Mathematics and contemporary music 9786010400368

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MINISTRY OF EDUCATION AND SCIENCE OF THE REPUBLIC OF KAZAKHSTAN

The Y. Altynsarin Gymnasium №159

Abdyssagin Rakhat-Bi

MATHEMATICS AND CONTEMPORARY MUSIC

Almaty “Kazak Universiteti” 2013

UDС 51:78 BBK 22.1 + 85.31 А 139

Scientific advisor: Duisebayeva Bakytzhamal Askarbekovna, teacher of mathematics

Abdyssagin Rakhat-Bi Tolegenuly A 139 Mathematics and contemporary music. – Almaty: Kazak Universiteti, 2013. – 24 p. ISBN 978–601–04–0036–8 This paper is specifically designed to study the basic principles of mathematics and music correlation from ancient to modern times. The role of mathematics in the formation of compositional technique has been determined. On their basis, the young author created original music compositions in the modern academic mainstream. This study took the 1st place at the VII International Competition of Research Projects “Mathematics and Projecting” in the category “Mathematics and Art” (Moscow, 2013).

В работе изучены основные принципы взаимосвязи математики и музыки от древних до современных исследований. Установлена роль математики в формировании композиторской техники. И на их основе юным автором созданы оригинальные музыкальные произведения в современном академическом направлении. Данное исследование заняло 1-е место на VII Международном конкурсе научных проектов «Математика и проектирование» в номинации «Математика и искусство» (г. Москва 2013 г.).

UDС 51:78 BBK 22.1 + 85.31

ISBN 978–601–04–0036–8

© Abdyssagin Rakhat-Bi, 2013

Mathematics and contemporary music

ABSTRACT The aim of the paper is to study the key principals of mathematics and music correlation and to create musical compositions in the contemporary academic mainstream. The objectives are set to study the principals of mathematics and music correlation, to determine the features of mathematics and contemporary classical music interrelation, to show their practical application in writing my own works and to analyze the results of the study. Summary. The key principals of the relationship of mathematics and music from ancient to modern times have been studied thoroughly. It is assumed that these principals are originally founded and determined by the nature of the sound, its wave’s origin and the psychology of sound perception, which is defined by the anatomical constitution of the human ear. The role of mathematics in the formation of the basic fundamental musical concepts like interval, tuning, etc. which organically penetrate into the musical “tissue”, was determined. Individual characteristics of application of mathematical methods and achievements in the development of the theory of music as well as their importance in the formation of modern composition techniques were investigated. On the basis of my own research, I’m the first composer who independently wrote the works: “Light through fog” for solo flute in a sonoristic manner and “Petrogliff” for two pianos in a dodecaphonic style, in the framework of “musical mathematics” method by Arnold Schoenberg, and conducted their structural analysis. I underscores the efficiency of new information technologies application to the work of a musician-composer. The novelty and independence of this research was supported by the patents №935 of July 5, 2012, and №393 of April 9, 2013, issued by the Ministry of Justice of the Republic of Kazakhstan. The findings of the study are my creative contribution to the modern musical trend, and they might turn out to be advantages in the development of musical science in general. 3

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INTRODUCTION Mathematics and Music ... If in mathematics source components are the numbers, in music they are sounds. What is a sound? And what is its relationship with mathematics? As is well known, sound is elastic waves. Sound settings are wavelength, period and frequency, amplitude of oscillation which are determined empirically or by the decision of mathematical equations. Elastic waves are identified in the brain as sound, although they are only waves in nature. The person can perceive sounds in a wide range: 16 to 20,000 Hz. Thus, the correlation of mathematics with music is originally founded by the nature of sound, its wave’s origin and by the anatomical constitution of the human ear. Some animals, such as bats, dolphins and whales detect objects and search for nutrition with the help of sounds. They send short pulses of high frequency sound and capture the reflection echo which is called echolocation. Infrasound represents sound waves with a frequency below 16 Hz. Ultrasound is above 20 kHz. There are specific physical concepts applied to characterize sound associated with the perception of sound: volume, pitch, timbre. The sound volume is determined by the amplitude of oscillations of “sounding” body and the pitch – by its frequency. It is well known that sounds of the human voice by the pitch are divided into several ranges: bass (80-150 Hz), baritone (110-149 Hz), tenor (130 -520 Hz), Treble (260 – 1050 Hz), a coloratura soprano (to 1400 Hz). An important characteristic of sound is its color, or as musicians say, its timbre. Timbre is a sound quality that gives an individual color to the human voice or sound of the instrument. It depends on the composition of sound. Thanks to the timbre of the voice, we recognize a familiar person, even if we don’t see him, or distinguish musical instruments by their sounds. For example, sounds with one tonality and the same volume speak in different ways on different instruments; one and the same note, though recognized as one from a mathematical point of view, 4

