107–109); a conducting plan for it was given in Chapter 6 as Example 6.8. Once again Brahms varies his repetition of the A section so much that a separate map of it is included. As in Variations IV and V, two-note slurs articulate the fastest moving dissonant layer, the D2+1. The relative metrical stability of each part within the rounded binary form matches the expectations associated with formal function, with the exception of the ending cadential function; the ending is once again more dissonant than music serving a cadential function typically is.

EXAMPLE 8.15

Brahms, Variations on a Theme by Haydn, Variation VIII.

Compare the map of the Finale, given as Example 8.16, to the score (see Anthology, pp. 110–121). The finale is where Brahms allows his Romanticism to override his commitment to classical-period conventions. While a classicalperiod sectional variations form would most likely end with a rounded binary finale, Brahms writes a continuous variation form instead. The entire finale up to m. 446 is based rather strictly on a five-bar bass ostinato, one that is a combination of the melody and the bass line from the first five bars of the theme (the second and third bars match mm. 2–3 of the melody, while the other bars

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match mm. 1, 4, and 5 of the bass line). The continuous variation form unfolds strictly in five-bar phrases that match the ostinato until m. 446, as is shown in Example 8.16.

EXAMPLE 8.16

Brahms, Variations on a Theme by Haydn, Finale.

A number of questions naturally arise when studying the maps of the theme, the variations, and the finale. First, one might ask, “why not use numbers in each map to reflect how many voices articulate each antimetrical dissonance?” The first draft of these maps did actually include such numbers, but there are many places in the maps where the numbers did not seem to reflect how strongly a dissonance was heard, and because the numbers only reflect how many voices articulate a given dissonance, we should not be surprised by this. As Chapter 4 explained (see pp. 68–69), there are a number of other factors that affect how strongly we hear a given metrical dissonance: how long it lasts, how many unaligned pulses there are between conflicting layers, and how strong the

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changes in harmonic rhythm, melodic grouping, dynamics, articulation, texture, and register are. The number of voices is just one factor, and often not the strongest, so it’s questionable whether including it in the map would be useful, or just misleading. This book proposes that in some cases it is more useful than it is misleading, but also proposes that musicians need not adhere strictly to any one method of constructing metrical maps demonstrated in this book after the course is over. Thoughtful and creative musicians should always tailor analytic methods and tools to suit their own needs, and to suit the music that they are studying. Dissonance logs could be constructed where numbers reflect not just the number of voices articulating each dissonance, but the total number of notes in the score, so that analyses of music like the finale discussed here would produce higher numbers than those given in Examples 8.3–8.5 due to all of the octave doubling. Or dissonance logs could be constructed where the numbers represent more than just the number of voices or number of notes, and attempt to account for other factors as well, such as dynamic accents. For example, the numbers might result from a formula that counts the total number of notes, but then doubles that number if the dynamic is forte or louder. Metrical maps are relatively new (the first influential book to include them was by Harald Krebs, published in 1999), so the student should not assume that there is a standardized way of writing them outside of this textbook—there is not. A second question that arises from the study of Example 8.16: “How does one choose the best label for a metrical dissonance?” This was addressed earlier in Chapter 5 (see pp. 82–83), but is such an important topic that it is worth revisiting here. To begin answering this question, first consider two possible labels for the second measure of the finale (m. 362): D4+2 (‰) and D8+6 (‰). Context is always essential in deciding between labels. The interpretation given as Example 8.16 calls this dissonance D4+2 because of what follows it: the half note in the second violin part of m. 363 clearly marks the ending of the tied note that articulates this dissonance. This context for the dissonance confirms that it is a D4+2 by establishing that the antimetrical layer it begins is moving in pulses that are four eighth notes apart (since both the tied pair of quarters and the half note that follows it are four eighth notes long). D8+6 is not as accurate a label because it doesn’t really capture how fast the antimetrical layer is moving. A better example of a true D8+6 occurs just two measures later in m. 364: now the tied note on beat 4 is tied to a dotted half and thus defines an antimetrical layer with pulses that are eight eighth notes apart. It is a better example, but not completely unambiguous, because the note that follows the dissonance in m. 364 (the C5 in m. 365) is not sustained for eight pulses,

