References. 18. Andreatta, M., Agon, C., (guest eds), Special Issue Tiling Problems in Music, Journal of Mathematics and Music, July, 3 2, 2009.

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1 References 1. Agon, C., Amiot, E., Andreatta, M., Tiling the line with polynomials, Proceedings ICMC Agon, C., Amiot, E., Andreatta, M., Ghisi, D., Mandereau, J., Z-relation and Homometry in Musical Distributions, JMM 5 2, Amiot, E., Why Rhythmic Canons Are Interesting, in: E. Lluis-Puebla, G. Mazzola and T. Noll (eds.), Perspectives of Mathematical and Computer-Aided Music Theory, EpOs, , Universität Osnabrück, Amiot, E., Autosimilar Melodies, Journal of Mathematics and Music, July, 2 3, 2008, pp Amiot, E., Pour en finir avec le désir, Revue d Analyse Musicale XXII, 1991, pp Amiot, E., Rhythmic canons and Galois theory, Grazer Math. Ber., 347, 2005, pp Amiot, E., À propos des canons rythmiques, Gazette des Mathématiciens, SMF Ed., 106, 2005, pp Amiot, E., New perspectives on rhythmic canons and the spectral conjecture, in Special Issue Tiling Problems in Music, Journal of Mathematics and Music, July, 3 2, Amiot, E., Can a scale have 14 generators?, Proceedings of MCM, London, Springer 2015, pp Amiot, E., David Lewin and Maximally Even Sets, JMM, 1 3, 2007, pp Amiot, E., Structures, Algorithms, and Algebraic Tools for Rhythmic Canons, Perspectives of New Music 49 (2), 2011, pp Amiot, E. Viewing Diverse Musical Features in Fourier Space: A Survey, presented at ICMM Puerto Vallarta 2014, to appear. 13. Amiot, E., Sethares, W., An Algebra for Periodic Rhythms and Scales, JMM 5 3, Amiot, E.: Sommes nulles de racines de l unité, in: Bulletin de l Union des Professeurs de Spéciales 230, 2010, pp Amiot, E., The Torii of phases, Proceedings of MCM, Montreal, Springer Amiot, E., Discrete Fourier Transform and Bach s Good Temperament, Music Theory Online (2), Andreatta, M., On group-theoretical methods applied to music: some compositional and implementational aspects, in: E. Lluis-Puebla, G. Mazzola, T. Noll (eds.), Perspectives of Mathematical and Computer-Aided Music Theory, EpOs, , Universität Osnabrück, Andreatta, M., Agon, C., (guest eds), Special Issue Tiling Problems in Music, Journal of Mathematics and Music, July, 3 2, Ó Springer International Publishing Switzerland 2016 E. Amiot, Music Through Fourier Space, Computational Music Science, DOI /