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sounds different. This is the reason why we have a variety of musical instruments and perception of image. Noise is a combination of random sounds of different frequencies. Such noises as quietly rustling leaves, the voice of a stream, bird twittering, and light splash of water are always pleasing to the ear. They soothe people, relieve stress. There are other noises that cause the same harm to a person such as blinding light, poisonous gas, burn. The volume of the noise should not exceed 30-40 dB. Being aware of even such simple mathematical rules is essential to every composer. They define the “corridor” for his work.

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“TIME” SCALE Another important characteristic of the sound is duration: Lunga, Brevis, semibreve, half-note, quarter-note, quaver, semiquaver, demisemiquaver, semi-demisemiquaver, etc. which depend on the usual rules of arithmetic and together with a pause constitute time axis. Semibreve

Duration of notes

half-note quarter-note quaver

semiquaver

Pauses semibreve

half-note

quarter-note

quaver

semiquaver

This relationship can be expressed with mathematical formula: Xn+1=1/2Xn, where n = 0, ± 1, ± 2, ± 3 ..., etc. and Xo = 1 semibreve. “MULTIDIMENSIONALITY” OF STRUCTURE Before the late-romanticism period, i.e. until the end of the 19th century, music existed in two-dimensional space. The coordinates of musical space were sound pitch and metrorhythm which define the stream of music time. In case of introducing other different sound characteristics, such as timbre or sonoristic characteristics, the number of dimensions increases. The basic principles of music can indicate an obvious connection with mathematics. Certain relations are also identifiable in basic concepts of musical language, such as interval, tuning, etc. 6

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A SHORT HISTORY OF THE THEORY OF MUSIC That over centuries attributed to the heart actually exists in brain. L.D. Landau

The studies of the correlation of mathematics and music have been done since ancient times. Of particular note is the contribution of the ancient Greek mathematician Pythagoras (570-490 gg. BC) and his followers to the development of the basics of music theory. The greatest mathematicians have dedicated their investigations to music. They are: Rene Descartes, Gottfried LeibFig.1 niz, Christian Goldbach, Jean-Baptiste le Rond d’Alembert, Leonhard Euler, Daniel Bernoulli. The first work of Rene Descartes – “Compendium Musicae” («Treatise on Music”), the first major work of Leonhard Euler – “Thesis of the sound,” etc. It is widely accepted that Pythagoras found that pleasing to the ear harmonies come into being if the length of sounding strings is like the first four integers: 1:2, 2:3, 3:4. This discovery shocked Pythagoras and inspired his followers to seek for new numerical regularity in nature. However, Pythagoras is more popular as the author of “Pythagoras’ Theorem”, known to every student as the equality of square of the hypotenuse to the sum of squares of cathetuses (Fig. 1). PYTHAGORAS GAMUT However, it was Pythagoras who introduced accord, now called the Pythagoras gamut [1]. Gamut, or scale, is a sequence of sounds of some music system arranged, starting from the basic sound in an ascending or descending order. 7

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Consistent combination of the two sounds is called consonance, inconsistent – dissonance. Octaves, fifths, fourths and thirds, which have appeared historically, traditionally refer to consonance. Tonality is an interrelationship system of scale degrees determined by the supremacy of the main reference tone (the tonic) and dependence of the rest of degrees on it. Interval is a correlation between the two sounds of a certain pitch, or a distance expressed in semi-tones, which are the minimum semantic interval unit. And interval coefficient I21 of two tones is a relation of the oscillation frequency of the upper tone to that of the lower:

The tone pitch (frequency f) of sounding string inversely proportional to its length:

where a is a coefficient of proportionality that depends on the physical properties of the string (thickness, material, etc.). Two sounding strings provide consonance (as noted above) if their length is related as integers consisting triangular number 10 = 1 + 2 + 3 + 4, i.e. as 1:2, 2:3, 3:4. 3:4.