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but only for four. Because there is really only one pulse in the antimetrical D8+6 layer, it is impossible to say how fast it is moving, but the listener will unconsciously compare it to the two pulses from D4+2 in the preceding two measures and hear it as moving at half speed, an observation that the label D8+6 effectively captures. A third question that arises from a comparison of Example 8.16 and the score is this: “What circumstances allow an instance of imitation to create a metrical dissonance?” The very first hypermetrical dissonance logged in Example 8.16, D40+8, is created by the stretto between the first three bars of the ostinato in the bass and the viola part in mm. 2–4. Though m. 364 in the viola is a varied repetition of m. 363 in the bass line, the unvaried repetition of the bass line’s first two measures is enough to make the stretto audible. Compare this case to the use of imitation in mm. 391–394. In these measures, the ostinato in the bass is presented simultaneously with its first three measures in diminution played by the first violins. It is actually the form presented by the violins that is imitated clearly by the flutes in m. 393, and by the bassoons less clearly in m. 392. The interpretation given as Example 8.16 hears the flute entrance in m. 393 as creating a dissonance of D40+16, but does not hear the bassoon entrance in m. 392 as creating a dissonance of D40+8 for several reasons. First, the flutes imitate the first two bars of the Violin I part in mm. 391–392, not just the first bar as in the case of the bassoons, and so there is an overlap with the fourth bar of the fivebar hypermeasure, one that cuts across the 3+2 bar division of the hypermeasure established in the theme itself. Second, while neither the flutes nor the bassoons imitate the pitches in the Violin I part, the flutes do imitate its contour, while the bassoons do not. Third, while the bassoons enter forcefully, their entrance is buried between other forceful entrances in mm. 391 and 393, and registrally buried between the contrabassoon and contrabasses on the one hand and the upper strings and horns on the other, and so their fortissimo dynamic does not actually stand out as an accent, as opposed to the fortissimo entrance of the flutes, which is accompanied by simultaneous fortissimo entrances in the oboes and clarinets, and quickly skips up to the registral high point of the passage. Here and everywhere else, context must be taken into account whenever one decides whether or not an instance of imitation constitutes a hypermetrical accent. A fourth question that arises from the study of Example 8.16 brings us back to the subject of this chapter: “What role does metrical dissonance play in articulating the form of the finale?” The finale is in a continuous variation form based on a five-bar ostinato, which is repeated sixteen times after its initial presentation. In mm. 446–447, there is a link that quickly leads to a recapitulatory expression of the theme’s first ten bars in mm. 448–457, followed by a

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fifteen-bar coda. The formal functions within the finale are therefore arranged as follows: mm. 361–365, expository; mm. 366–447, developmental; mm. 448–457, recapitulatory; and mm. 457–471, cadential. Metrical dissonance in the finale articulates its form in many ways. First, it often (though not always) plays a role in articulating the five-bar hypermeter. Specific dissonances are often either introduced at the beginning of an ostinato repetition or disappear with the ending of one, and sometimes both, as in the case of D8+4 in mm. 381–390, and the subtactus-level G6/4 in mm. 411–420. There is a compositional danger, however, in always beginning and ending a dissonance with the beginning and ending of an ostinato: if all of the groupings are always exactly the same length and there are no musical accents that challenge those groupings, the music might start to sound plodding, like a relentless and unimaginative march to nowhere (William Rothstein, the inspiration for Chapters 2 and 3, has labeled a similar phenomenon as “The Great Nineteenth-Century Rhythm Problem” that he defines as “the danger . . . of too consistent and unvarying a phrase structure”). Brahms often allows metrical dissonance to overlap with the five-bar phrasing and hypermeter that is dictated by the repetition of the ostinato, as in the case of the D4+2 in mm. 362–374, and in the case of the subtactus-level G3/2 in mm. 375–380. These overlaps help to soften the effect of what otherwise might be heard as too many five-bar phrases/hypermeasures. The hypermetrical dissonance also helps to articulate the finale’s formfunctional shifts at the end. The two-bar link in mm. 446–447 is the first break from a five-bar hypermeter since the finale’s beginning 85 bars earlier, and it thus serves to highlight the recapitulatory function of what follows it, starting in m. 448. In a similar way, the shift to a cadential function is marked by the Finale’s first (and only) hypermetrical reinterpretation in m. 457: the measure serves as both the fifth bar of the theme’s consequent phrase, and as the first bar of the theme’s cadential extension (it is essentially a varied repetition of the same hypermetrical reinterpretation in m. 23 of the theme). Finally, the relative balance of dissonance in the finale as a whole helps to communicate its formal functions. There is an immediate increase in the amount of dissonance as the music moves from the conventionally stable expository function to the conventionally unstable developmental function. This increased level of dissonance doesn’t really taper off until m. 426, but from there on the music is less metrically dissonant, matching the expectations of both its final recapitulatory and cadential functions. While there is a brief resurgence of dissonance in the final ten bars, the cadential material in m. 457–462 provides the metrical stability one expects of a section with a cadential function.

