Music Through Fourier Space Discrete Fourier Transform in Music Theory

• Main Author: Amiot, Emmanuel
• Format: eBook
• Language: English
• Published: Cham Springer Nature 2016; Springer International Publishing AG; Springer International Publishing
• Edition: 1
• Series: Computational Music Science
• Subjects: Computer Appl. in Arts and Humanities; Computer Science; Data processing Computer science; Fourier transformations; Mathematics in Music; Mathematics of Computing; Music; Music-Mathematics; Signal, Image and Speech Processing; User Interfaces and Human Computer Interaction
• ISBN: 9783319455815; 3319455818; 9783319455808; 331945580X
• ISSN: 1868-0305; 1868-0313
• DOI: 10.1007/978-3-319-45581-5

This book explains the state of the art in the use of the discrete Fourier transform (DFT) of musical structures such as rhythms or scales. In particular the author explains the DFT of pitch-class distributions, homometry and the phase retrieval problem, nil Fourier coefficients and tilings, salienc...

Table of Contents:

• 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
• Intro -- 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