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Computational Music Science Emmanuel Amiot Music Through Fourier Space Discrete Fourier Transform in Music Theory

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Computational Music Science Series editors Guerino Mazzola Moreno Andreatta

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More information about this series at http://www.springer.com/series/8349

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Emmanuel Amiot Music Through Fourier Space Discrete Fourier Transform in Music Theory 123

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Emmanuel Amiot Laboratoire de Mathématiques et Physique Université de Perpignan Via Domitia Perpignan France ISSN 1868-0305 ISSN 1868-0313 (electronic) Computational Music Science ISBN 978-3-319-45580-8 ISBN 978-3-319-45581-5 (eBook) DOI 10.1007/978-3-319-45581-5 Library of Congress Control Number: 2016954630 © Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

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Introduction This book is not about harmonics, analysis or synthesis of sound. It deals with harmonic analysis but in the abstract realm of musical structures: scales, chords, rhythms, etc. It was but recently discovered that this kind of analysis can be per- formed on such abstract objects, and furthermore the results carry impressively meaningful signiﬁcance in terms of already well-known musical concepts. Indeed in the last decade, the Discrete Fourier Transform (DFT for short) of musical struc- tures has come to the fore in several domains and appears to be one of the most promising tools available to researchers in music theory. The DFT of a set (say a pitch-class set) is a list of complex numbers, called Fourier coefﬁcients. They can be seen alternatively as pairs of real numbers, or vectors in a plane; each coefﬁcient provides decisive information about some musical dimensions of the pitch-class set in question. For instance, the DFT of C♯EGB♭ is 4iπ/3 2iπ/3 (4,0,0,0,4e ,0,0,0,4e ,0,0,0) where all the 0’s show the periodic character of the chord, the sizes of the non-nil coefﬁcients mean that the chord divides the octave equally in four parts, and the angles (2iπ/3,4iπ/3) specify which of the three diminished sevenths we are looking at. From David Lewin’s very ﬁrst paper (1959) and its revival by Ian Quinn (2005), it came to be known that the magnitude of Fourier coefﬁcients, i.e. the length of these vectors, tells us much about the shape of a musical structure, be it a scale, chord, or (periodic) rhythm. More precisely, two objects whose Fourier coefﬁcients have equal magnitude are homometric, i.e. they share the same interval distribution; this generalization of isometry was initially studied in crystallography. Saliency, i.e. a large size of some Fourier coefﬁcients, characterises very special scales, such as the diatonic, pentatonic, whole-tone scales. On the other hand, ﬂat distributions of these magnitudes can be shown to correspond with uniform intervallic distributions, showing that these magnitudes yield a very concrete and perceptible musical mean- ing. Furthermore, nil Fourier coefﬁcients are highly organised and play a vital role in the theory of tilings of the line, better known as “rhythmic canons.”