Form and Meter

Chapter Review 1

There are many ways of articulating formal boundaries, but moving decisively from one fairly stable metrical consonance to another is one of the more effective in terms of being easily identifiable by the listener. Another way in which a composer can use metrical changes to articulate a formal boundary is to insert a short passage featuring a change of metrical state in between two parts of the form; the passage serves as either a transition or introduction in itself, too short to be considered a part of the form in its own right.

2

In addition to marking formal boundaries, metrical changes can also be used to help establish the character and formal function of a given section within a musical form, and metrical processes can be used to manipulate the sense of momentum in moving from one section to another.

3

There are six types of formal functions: expository, recapitulatory, introductory, developmental, transitional, and cadential. The expository, recapitulatory, and cadential functions are relatively stable (or at least start that way) compared to the developmental and transitional functions, which are relatively unstable (this chapter makes no claims regarding the stability of introductory sections).

4

Just because a section is metrically more consonant or dissonant than other sections, that does not mean that it is necessarily more stable or unstable overall, because tonal stability is just as important to our sense of overall stability as metrical stability, and other factors contribute to that sense as well (e.g. an imitative texture might be heard as more unstable than a homophonic one, since there is not a clear hierarchy among the voices).

5

The tonal distance is the distance between the primary key and a secondary key, as measured by the number of accidentals that are different between their key signatures, with the raised leading tone counted as part of the key signature in minor keys for counting purposes (so relative keys differ consistently by one accidental, and parallel keys differ consistently by two). The tonal distance does not account for all accidentals, but only those that are the product of tonicization or modulation (i.e. chromatic notes that are not diatonic to the current key will be ignored).

157

158

Form and Meter

Homework Assignment 8.1 Listen to the first movement of Haydn’s Symphony No. 103 while following along with the score (see Anthology, pp. 182–199); then create a conducting plan for the Allegro starting with its first note in m. 39 and ending in m. 201. Interpret its first note as a downbeat of a four-beat pattern in which the dotted quarter note = tactus (i.e. the first note after the time signature change on p. 184 should be treated as the sounding downbeat). Label the parts of the sonata form in your conducting plan, then write a paragraph that discusses how metrical changes in the movement affect how one hears the form. Use measure numbers to cite specific locations that either help articulate a formal division, or create a sense of instability that helps to establish the character of a specific section’s formal function. BONUS: Listen to the movement’s introduction a number of times without the score while conducting along in three with the pulse stream articulated by every two notes, starting with the first note played by the bassoons, cellos, and double basses. Then consult the score and identify the dissonance you just conducted by type and write a one-page analysis in which you address the following questions. 1) What creates that dissonance? 2) Is this dissonance direct, indirect, or subliminal, and why? 3) Is that dissonance articulated throughout the entire introduction? If not, where is the notated meter unchallenged by the antimetrical layer? Cite measure numbers.

Homework Assignment 8.2 Listen to mm. 154–283 from the first movement of Beethoven’s Symphony No. 3 while conducting the notated meter along with the score (see Anthology, pp. 16–23); then complete the dissonance log for it given on p. 159. This constitutes roughly the first half of the development section for this sonata form movement. Calculating tonal distance at the end is particularly challenging, as there are a lot of fully diminished seventh chords, but go ahead and interpret the distance as though mm. 248–259 are in A minor, and mm. 260–283 are in E minor, as this is certainly one of the more convincing interpretations. Make a copy of your completed dissonance log, write a paragraph in which you discuss how the use of metrical dissonance in this development either reinforces or contradicts the section’s formal function, and turn both into your instructor. BONUS: analyzing parallel music in a parallel way, provide a dissonance log for the rest of the development section as well.

