Music Numinous and Mathematics. Form and Analysis Through Fourier Space

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Cover Computational Music Science Music: Numinous and Mathematics. Form Analysis Through Fourier Space © Contents Part I. Music and the Numinous Contents Preface Introduction 1 The Transcendental and Rational Discourse 2 Music as Sublime Organism 3 Process Philosophy 4 Music and Process 5 A Whiteheadian Aesthetic and a Musical Paradigm 6 Music, the Other Arts and Process Conclusion Selected Bibliography Index of Names Part II. Music and Mathematics: From Pythagoras to Fractals Preface CONTENTS Music and mathematics: an overview I Music and mathematics through history 1 Tuning and temperament: closing the spiral 2 Musical cosmology: Kepler and his readers II The mathematics of musical sound 3 The science of musical sound 4 Faggofs fretful fiasco 5 Helmholtz: combinational tones and consonance III Mathematical structure in music 6 The geometry of music 7 Ringing the changes: bells and mathematics 8 Composing with numbers: sets, rows and magic squares IV The composer speaks 9 Microtones and projective planes 10 Composing with fractals NOTES ON CONTRIBUTORS NOTES, REFERENCES, AND FURTHER READING ACKNOWLEDGEMENTS INDEX Part III. Musical Form and Analysis: Time, Pattern, Proportion CONTENTS PREFACE PRELUDE: Why Analysis? I - TIME: The Motivating Forces CHAPTER 1: Basic Factors in Form I. Rhythm II. Melody III. Harmony and Tonality IV. Tension and Relaxation V. Articulation of Structural Units CHAPTER 2: The Phrase I. General Characteristics II. The Additive Process III. Phrase Length IV. The Semiphrase V. The Rhythmic Structure of the Phrase VI. The Harmonic Cadence VII. The Harmonic Structure of the Phrase VIII. Phrase Extension IX. Motivic Structure of the Phrase CHAPTER 3: Phrase Groupings I. The Two-Phrase Period II. Symmetrical and Asymmetrical Periods III. Three- and Four-Phrase Periods IV. The Repeated Period V. The Independent Phrase VI. The Repeated Phrase VII. The Phrase Group VIII. The Connection of Phrases IX. Disguised Phrase Endings X. Duple Measure Groupings XI. Description of the Period II - PATTERN: The Shaping Factors CHAPTER 4: The Simple Part Forms I. One-Part Form II. Two-Part Form III. Macrorhythm and Macrotonal Plan IV. Three-Part Form V. Four- and Five-Part Forms VI. Simple Part Forms as Internal Units VII. Simple Part Forms in Vocal Music VIII. The Connection of Parts INTERLUDE: Auxiliary Members I. Beginnings II. Connections III. Conclusions CHAPTER 5: Composite Part Forms I. Composite Three-Part Form II. Composite Five-Part Form III. Composite Part Forms in Vocal Music IV. Other Composite Part Forms CHAPTER 6: The Rondo I. Historical Background II. The Classical Rondo: General Characteristics III. Treatment of the Principal Theme IV. The Episodes V. The Transition VI. The Coda VII. The Five-Part Rondo VIII. The Seven-Part Rondo IX. Rondo and Part Forms: Similarities and Differences CHAPTER 7: Variation Forms I. Theme and Variations II. Ostinato Variations CHAPTER 8: Fugue I. The Exposition II. The Episode III. The Middle Entries IV. The Conclusion V. The "Form" of the Fugue VI. Fugues with Multiple Subjects VII. Related Types CHAPTER 9: Sonata Form I. General Characteristics; Historical Background II. The Introduction III. The Exposition IV. The Development V. The Recapitulation VI. The Coda VII. Sonata Form in the Concerto VIII. Related Forms III - PROPORTION: The Distinguishing Features CHAPTER 10: Multimovement Form I. Classical Models II. The Concerto: Special Considerations III. Cyclical Design IV. Away from the Classical Model CHAPTER 11: Broader Horizons I. Rhythmic Delineation II. Parametric Change and Expansion of the Phrase Concept POSTLUDE: METAFORM: Beyond Formal Analysis I. The Musical Idea II. Musical Character III. Proportional Aspects IV. Under the Magnifying Glass: The Copyist ANTHOLOGY: MUSIC FOR ANALYSIS SELECTED BIBLIOGRAPHY CREDITS INDEX Part IV. Music Through Fourier Space: Discrete Fourier Transform in Music Theory Introduction Historical Survey and Contents A Couple of Examples Public Acknowledgements Notations Contents 1 Discrete Fourier Transform of Distributions 1.1 Mathematical definitions and preliminary results 1.1.1 From pc-sets to an algebra of distributions 1.1.2 Introducing the Fourier transform 1.1.3 Basic notions 1.2 DFT of subsets 1.2.1 What stems from the general definition 1.2.2 Application to intervallic structure 1.2.3 Circulant matrixes 1.2.4 Polynomials Exercises 2 Homometry and the Phase Retrieval Problem 2.1 Spectral units 2.1.1 Moving between two homometric distributions 2.1.2 Chosen spectral units 2.1.3 Rational spectral units with finite order 2.1.4 Orbits for homometric sets 2.2 Extensions and generalisations 2.2.1 Hexachordal theorems 2.2.2 Phase retrieval even for some singular cases 2.2.3 Higher order homometry Exercises 3 Nil Fourier Coefficients and Tilings Cyclotomic polynomials 3.1 The Fourier nil set of a subset of 3.1.1 The original caveat 3.1.2 Singular circulating matrixes 3.1.3 Structure of the zero set of the DFT of a pc-set 3.2 Tilings of Zn by translation 3.2.1 Rhythmic canons in general 3.2.2 Characterisation of tiling sets 3.2.3 The Coven-Meyerowitz conditions 3.2.4 Inner periodicities 3.2.5 Transformations 3.2.6 Some conjectures and routes to solve them 3.3 Algorithms 3.3.1 Computing a DFT 3.3.2 Phase retrieval 3.3.3 Linear programming 3.3.4 Searching for Vuza canons Exercises 4 Saliency 4.1 Generated scales 4.1.1 Saturation in one interval 4.1.2 DFT of a generated scale 4.1.3 Alternative generators 4.2 Maximal evenness 4.2.1 Some regularity features 4.2.2 Three types of ME sets 4.2.3 DFT definition of ME sets 4.3 Pc-sets with large Fourier coefficients 4.3.1 Maximal values 4.3.2 Musical meaning 4.3.3 Flat distributions Exercises 5 Continuous Spaces, Continuous FT 5.1 Getting continuous 5.2 A DFT for ordered collections of pcs on the continuous circle 5.3 ‘Diatonicity’ of temperaments in archeo-musicology 5.4 Fourier vs. voice leading distances 5.5 Playing in Fourier space 5.5.1 Fourier scratching 5.5.2 Creation in Fourier space 5.5.3 Psycho-acoustic experimentation Exercises 6 Phases of Fourier Coefficients 6.1 Moving one Fourier coefficient 6.2 Focusing on phases 6.2.1 Defining the torus of phases 6.2.2 Phases between tonal or atonal music 6.3 Central symmetry in the torus of phases 6.3.1 Linear embedding of the T/I group 6.3.2 Topological implications 6.3.3 Explanation of the quasi-alignment of major and minor triads Exercises 7 Conclusion 8 Annexes and Tables 8.1 Solutions to some exercises 8.2 Lewin’s ‘special cases’ 8.3 Some pc-sets profiles 8.4 Phases of major/minor triads 8.5 Very symmetrically decomposable hexachords 8.6 Major Scales Similarity References Index