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VI Finally, the cutting-edge research is currently focused on the other component of Fourier coefﬁcients, their directions (called phases). These phases appear to model some aspects of tonal music with unforeseen accuracy. Most of these aspects can be extended from the discrete to the continuous domain, allowing the consideration of microtonal music or arbitrary pitch, and interesting links with voice-leading theory. This type of analysis can also be deﬁned for ordered collections of non-discrete pitch classes, enabling, for instance, comparisons of tunings. Historical Survey and Contents Historically, the Discrete Fourier Transform appeared in D. Lewin’s very ﬁrst paper in 1959 [62]. Its mention at the very end of the paper was as discreet as possible, anticipating an outraged reaction at the introduction of “high-level” mathematics in a music journal – a reaction which duly occurred. The paper was devoted to the inter- 1 esting new notion of the Intervallic Relationship between two pc-sets , and its main result was that retrieval of A knowing a ﬁxed set B and IFunc(A,B) was possible, provided B did not fall into a hodgepodge of so-called special cases – actually just those cases when at least one of the Fourier coefﬁcients of B is 0. These were the times when Milton Babbitt proved his famous hexachordal theorem, probably with young Lewin’s help. As we will see, its expression in terms of Fourier coefﬁcients allows one to surmise that the perception of missing notes (or accents, in a rhythm) completely deﬁnes the motif’s intervallic structure. These questions, together with any relevant deﬁnitions and properties (with some modern solutions to Lewin’s and others’ problems), are studied in Chapter 1. Lewin himself returned to this notion in some of his last papers [63], which in- ﬂuenced the brilliant PhD research of I. Quinn, who encountered DFT and espe- cially large Fourier coefﬁcients as characteristic features of the prominent points of his “landscape of chords” [72], see Fig. 4.1. Since he had voluntarily left aside for readers of the Journal of Music Theory the ‘stultifying’ mathematical work involved in the proof of one of his nicer results, connecting Maximally Even Sets and large Fourier coefﬁcients, I did it in [10], along with a complete discussion of all maxima of Fourier coefﬁcients of all pc-sets, which is summarised and extended in Chapter 4. Lacklustre Fourier coefﬁcients, with none showing particular saliency, are also studied in that chapter. Meanwhile, two apparently extraneous topics involved a number of researchers in using the very same notion of DFT: homometry which is covered in Chapter 2 (see the state of the art in [2, 64] and Tom Johnson’s recent compositions Intervals or Trichords et tetrachords); and rhythmic canons in Chapter 3 – which are really algebraic decompositions of cyclic groups as direct sums of subsets. The latter can be used either in the domain of periodic rhythms or pitches modulo some ‘octave,’ and 2 were ﬁrst extensively studied by Dan Tudor Vuza [94] , then connected to the general 1 I use the modern concept, though the term ‘pitch-class set’ had not yet been coined at the time. IFunc(A,B) is the histogram of the different possible intervals from A to B. 2 At the time, probably the only theorist to mention Lewin’s use of DFT.

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VII theory of tiling by [19, 17] and developed in numerous publications [8, 18, 73] which managed to interest some leading pure mathematician theorists in the ﬁeld (Matolcsi, 3 Kolountzakis, Szabo´) in musical notions such as Vuza canons. There were also improbable cross-overs, like looking for algebraic decomposi- tions of pc-collections (is a minor scale a sum and difference of major scales?) [13], or an incursion into paleo-musicology, quantifying a quality of temperaments in the search for the tuning favoured by J.S. Bach [16], which unexpectedly warranted the use of DFT. Aware of the intrinsic value of DFT, several researchers commented on it, trying to extend it to continuous pitch-classes [25] and/or to connect its values to voice- leadings [89, 88]. These and other generalisations to continuous spaces are studied in Chapter 5. Another very original development is the study of all Fourier coefﬁ- cients with a given index of all pc-sets [50], also oriented towards questions of voice- leadings. On the other hand, consideration of the proﬁle of the DFT enables charac- terisation of pc-sets in diverse voices or regions of tonal and atonal pieces [98, 99] as we will see in Chapter 6, which takes up the dimension that Quinn had left aside, the phase (or direction) of Fourier coefﬁcients. The position of pairs of phases (angles) on a torus was only recently introduced in [15] but has known tremendously inter- esting developments since, for early romantic music analysis [96, 97] but also atonal compositions [98, 99]. Published analyses involve Debussy, Schubert, Beethoven, Bartok, Satie, Stravinsky, Webern, and many others. Other developments include, for instance, comparison of intervals inside chromatic clusters in Łutoslawski and Carter, using DFT of pitches (not pitch classes) by Cliff Callender [25]. A Couple of Examples I must insist on the fact that DFT analysis is no longer some abstract considera- ◦ tion, but is done on actual music: consider for instance Chopin’s Etude op. 10, N 5, wherein the pentatonic (black keys) played by the right hand is a subset of G ♭ ma- jor played by the left hand; but so are many other subsets (or oversets). I previously pointed out in [10] that, because the pentatonic and diatonic scales are complemen- tary Maximally Even Sets, one is included in the other up to transposition (warrant- ing the name ‘Chopin’s Theorem’ for this property of ME sets); however, it is much more signiﬁcant to observe that these two scales have identical Fourier coefﬁcients 4 with odd indexes , which reﬂects spectacularly their kinship (see Chapter 6 and Fig. 4.7). I cannot wait to exhibit another spectacular example of the ‘unreasonable efﬁ- ciency’ of DFT: Jason Yust’s discovery [98] that in Bartok’s String Quartet 4 (iv), the accompaniment concentrates its energy in the second Fourier component while this component vanishes for the melody, and conversely for the sixth component (associated with the whole-tone character). This is again vastly superior to classic 3 The musical aspect lies in the idea that a listener does not hear any repetition either in the motif nor in the pattern of entries of a Vuza canon. 4 The other coefﬁcients, with even indexes, have the same magnitude, but different directions.