HOMEWORK ASSIGNMENT 8.2

Map of Beethoven, Symphony No. 3, mm. 154–283.

HOMEWORK ASSIGNMENT 8.3

Map of Mozart, Piano Sonata, K. 333, III, mm. 1–112.

Form and Meter

161

Homework Assignment 8.3 Listen to mm. 1–112 from the third movement of Mozart’s Piano Sonata in Bˉ Major, K. 333 while conducting the notated meter along with the score (see Anthology, pp. 208–210); then complete the dissonance log for it given on p. 160. The movement is in a sonata-rondo form. Note that the dissonance log you complete doesn’t track hypermetrical dissonances, such as the indirect grouping dissonances that are heard as disruptions to the fairly consistent four-bar hypermeter that governs most of the movement (see mm. 29–31, 36–40, 89–90, and 107–111); write a paragraph in which you explain how each of these changes in hypermeter can be understood as the result of either elisions or phrase expansions (cite measure numbers). Make a copy of your completed dissonance log, then write another paragraph in which you discuss how the use of metrical dissonance in these measures articulates formal boundaries, or reinforces formal functions, and submit your paragraphs and your dissonance log to your instructor. BONUS: analyzing parallel music in a parallel way, provide a dissonance log for the rest of the movement as well (you don’t need to create a log for mm. 112–131, as these are a literal repetition of mm. 1–20).

Homework Assignment 8.4 Choose a movement or work not included in the Anthology that moves through at least three different formal functions, and uses at least three different metrical dissonances (if you have trouble finding one, look for movements in sonata form, sonata-rondo, or rounded binary form by Classical or Romantic-period composers). Then create a dissonance log for it using a spreadsheet program like Microsoft Excel (use those found in the chapter as models). Finally, write a paragraph or two in which you explain how the dissonances articulate formal boundaries, and either reinforce or contradict expectations of stability based on formal function (cite measure numbers). Submit your dissonance log, your paragraphs, and a copy of the score with measure numbers added to your instructor.

Further Reading Caplin, William. “Tonal Function and Metrical Accent: A Historical Perspective.” Music Theory Spectrum 5/1 (1983), 1–14. Caplin, William. Classical Form: A Theory of Formal Functions for the Instrumental Music of Haydn, Mozart, and Beethoven. Oxford: Oxford University Press, 1998.

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Cone, Edward T. Musical Form and Musical Performance. New York: Norton Press, 1968. Cone, Edward T. “Musical Form and Musical Performance Reconsidered.” Music Theory Spectrum 7/1 (1985), 149–158. Cook, Nicholas. “The Perception of Large-Scale Tonal Closure.” Music Perception 5/2 (1987), 197–205. Hepokoski, James, and Warren Darcy. Elements of Sonata Theory: Norms, Types, and Deformations in the Late Eighteenth-Century Sonata. New York: Oxford University Press, 2006. Krebs, Harald. Fantasy Pieces: Metrical Dissonance in the Music of Robert Schumann. Oxford: Oxford University Press, 1999. Mirka, Danuta. Metric Manipulations in Haydn and Mozart: Chamber Music for Strings, 1787–1791. Oxford: Oxford University Press, 2009. Smith, Peter. “Brahms and the Shifting Barline: Metric Displacement and Formal Process in the Trios With Wind Instruments.” Brahms Studies 3 (2001), 191–229.

Appendix This appendix provides a matrix that calculates the tonal distance between any two keys, which is measured here by the number of accidentals that are different between their key signatures, with the raised leading tone counted as part of the key signature in minor keys for counting purposes.