2 200 References 19. Andreatta, M., De la conjecture de Minkowski aux canons rythmiques mosaïques, L Ouvert, n 114, p , March Babinet, J. Babinet s principle is explained online at org/wiki/babinet s_principle. 21. Ballinger, B., Benbernou, F., Gomez, F., O Rourke, J., Toussaint, G., The Continuous Hexachordal Theorem, MCM Conference, New Haven, 2009, pp Bartlette, C.A., A Study of Harmonic Distance and Its Role in Musical Performance, PhD diss., Eastman School of Music, Beauguitte, P., Transformée de Fourier discrète et structures musicales, Master s thesis, IRCAM 2011 for ATIAM Master. Available online: Cafagna V., Vicinanza D., Myhill property, CV, well-formedness, winding numbers and all that, Keynote adress to MaMuX seminar in IRCAM - Paris, May 22, Callender, C., Continuous Harmonic Spaces, Journal of Music Theory Volume 51 2, Callender, C., Quinn, I., Tymoczko, D., Generalized Voice-Leading Spaces, Science , 2008, pp Caure, H., From covering to tiling modulus p (Modulus p Vuza canons: generalities and resolution of the case { 0,1,2 k} with p = 2.), JMM, Carey, N., Clampitt, D., Aspects of Well Formed Scales, Music Theory Spectrum, 112, 1989, pp Chmelnitzki, A., Some problems related to Singer sets, available online: maslan/docs/ AtiyahPrizeEssay.pdf 30. Clough, J., Douthett, J., Maximally even sets, Journal of Music Theory, 35, 1991, pp Clough, J., Myerson, G., Variety and Multiplicity in Diatonic Systems, Journal of Music Theory, 29: 1985, pp Clough, J., Myerson, G., Musical Scales and the Generalized Circle of Fifths, AMM, 93:9, 1986, pp Clough, J., Douthett, J., Krantz, R., Maximally Even Sets: A Discovery in Mathematical Music Theory Is Found to Apply in Physics, Bridges: Mathematical Connections in Art, Music, and Science, Conference Proceedings, ed. Reza Sarhangi, 2000, pp Cohn, R., Properties and Generability of Transpositionally Invariant Sets, Journal of Music Theory, 35:1 1991, pp Coven, E., Meyerowitz, A. Tiling the integers with one finite set, in: J. Alg. 212, 1999, pp Davalan, J.P., Perfect rhythmic tilings, PNM special issue on rhythmic tilings, de Bruijn, N.G., On Number Systems, Nieuw. Arch. Wisk. 3 4, 1956, pp Douthett, J., Krantz, R., Maximally even sets and configurations: common threads in mathematics, physics, and music, Journal of Combinatorial Optimization, Springer, Online: Fidanza, G., Canoni ritmici, tesa di Laurea, U. Pisa, Jedrzejewski, F., A simple way to compute Vuza canons, MaMuX seminar, January 2004, Jedrzejewski, F., Johnson, T., The Structure of Z-Related Sets, Proceedings MCM 2013, Montreal, 2013, pp Forte, A., The Structure of Atonal Music, Yale University Press, 1977 (2nd ed).

3 References Fripertinger, H. Remarks on Rhythmical Canons, Grazer Math. Ber., 347, 2005, pp Fripertinger, H. Tiling problems in music theory, in: E. Lluis-Puebla, G. Mazzola, T. Noll (eds.), Perspectives of Mathematical and Computer-Aided Music Theory, EpOs, Universität Osnabrück, 2004, pp Fuglede, H., Commuting Self-Adjoint Partial Differential Operators and a Group Theoretic Problem, J. Func. Anal. 16, pp Gilbert, E., Polynômes cyclotomiques, canons mosaïques et rythmes k-asymétriques, mémoire de Master ATIAM, Ircam, May Hajós, G., Sur les factorisations des groupes abéliens, in: Casopsis Pest. Mat. Fys. 74, 1954, pp Hall, R., Klinsberg, P., Asymmetric Rhythms and Tiling Canons, American Mathematical Monthly, Volume 113, Number 10, December 2006, pp Hanson, H., Harmonic Materials of Modern Music. Appleton-Century-Crofts, Hoffman, J., On Pitch-Class Set Cartography Relations Between Voice-Leading Spaces and Fourier Spaces, JMT, 52 2, Johnson, T., Tiling the Line, Proceedings of J.I.M., Royan, Johnson, T., Permutations of 1234, rhythmic canons, block designs, etc, Curtat Tunnel et Forde, Lausanne, Johnson, T., Tiling in My Music, Perspectives of New Music 49 2, 2011, pp Johnson, T., Self-Similar Melodies, Two-Eighteen Press, NY 1996 (2nd ed). 55. Kolountzakis, M. Translational Tilings of the Integers with Long Periods, Elec. J. of Combinatorics (10)1, R22, Kolountzakis, M. Matolcsi, M., Complex Hadamard Matrices And the Spectral Set Conjecture, Collectanea Mathematica 57, 2006, pp Draft available online:// Kolountzakis, M. Matolcsi, M., Algorithms for translational tiling, in Special Issue Tiling Problems in Music, Journal of Mathematics and Music, July, 3 2, Krumhansl, C., Kessler, E., Tracing the Dynamic Changes in Perceived Tonal Organization in a Spatial Representation of Musical Keys, Psychological Review 89 4, 1982, pp Łaba, I., The spectral set conjecture and multiplicative properties of roots of polynomials, J. London Math. Soc. 65, 2002, pp Łaba, I., and Konyagin, S., Spectra of certain types of polynomials and tiling of integers with translates of finite sets, J. Num. Th , 2003, pp Lagarias, J., and Wang, Y. Tiling the line with translates of one tile, in: Inv. Math. 124, 1996, pp Lewin, D., Intervallic Relations Between Two Collections of Notes, JMT 3, Lewin, D., Special Cases of the Interval Function Between Pitch-Class Sets X and Y, JMT, , pp Mandereau, J., Ghisi, D., Amiot, E., Andreatta, M., Agon, C., Discrete Phase Retrieval in Musical Distributions, JMM, 5, Mazzola, G., The Topos of Music, Birkhäuser, Basel, Mazzola, G., Gruppen und Kategorien in der Musik: Entwurf einer mathematischen Musiktheorie, Heldermann, Lemgo 1985, pp Milne, A., Bulger, D., Herff, S., Sethares, W., Perfect Balance: A Novel Principle for the Construction of Musical Scales and Meters, in Proceedings of MCM 5th international conference, London (Springer) 2015, pp Milne, A., Carlé, M., Sethares, W., Noll, T., Holland, S., Scratching the Scale Labyrinth, Proceedings MCM 2011 Paris, Springer, 2001, pp

