Generalized Musical Intervals and Transformations 0195317130, 9780195317138
381 56 753KB
English Pages 280 [291] Year 2007
Table of contents :
Contents......Page 8
Foreword......Page 10
Preface......Page 14
Acknowledgments......Page 28
Introduction......Page 30
1. Mathematical Preliminaries......Page 34
2. Generalized Interval Systems (1): Preliminary Examples and Definition......Page 49
3. Generalized Interval Systems (2): Formal Features......Page 64
4. Generalized Interval Systems (3): A Non-Commutative GIS; Some Timbral GIS models......Page 93
5. Generalized Set Theory (1): Interval Functions; Canonical Groups and Canonical Equivalence; Embedding Functions......Page 121
6. Generalized Set Theory (2): The Injection Function......Page 156
7. Transformation Graphs and Networks (1): Intervals and Transpositions......Page 190
8. Transformation Graphs and Networks (2): Non-Intervallic Transformations......Page 208
9. Transformation Graphs and Networks (3): Formalities......Page 226
10. Transformation Graphs and Networks (4): Some Further Analyses......Page 253
Appendix A: Melodic and Harmonic GIS Structures; Some Notes on the History of Tonal Theory......Page 278
Appendix B: Non-Commutative Octatonic GIS Structures; More on Simply Transitive Groups......Page 284
E......Page 288
J......Page 289
R......Page 290
Z......Page 291
Contents......Page 8
Foreword......Page 10
Preface......Page 14
Acknowledgments......Page 28
Introduction......Page 30
1. Mathematical Preliminaries......Page 34
2. Generalized Interval Systems (1): Preliminary Examples and Definition......Page 49
3. Generalized Interval Systems (2): Formal Features......Page 64
4. Generalized Interval Systems (3): A Non-Commutative GIS; Some Timbral GIS models......Page 93
5. Generalized Set Theory (1): Interval Functions; Canonical Groups and Canonical Equivalence; Embedding Functions......Page 121
6. Generalized Set Theory (2): The Injection Function......Page 156
7. Transformation Graphs and Networks (1): Intervals and Transpositions......Page 190
8. Transformation Graphs and Networks (2): Non-Intervallic Transformations......Page 208
9. Transformation Graphs and Networks (3): Formalities......Page 226
10. Transformation Graphs and Networks (4): Some Further Analyses......Page 253
Appendix A: Melodic and Harmonic GIS Structures; Some Notes on the History of Tonal Theory......Page 278
Appendix B: Non-Commutative Octatonic GIS Structures; More on Simply Transitive Groups......Page 284
E......Page 288
J......Page 289
R......Page 290
Z......Page 291
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