C G D A E B Gˉ Dˉ Aˉ Eˉ Bˉ F c g d a e b f˜ c˜ g˜ eˉ bˉ F

C 0 1 2 3 4 5 6 5 4 3 2 1 2 3 2 1 2 3 4 5 6 5 4 3

G 1 0 1 2 3 4 5 6 5 4 3 2 3 2 3 2 1 2 3 4 5 6 5 4

D 2 1 0 1 2 3 4 5 6 5 4 3 4 3 2 3 2 1 2 3 4 5 6 5

A 3 2 1 0 1 2 3 4 5 6 5 4 5 4 3 2 3 2 1 2 3 4 5 6

E 4 3 2 1 0 1 2 3 4 5 6 5 6 5 4 3 2 3 2 1 2 3 4 5

B 5 4 3 2 1 0 1 2 3 4 5 6 5 6 5 4 3 2 3 2 1 2 3 4

Gˉ Dˉ Aˉ Eˉ 6 5 4 3 5 6 5 4 4 5 6 5 3 4 5 6 2 3 4 5 1 2 3 4 0 1 2 3 1 0 1 2 2 1 0 1 3 2 1 0 4 3 2 1 5 4 3 2 4 3 2 1 5 4 3 2 6 5 4 3 5 6 5 4 4 5 6 5 3 4 5 6 2 3 4 5 3 2 3 4 2 3 2 3 1 2 3 2 2 1 2 3 3 2 1 2

Bˉ 2 3 4 5 6 5 4 3 2 1 0 1 2 1 2 3 4 5 6 5 4 3 2 3

F 1 2 3 4 5 6 5 4 3 2 1 0 3 2 1 2 3 4 5 6 5 4 3 2

c 2 3 4 5 6 5 4 3 2 1 2 3 0 3 4 3 5 4 5 4 4 3 4 3

g 3 2 3 4 5 6 5 4 3 2 1 2 3 0 3 4 3 5 4 5 4 4 3 4

d 2 3 2 3 4 5 6 5 4 3 2 1 4 3 0 3 4 3 5 4 5 4 4 3

a 1 2 3 2 3 4 5 6 5 4 3 2 3 4 3 0 3 4 3 5 4 5 4 4

e 2 1 2 3 2 3 4 5 6 5 4 3 4 3 4 3 0 3 4 3 5 4 5 4

b 3 2 1 2 3 2 3 4 5 6 5 4 4 4 3 4 3 0 3 4 3 5 4 5

f˜ 4 3 2 1 2 3 2 3 4 5 6 5 5 4 4 3 4 3 0 3 4 3 5 4

c˜ 5 4 3 2 1 2 3 2 3 4 5 6 4 5 4 4 3 4 3 0 3 4 3 5

g˜ 6 5 4 3 2 1 2 3 2 3 4 5 5 4 5 4 4 3 4 3 0 3 4 3

eˉ 5 6 5 4 3 2 1 2 3 2 3 4 3 5 4 5 4 4 3 4 3 0 3 4

bˉ 4 5 6 5 4 3 2 1 2 3 2 3 4 3 5 4 5 4 4 3 4 3 0 3

f 3 4 5 6 5 4 3 2 1 2 3 2 3 4 3 5 4 5 4 4 3 4 3 0

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Index

accompanimental patterns 59, 73 Afro-Cuban music 57 agogic accent 10–12, 18, 66, 68 Alberti basses 59, 73 antimetrical layer 60–64, 67, 72–73, 78, 82, 89–90, 93, 101, 143, 154 asymmetrical meters 65 augmentation 77–80, 89, 95, 138–139, 144, 148

diminution 77–80, 89, 95, 138, 155 direct metrical dissonances 63–64, 67–68, 73, 78 displacement dissonances 53–54, 57–60, 62–64, 67–68, 71–73, 78–81, 86, 89, 106, 143 dissonance logs 93–101, 106, 109, 124–125, 154 downshifting 83–86, 90 duple meter 5–6, 9, 13, 16–17, 119, 152 Dvořák, Anton 16

Bach, Johann Sebastian 59, 70–72, 77–78, 88, 119 Baroque period 16 beat, definition of 5 Beatles, The 65 Beethoven, Ludwig van 2–4, 10, 21–22, 27–31, 38–39, 44–45, 56, 63, 81–85, 119, 134 binary regularity 9–11, 18, 28 Brahms, Johannes 12, 16, 44, 55, 57, 64–65, 104–106, 131–135, 143–156 broken chords 59, 73 Brubeck, Dave 65

elision 31, 33, 105, 121 expository function 133, 135, 138–139, 143, 148–149, 156, 157

cadential extensions 31, 43–44, 47–49, 143, 147, 151, 156 cadential function 135, 143–144, 147–148, 150, 152, 156, 157 Chopin, Frederic 56 composed-out fermatas 44, 48 composed-out ritardandos 44, 48 compound meter 5–6, 9, 16–17 conducting plans 93, 104–109 continuity 10, 14, 18, 101