Octave:

Fifth:

Fourth:

– Fifth is a harmonic mean of the length of strings of main tone l1 and octave l2. – Quart is an arithmetic average l1 and l2. – Octave is a product of fifths per quart. – Octave is divided into two unequal consonance intervals – the fifth and quart

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These correlations are easily proved by application of formulas for the mean: – arithmetic average

– geometrical average

– harmonic average

It is worth noting that harmonic average of a, b, c values is arithmetic average of 1/a, 1/b, 1/c reversal values. Any music structure contains the basic musical intervals: octave, fifth, quart, and their various combinations. This is a clear evidence of organic penetration of mathematics into music. It should be noted that the first four numbers “tetractys” played a special role for Pythagoreans. It was believed that one – point, “geometric atom”, is the basis of everything. Two – two points, through which only one straight, one measurement can be drawn. Three – triangle: one plane, two measurements. Four – pyramid, volume, space, and three dimensions. The sum of tetractys numbers equal to ten was considered sacred and represented the entire universe. However, please note that Pythagoreans have no zero. The role of zero and the universe will be discussed below.

Musical pitch is a mathematical expression of a certain system of pitch relations. Pitch applies to the tuning of musical instruments with a fixed pitch of sounds, such as the accordion, piano, organ, etc. Pythagoras pitch of natural major [2]: 9

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Do

Re

Mi

Fa

Sol

La

Si

Do

where the numbers at the bottom indicate the interval coefficients of neighboring levels of gamut; recall that 9/8 is the tone and the 256/243 is the semitone. He determined the development of musical culture, not only in Europe but also in the East for thousands of years to come. Thus the role of mathematics can clearly be seen in the structure of the basic concepts (interval, pitch) of music. However, this pitch contains a problem known as “Pythagoras comma”. The point is that on the scale built according to its formula, integer fifths does not fit the whole number of octaves. “Pythagoras comma” is not only an apparent mathematical paradox. The main thing is that with the Pythagoras system, it is impossible to play in random key without being off key. Centuries later, the problem was solved by organist Andreas Werckmeister. An organist, not a mathematician, Andreas Werckmeister slightly improved the invention of Pythagoras. However, he could not do it without mathematics [3]. Werckmeister created his own, instead of natural scale putting of three mathematical postulates to form the basis of the system: – the relation of same notes frequencies in neighboring octaves should be equal to two; – according to the number of semitones in an octave; – according to the number of semitones in an octave between these frequencies should lie only twelve notes. – all semitones should be equal. In accordance with these postulates Werckmeister divided an octave into twelve absolutely equal semitones, which was called the tempered scale.

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THE MODERN FORMS OF COMPOSITION TECHNIQUES “The music of the first nature is just a pleasure, the music of the second nature evokes (expresses) passion, and the music of the third nature excites our imagination” Abu Nasr Al-Farabi

A modern example of bringing the mathematical apparatus to the music is the approach based on the technique of serial production. Series is a central element of dodecaphonic composition, the center of its intonation resources. If we speak about the essence of the method, it all comes down to the fact that the serial music composition is based on repetition in different versions of the same (elected individually for a given composition) sound tracks, which is a sequence of all the available 12 tones. Dodecaphony is the foundation of the theory of writing music for the twelve equal and independent tones. Arnold Schoenberg is the founder of this “musical mathematics”. The basic principle of the dodecaphony theory is the non-repetition of similar sounds in time until, after the exhaustion of all 12 sounds, into which the octave is divided within the framework of tempered scale. The sequence of the 12 non-repeating sounds forms series. Sounds of series form the harmonies and the set of sounds may compose a melody. The basic law of melody is the prohibition of changes of sound sequences in series as well as the violation of their order. The following types of series are distinguished: retrogrades (series, read from the last to the first sound), inversion (series derived from the original sequence by replacing all of the intervals in their treatment), and the retrograde inversion (the same as the retrograde inversion right up to transposition). The development of formalization of serial techniques is associated with the name of Anton Webern. A major focus of his experiments was to find the formative regularities at a level finer than a series. Webern picked up the series with an internal symmetry. At the same time, he organized large compositions that also had symmetry. While writing their works, these composers and their followers didn’t use mathematical terms. But any mathematician will certainly 11

