Music Through Fourier Space

Discrete Fourier Transform in Music Theory

verfasst von: Emmanuel Amiot

Verlag: Springer International Publishing

Buchreihe : Computational Music Science

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Über dieses Buch

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, saliency, extrapolation to the continuous Fourier transform and continuous spaces, and the meaning of the phases of Fourier coefficients.

This is the first textbook dedicated to this subject, and with supporting examples and exercises this is suitable for researchers and advanced undergraduate and graduate students of music, computer science and engineering. The author has made online supplementary material available, and the book is also suitable for practitioners who want to learn about techniques for understanding musical notions and who want to gain musical insights into mathematical problems.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Discrete Fourier Transform of Distributions

Summary

This chapter gives the basic definitions and tools for the DFT of subsets of a cyclic group, which can model for instance pitch-class sets or periodic rhythms. I introduce the ambient space of distributions, where pc-sets (or periodic rhythms) are the elements whose values are only 0’s and 1’s, and several important operations, most notably convolution which leads to ‘multiplication d’accords’ (transpositional combination), algebraic combinations of chords/scales, tiling, intervallic functions and many musical concepts. Everything is defined and the chapter is hopefully self-contained, except perhaps Section 1.2.3 which uses some notions of linear algebra: eigenvalues of matrices and diagonalisation. Indeed it is hoped that the material in this chapter will be used for pedagogical purposes, as a motivation for studying complex numbers and exponentials, modular arithmetic, algebraic structures and so forth. The important Theorem 1.11 proves that DFT is the only transform that simplifies the convolution product into the ordinary, termwise product.

Emmanuel Amiot

Chapter 2. Homometry and the Phase Retrieval Problem

Summary

This chapter studies in depth the notion of homometry, i.e. having identical internal shape, as seen from Fourier space, where homometry can be seen at a glance by the size (or magnitude) of the Fourier coefficients. Finding homometric distributions is then a question of choosing the phases of these coefficients, hence this problem is often called phase retrieval in the literature. Such a choice of phases is summed up in the objects called spectral units, which connect homometric sets together. I included the original proof of the one difficult theorem of this book (Theorem 2.10), which non-mathematicians are quite welcome to skip. Some generalisations of the hexachord theorem are given, followed by the few easy results on higher-order homometry which deserve some room in this book because they rely heavily on DFT machinery. An original method for phase-retrieval with singular distributions (the difficult case) is also given. Some knowledge of basic linear algebra may help in this chapter.

Emmanuel Amiot

Chapter 3. Nil Fourier Coefficients and Tilings

Summary

Originally, vanishing Fourier coefficients appeared as an obstruction: they impede phase retrieval and prevent, for instance, the solution of Lewin’s problem (find A knowing B and IFunc(A,B)). But recent research and problems shed a more positive light: for instance the set Z(A) of indexes k such that \( a_{k} = \hat{1_{A}} (k) \, = 0 \), a highly organised subset of \( {\mathbb{Z}}_{n} \), is now the fashionable introduction to a definition of tilings. The theory of tilings is a crossroad of geometry, algebra, combinatorics, topology; and one of those privileged domains where musical ideas enable us to make some headway in non-trivial mathematics. Here the notion of Vuza canon together with transformational techniques (often introduced by composers) allowed some progress on difficult conjectures. More generally, tiling situations provide rich compositional material as we will see later in this book, cf. Section 4.3.3. In that respect, I included in this chapter Section 3.3 on algorithms (for practical purposes, though there are some interesting theoretical implications in there too. We will need some additional algebraic material on polynomials, which is introduced in the preliminary section. A few more technical results of Galois theory are recalled and admitted without proof.

Emmanuel Amiot

Chapter 4. Saliency

Summary

In the seminal [72], Ian Quinn tries to define a ‘landscape of chords’ starting from cultural/intuitive knowledge of the most ‘salient’ chords, and from there infers in a prodigious leap of intuition the existence of a measurable ‘chord quality’, or saliency, maximal for the prototypical chords.

Emmanuel Amiot

Chapter 5. Continuous Spaces, Continuous FT

Summary

The formula for the Fourier Transform \( \hat{f}\left( t \right) = \sum {f\left( k \right)} \;{\text{e}}^{ - 2ik\pi t/n} \) can be extended to continuous settings in several ways: transforming the discrete \( \sum \) into a continuous\( \smallint \) with some appropriate (usually Lebesgue) measure, i.e. summing on an infinite set; or having the variable t move on the real line instead of a cyclic group; or having the frequency \( \frac{2\pi }{n} \) become infinitesimal, perhaps keeping value \( \frac{2k\pi }{n} \) constant while both k and n grow to infinite. A last variant considers ordered collections of pcs with fixed cardinality, leading unexpectedly to a good measure of quality for temperaments such as might have been used by J.S. Bach. All these changes, advertised by various researchers [25, 88], require however precise definitions of their contexts and limitations, which will be scrupulously enunciated hereafter. Several practical situations of music devised by playing directly in some continuous Fourier space have occurred in recent years, and are reviewed in the last section.

Emmanuel Amiot

Chapter 6. Phases of Fourier Coefficients

Summary

We have explored in great depth one dimension of Fourier coefficients, their magnitude. This has proved a worthwhile journey, with incontrovertible musical meaning; it allows the painting of nice pictures of scales/chords landscapes, though with the major and embarrassing restriction that scales must share their cardinality in pictures such as Fig. 5.3; also the phase component had to be discarded because it did not make sense in most orbifold universes. It is now time to get back to genuine, ordinary pc-sets and look at the entirety of Fourier coefficients, taking into account not only their magnitudes but also their directions (or ‘phases’). This has been tackled in different ways, the first comprehensive try being Justin Hoffman’s in [50], developing upon a remark of Joseph Strauss. However I will devote the bulk of this chapter to the study of phases per se, since the magnitude has been previously covered extensively. I will only provide a few chosen musical examples, the purpose of this book being rather a clean and comprehensive exposition of the theoretical background necessary for such endeavours. The torus of phases was introduced in [15], but I refer the reader to Yust’s pioneering work for many far more convincing analyses, cf. [96, 97, 98, 99, 100].

Emmanuel Amiot

Chapter 7. Conclusion

Summary

The use of DFT in music theory really soared after the notion was resuscitated from Lewin’s work by Quinn [72] in 2005. As we have seen, it bears the tremendous advantage that each coefficient, and moreover each polar coordinate of each coefficient, yields dramatically important musical information (say, the phase of a5 shows which diatonic universe is closest to the pc-set in question). Some musical qualities are immediately visible in Fourier space whereas they require computations in the original musical domain (say, pc-distributions); Fourier space, with this minimised computational complexity, is closest to our perception of music. Indeed, psycho-acoustic experiments on the perception of saliency (and its evil twin, low saliency including nullity of a coefficient) should be enhanced and furthered, since Furier qualities seem to mirror exactly musical features processed by the human brain.

Emmanuel Amiot

Chapter 8. Annexes and Tables

Summary

This chapter features solutions to selected exercises, some pictures chosen from the online database of all profiles of pc-sets http://canonsrythmiques.free.fr/MaRecherche/photos-2/ which have been included here because they are mentioned in the main text, and, for reference, tables of singular pc-sets, phases of triads, enumeration of the most symmetrically pc-sets in the sense of Proposition 6.10, and values of Major Scale Similarity for a large panel of historical temperaments.

Emmanuel Amiot

Backmatter

Titel: Music Through Fourier Space
verfasst von: Emmanuel Amiot
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-45581-5
Print ISBN: 978-3-319-45580-8
DOI: https://doi.org/10.1007/978-3-319-45581-5