Handel, George Frederic 119 harmonic rhythm 10–12, 18, 23, 28, 55, 64, 66, 69, 72–74, 84, 116, 154 Haydn, Franz Joseph 49–42, 44–46, 83–84, 119, 136 hemiola 16, 46, 54–56, 61–62, 73, 78, 114, 132 Hensel, Fanny Mendelssohn 121–122, 124–126 hypermeasure 21, 25–29, 31–33, 43, 46, 55, 84, 93–97, 100, 105–106, 116, 122, 127, 133, 144, 147, 151, 155–156 hypermeter, definition of 21, 32 hypermetrical dissonances 63, 73, 155–156 hypermetrical reinterpretation 31, 33, 156

Davis, Miles 57 developmental function 133–135, 138, 143, 147–148, 150, 156, 157

formal functions 132–135, 142–144, 148–150, 152, 156–157 fragmentation 48, 86–87, 90, 147 fugal exposition 22, 139 Gavotte 16 grouping dissonances 53–57, 61–63, 65, 67, 71–73, 79, 86, 89, 105, 124, 126, 133, 143

174

Index

indirect metrical dissonances 63–68, 73–74, 86, 97, 105, 124, 143, 146, 148 internal repetition 40–42, 44–45, 47–49, 117, 147 introductory function 134–135, 138, 143, 157

Radiohead 65 recapitulatory function 133, 135, 138, 143, 150, 155–157 resultant rhythm 59, 73 Rothstein, William 23, 116, 156

jazz 57 Krebs, Harald 15, 60, 79, 81, 93, 154 Latin American music 57 links 46–48, 96, 155–156 loosening 81, 89, 95 loose relative 80–81, 89, 139 lyric song 119–121, 127 melisma 114, 119–122, 127–128 melodic grouping 10–12, 16, 18, 22–23, 26–28, 54–56, 64, 66, 68–69, 72–74, 101, 105, 154 meter, definition of 4, 17 metrical consonance, definition of 53, 72 metrical dissonance, definition of 53, 72 metrical layer 60–62, 64, 67, 71, 73, 78, 82, 89, 96 metrical maps 93–109, 124–126, 128, 154 metric modulation 86, 90 Mozart, Wolfgang Amadeus 7–8, 25–26, 43–44, 47, 57–58, 87 natural pace 7, 17, 21, 32–33, 61 non-thematic material 37, 47–49, 133 parenthetical insertions 45, 48 phenomenal accents 9–10, 14, 18 phrase, definition of 23, 32–33 phrase expansion 37–49 phrase group 27, 33, 43, 48 phrase prefix 37–39, 46–49, 69, 116 phrase rhythm 21–28, 33, 37, 48, 95, 116 polymeter 14–15, 19 primacy effect 9–10, 18 Prokofiev, Sergei 86 Purcell, Henry 119 quadruple meter 5–6, 9, 13, 17, 28–29, 118

Schubert, Franz 38–40, 114–118 Schumann, Robert 1–2, 13–16, 29, 55, 58–60, 66, 68, 79, 93–104, 106–109, 134–143 simple meter 5–6, 9, 16–17 speech song 119–121, 127 Strauss, Johann II 24 Strauss, Richard 121 stretto 88, 90, 149, 155 subliminal dissonances 66–67, 73–74, 78, 96, 100, 108, 151 subphrase 23, 25, 27–28, 33, 39–43, 47, 113, 115, 126 Sub-Saharan African music 57 subtactus-level dissonances 63, 73, 144–145, 156 symmetrical phrasing 43, 48 syncopation 7, 14, 54, 57–58, 62, 69–72, 123, 138 tactus 6–7, 9, 14, 16–17, 21, 28, 32–33, 46, 54–55, 61, 63, 65, 69, 73, 82, 84–86, 90, 105–106, 109, 146, 151, 156 Tchaikovsky, Pyotr Ilyich 16 tempo modulation 86, 90 text painting 116–119, 124, 127 text setting 113–122, 126–128 thematic material 47–49, 133 tightening 81, 89, 95 tight relative 79–81, 89, 138 tonal distance 136, 139, 142, 157 transitional function 134–135, 157 triple meter 5–6, 9, 16–17, 119, 147 upshifting 83–86, 90 Verdi, Giuseppe 116, 120 verse 119, 124, 127 Vivaldi, Antonio 119 Wolf, Hugo 114, 122–123