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VIII ‘Set-Theory’ subset-relationships (parts of this analysis and others are reproduced in this book), cf. Fig. 0.1 (further commented on in Chapters 4 and 6). 7 6 5 {0,2,3,5,6,8,10} {2,3,7,8} 4 3 2 1 1 2 3 4 5 6 7 8 9 10 11 Fig. 0.1. DFT magnitudes of melody and accompaniment in Bartok One explanation of the efﬁciency of DFT in music theory may well be Theorem 1.11. As we will see throughout this book, many music theory operations can be expressed in terms of convolution products. Not only is this product signiﬁcantly simpler in Fourier space (i.e. after Fourier transform, cf. Theorem 1.10), but the aforementioned theorem proves that Fourier space is the only one where such a simpliﬁcation occurs. This means that, for instance, interval functions or vectors, which are essential in the perception of the shape of musical objects, are more easily constructed and even perceived in Fourier space. Idem for the property of tiling – ﬁlling the space with one motif according to another – which is completely obvious when glancing at nil Fourier coefﬁcients. Furthermore, we will see and understand how each and every polar coordinate in Fourier space carries rich musical meaning, not requiring any further computation. Public This book aims at being self-contained, providing coherent deﬁnitions and proper- ties of DFT for the use of musicians (theorists and practitioners alike). A wealth of examples will also be given, and I have chosen the simplest ones since my purpose is clarity of exposition. More sophisticated examples can be found in the already abun- dant bibliography. I have also added a number of exercises, some with solutions, because the best way to make one’s way through new notions is always with pen and pencil. Professional musicians, researchers and teachers of music theory are of course the privileged public for this monograph. But I tried to make it accessible at pre-

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IX graduate level, either in music or in mathematics. In the former case, besides in- troducing the notion of DFT itself for its intrinsic interest, it may help the student progress through useful mathematical concepts that crop up along the way. In the latter case, I hope that maths teachers may ﬁnd interesting material for their classes, and that the musical angle can help enlighten those students who need a purpose before a concept. It is even hoped, and indeed expected, that hardened pure mathe- maticians will ﬁnd in here a few original results worth their mettle. Some general, elementary grounding in mathematics should be useful: knowl- edge of simple number sets (integers, rationals, real and complex numbers), basics of group theory (group structure, morphism, subgroups) which are mostly applied to the group Z12 of integers modulo 12; other simple quotient structures make furtive appearances in Chapters 1 and 3; vector spaces and diagonalization of matrixes are mentioned in Chapter 1 and used once in Chapter 2, providing sense to the other- wise mysterious ‘rational spectral units’. The corresponding Theorem 2.10 is the only really difﬁcult one in this book: many proofs are one-liners, most do not exceed paragraph length. All in all, I hope that any cultured reader with a smattering of sci- entiﬁc education will feel at ease with most of this book (and will be welcome to skip the remaining difﬁculties). On the other hand, mathematically minded but non- musician readers who cannot read musical scores or are unfamiliar with ‘pc-sets’ or ‘scales’ can rely on the omnipresent translations into mathematical terms. Last but not least, some online content has been developed speciﬁcally for the readers of this book, who are strongly encouraged to use it: for instance all ‘Fourier proﬁles’ of all classes of pc-sets can be perused at http://canonsrythmiques.free.fr/MaRecherche/photos-2/ while only a selection of the 210 cases is printed in Chapter 8, and software is avail- able for the computation of the DFT of any pc-set in Z12.

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