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notice that, for example, the inversion operator, retrograde, and their combination and the identity operator (the original series) form a group. In general, it appears that the methods and algorithms of algebra can be used to compose music [3]. Serialism is a technique running just behind the series (Schoenberg, Webern and Berg), which was founded by Olivier Messiaen (“Four studies in rhythm”), and developed in the 50s of the 20th century in the works of Pierre Boulez, Karlheinz Stockhausen and Luigi Nono. The founders of the Russian serialism, like all the Soviet avantgarde, were Vyshnegradsky, Roslavetz, Mosolov, Lurie, Abraham and others. In Kazakhstan, the elements of this technique are found in the works of Gaziza Zhubanova, Tiles Kazhgaliev, Aktoty Raimkulova, Bakhtiyar Amanzhol, Vladimir Strigotskiy-Pak, Serik Yerkimbekov and some other composers. This trend in music has been developed by spectral, electronic, noise-music and sonoristic music. For example, when writing works “Petrogliff” for two pianos, I used the principles of dodecaphony. It should be noted that this work has been awarded a patent №935 of July 5, 2012, issued by the Ministry of Justice of the Republic of Kazakhstan. If in mathematics the bases for the action are addition, subtraction, multiplication, and division, then all of music can be described as a sound scale (in the form of gamma), interval (in the form of jump), consonance (in the form of a chord). All numbers in mathematics are meaningless if zero is absent. In music the pause plays the role of zero. I specifically addressed this aspect in my work for two pianos “Petrogliff”. The pause has got a different color. In theory I should have called this work “Petrogliff of the universe”, as it shows my spectral perception of the universe. In this work, I tryed to convey multidimensional facets of space and huge scale of the universe. A simple example is that light overcomes 300,000 km for a second, and from the Sun to the Earth light takes a few minutes. From the one end of Galaxy to the other light travels millions of years. The Galaxy – in the visible part of the universe – is less than a grain of sand. The human’s feeling of sense is limited... Everything that we don’t see 12

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is a void for us. But is it? What we see is just a hundredth of a reality. For example, stars. But we do not see the stars themselves, but only their light. Light reaches us in millions of years. We can only see the state of the stars, in which they were millions of years ago! Petroglyphs are cave paintings that have come down to us and they can tell us a lot about the life and outlook of the tribes who created them years ago. That’s why I called this work “Petrogliff”. Know-how is the use of a pause in the form of a void. Below is the link to audition video on YouTube. “Petrogliff” is performed by pianists Associate Professor of the Kurmangazy Kazakh National Conservatory Arman Tleubergenov and Askar Kulsharipov: http://www.youtube.com/watch?v=RXdui8_8ISM&feature=youtu.be

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SPECTRAL MUSIC The main purpose of spectral music is to study sound in an environment independent of tonality where each sound is the center and basis. Spectralist-composers work with sound as if with light, passing it through a prism. In contrast to the large number of composition techniques emerged after the Second World War, the spectral composition is not an individual author’s idea. It is generally accepted that founders of spectral music are the composers of l’Itineraire, in particular, Gerard Grisey 14

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and Tristan Murail, whose views on the principles of musical composition differ significantly. However, many composers whose works are deemed to belong to the spectral music do not identify their own creative setting with the views of the founders of this trend. The methods of spectral composition allow the use of this music in any genres, any style trends of contemporary music. Spectral music does not become a separate musical style. Allowing for a myriad of stylistic decisions, the spectral composition provides a prospect of its own development, the boundaries of which are not currently possible to establish. Its particular feature is that the spectral world of sound and timbre is not isolated, not living alone, but in its interaction and relationship with the world of the more familiar sounds, and most importantly there is the possibility of moving from one system to another. Spectralist-composers: Group of L’Itineraire (France) – Gerard Grisey, Tristan Murail, Hugues Dufourt, Michael Levinas, Philippe Hurel, Marc-Andre Dalbavie, Kaija Saariaho, Francois Paris, Eric Tanguy. FeedbackStudios (Germany) – Clarence Barlow, Claude Vivier, Rolf Gelhaar, Joseph Fritsch, Mesias Maiguashca, Peter Eotvos. Hyperion (English-Romanian) – Iancu Dumitrescu, Ana-Maria Oram. As an example, let me present Tristan Murail’s score “Treize couleurs du soleil couchant”:

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ELECTRONIC MUSIC The development of new information technologies as a result of scientific discoveries of mathematicians and computer scientists greatly contributed to the development of new areas of music. At the beginning of the 20th century from the moment of using a variety of electronic devices, many composers aimed at examining its suitability for music writing. Advantages, i.e. “attractive” elements of electronic music, are based on the sound of computers that has improved dramatically that a good library of sounds and proper use of various special programs can allow to create almost perfect work, providing 98% similarity with the “live performance”, but, of course, not in all genres, especially in vocal performances. Also, it allows the composer to hear his own work and make adjustments well before the first public performance. The computer makes it possible to use almost any musical instruments of the world and greatly expands the horizons of composer search and fantasy. One of the main reasons for electronic technology popularity or electronic simulation of music is continuous spectrum of sounds, 16

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where the distance between the semitone (microtones) can be divided infinitely. It is worth noting that prior to the 20th century the “building material” of minimal meaningful musical units (motive or phrase) belonged to countable dozens. Tones of chromatic scale, namely they are “building material”, are countable dozens. Now we are working with the multitude of continuum power. – An example of Stockhausen’s electronic Studie II:

– The Creation of a huge number of various professional computer programs for working with sound has greatly facilitated the work of many composers. Compared with the equipment used by Pierre Schiffer, one of the pioneers in this area, there is no doubt that today’s technical base has become incomparably richer and wider, and it allows to make major discoveries and find new territories.

SONORISTIC MUSIC Sonoristics and many other trends of modern music are aimed at studying sound timbre, and for this purpose there have been invented new ways of playing classical instruments. A major breakthrough was made in the woodwind and brass instruments. 17

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In Russia the sonorism trend has been set by elite composers Avet Terteryan, Sofia Gubaidulina, Arvo Part, Alfred Schnittke, Yuri Kasparov and others. The pioneers in this field in Kazakhstan are five young composers, the so-called “The New Mighty Batch”: Sanzhar Baiterekov, Jamila Jazylbekova, Aigerim Seilova, Sergey (Sehyung) Kim, Rakhat-Bi Abdyssagin, by analogy with the “Mighty Batch” in 19th century in Russia (M. Mussorgsky, Rimsky-Korsakov, M. Balakirev, Ts. Kyui, A. Borodin). But I’m quite confident that the origins of the world’s spectral music dates back to the works of Korkyt Ata and his followers, extracted from the sacred instrument Kobyz. This idea is also supported by Maestros Tristan Murail, Hugues Dufourt, Zoltan Almashi, Yuri Kasparov, etc. There are three areas of studying sonoristics and timbre qualities of instruments [4,5,6]. – The search for new dissonances in a situation of total production of atonal music led to a deeper study of microintervalics and multiphonics. – The space of new sonorous effects has extended significantly with the advent of such techniques as multiphonics, tongue-ram, jet whistles, whistle-tones, bisbigliando and others. – The new ways of virtuoso performance have been discovered, such as circular breathing, significant expansion of the range of tools, in particular, due to playing with overtones, playing on cane or mouthpiece, “playing + singing” technique, etc. To play with such a technique I have written a musical work “Light through fog” for solo flute, which has a patent №393 of April 9, 2013, issued by the Ministry of Justice of the Republic of Kazakhstan. The quality of performance of this work on the audio-record is specially noted by Ivan Bushuyev, a famous flutist of the Russian Federation. The basis of dramaturgy lies in the contrast of multiphonics combination of jets and air noise. How can multiphonics be interpreted? In my particular case – it’s double notes. It is a kind of deformation of the ordinary flute sound. Improvising, I can give one of the possible versions. Like the sun, many times reflected in the shards of a broken mirror, a regular flute sound is distorted in our split space and is reflected in its parts. Whether it is really our space distorts normal sound of the flute, or it is all right with space, but the flute is in some extreme condi18

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tions and its response is similar if not to a cry, but, in any case, to the distorted voice – one way or another, at the very beginning there is an intrigue, necessary for any writing. This is followed by strong jets – with great energetic message sounds like a lash, and a low whistle of air as a symbol of reaction to a pain, or as a threat, like a hissing snake. Thus, the play begins with a deformed, split flute sound, strong and slash as a scourge, jets and concealed, like snakes hissing. This range of images served as a classical dualism in an understandable and familiar form of “question – answer”. These images immediately begin to develop, and this is what is not so important. Thus, regardless of the emotional interpretation of techniques, we have to deal with the bright, imaginative, contrasting methods, and they are quite similar to the different characters of the drama. The exhibition is extremely laconic and clear. Further development reveals the exposed images and expands the conflicting relationship between them, which leads to tremoling (i.e. vibrating like a train hurtling at top speed and closer to the one to fall apart on the go!) multifonics in growing up to “fortissimo” dynamics! It is the culmination of the first section, followed by another phase – quiet, as a response to or echo of distant dramatic events.

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Vector music is called so because it has a trend. This trend determines the shape and dramaturgy without the lyrical digressions, where all elements of the writing are subject to certain idea orientation. Vector music can exist in all genres and styles [7]. Stochastic music is a disorderly accumulation of images, which are then organized. Non-relativistic music is a music in which each element has a rationale and results from the previous one. Nanomusic is a work with microintervalics. Considering the interrelation of mathematics and music, I should point out the following: The structure of music writing is often very simple and is an alternation of some “block-modules” of certain extent [8]. Melodic parts have, as a rule, the division of motives, phrases, sentences and periods, and accompanying parts have a clearly expressed periodic character. And all of this is combined with harmony – a kind of matrices of normative sounds combination from some frequencies net. This has created some background for the preparation of special programs by mathematicians, in which there are blocks, tags, etc. too [9]. But, music text and sounding writing are different things. In fact, the score is the performer’s action plan only. From the acoustic point of view, sounding writing is an extremely complicated object, the uniqueness of which depends on an individual musician and his performance.

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SUMMARY In this research I have: – studied the closely knit interrelationship of mathematics and music and showed that it originated in and conditioned by sound nature, and acoustic effects associated with wave origin of the sound and perception psychology of the sound which is determined by the peculiarities of anatomical constitution of the human ear. – determined the role of mathematics in the formation of basic musical concepts like interval, pitch and others which organically penetrate to the music “tissue”. – showed the musical pitch, i.e. math formulas of Pythagoras, A. Werckmeister and others determined the music theory evolution for the next millennium. – studied the modern methods of music theory that has totally changed the art of composition. – composed: “Light through fog” for solo flute in a sonorism manner, and “Petrogliff” for two pianos in a dodecaphony style in the frame work of “Musical Mathematics” method of Arnold Schoenberg and fulfilled their structural analysis. – showed the efficiency of new information technologies application in the work of musician-composer. If the music does have laws and the composer is writing it according to the rules plus a little composing ingenuity, it’s just the beginning?! ...

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REFERENCES 1. Kholopov Yu.N. Garmonya: Teoreticheskiy kurs. Uchebnik. – SPb.: Izdatel’stvo “Lan”, 2003. – 544 s. 2. Voloshinov A.V. Matematika i Iskusstvo. – Moskva: Prosveschenie, 1992 – s.335 3. Petelin Yu. Matematika plus muzyka // http://www.petelin.ru/pcmagic/ math/math.htm 4. Personal correspondence of R.-B.Abdyssagin with Yu.S.Kasparov 5. Artaud, Pierre-Yves Flûtes au présent / Present Day Flutes. – Paris: Billaudot, 1995. 6. Bartolozzi, Bruno New Sounds for Woodwind. London: Oxford University Press, 1967. 7. Teoriya sovremennoi kompozicii: uchebnoe posobie / Otv.redaktor V.Cenova. – Muzyka, 2005. – 624 s. 8. Abdyssagin R.-B. Grani garmonii: Sbornik tvorcheskikh proizvedenii. – Almaty, 2011. – 28 s. 9. Ustinov А. Muzyka i matematika // http://virartech.ru/articles/musical_mathematics.php

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CONTENT Abstract ..........................................................................................................3 Introduction ....................................................................................................4 “Time scale” ...................................................................................................6 “Multidimensionality” of structure ................................................................6 A short history of the theory of music ...........................................................7 Pythagoras gamut ...........................................................................................7 The modern forms of composition techniques...............................................11 Spectral music ................................................................................................14 Electronic music.............................................................................................16 Sonoristic music .............................................................................................17 Summary ........................................................................................................21 References ......................................................................................................23

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MATHEMATICS AND CONTEMPORARY MUSIC Layout by G. Shakkozova Cover page designer R. Skakov IB №6556 Signed for publication 10.05.2013. Format 60x84 1/16. Offset paper. Digital printing. Volume 1,5 printer’s sheet. Edition 200 q. Order №800 “Kazak Universiteti” Publishing House Al-Farabi Kazakh National University 050040, Almaty, al-Farabi ave., 71, KazNU Printed at “Kazak Universiteti” Publishing House