4 202 References 69. Noll, T., Amiot, E., Andreatta, M., Fourier Oracles for Computer-Aided Improvisation, Proceedings of the ICMC: Computer Music Conference. Tulane University, New Orleans, Available online: Noll, T., Carle, M., Fourier scratching: SOUNDING CODE, Presented at the SuperCollider conference, Berlin Available online: fourier-scratching.pdf. 71. Perle, G., Berg s Master Array of the Interval Cycles, Musical Quarterly 63 1, January 2007, pp I. Quinn, General Equal-Tempered Harmony, Perspectives of New Music , Rahn, J., Amiot, E., eds, Perspectives of New Music, special issue 49 2 on Tiling Rhythmic Canons, Rahn, J., Basic Atonal Theory, New York, Longman, Rosenblatt, J., Seymour, P.D., The Structure of Homometric Sets, SIAM. J. on Algebraic and Discrete Methods Volume 3, Issue 3, Rosenblatt, J., Phase Retrieval, Communications in Mathematical Physics 95, , Sands, A.D., The Factorization of Abelian Groups, Quart. J. Math. Oxford, 10, Schramm, W., The Fourier transform of functions of the greatest common divisor, Electronic Journal of Combinatorial Number Theory A50 8 1, Available online: Sethares, W., Tuning, Timbre, Spectrum, Scale, Springer, Singer, J., A theorem in finite projective geometry and some applications to number theory, Trans. Amer. Math. Soc., 43, 1938, pp Szabó, S., A type of factorization of finite abelian groups, Discrete Math. 54, 1985, pp Tao, T., Fuglede s conjecture is false in 5 and higher dimensions, Mathematical Research Letters 11 2, July Available online: Tao, T., An uncertainty principle for cyclic groups of prime order, Math. Res. Lett., 12, 2005, pp Taruskin, R., Catching up with Rimsky-Korsakov, Music Theory Spectrum 33 2, 2011, pp Terhardt, E., Pitch, consonance, and harmony, Journal of the Acoustical Society of America 55 5, 1974, pp Tijdeman, R., Decomposition of the Integers as a direct sum of two subsets, in: Séminaire de théorie des nombres de Paris, 3rd ed., Cambridge University Press, 1995, pp Toussaint, G., The Geometry of Musical Rhythm, Chapman and Hall/CRC, January Tymoczko, D., Set-class Similarity, Voice Leading, and the Fourier Transform, Journal of Music Theory, , pp Tymoczko, D., Three conceptions of musical distance, Proceedings of MCM, Yale, Springer, 2009, pp Tymoczko, D., Geometrical Methods in Recent Music Theory, MTO161, Online: tymoczko.html 91. Tymoczko, D., A Geometry of Music, Oxford University Press, 2008, pp. 102 and others.

5 References Tymoczko, D., Colloquy: Stravinsky and the Octatonic: Octatonicism Reconsidered Again, Music Theory Spectrum 25 1, 2003, pp Van der Toorn, Colloquy: Stravinsky and the Octatonic: The Sounds of Stravinsky, Music Theory Spectrum 25 1, 2003, pp Vuza, D.T., Supplementary Sets and Regular Complementary Unending Canons, in four parts in Canons. Persp. of New Music, : n 29 2 pp ; 30 1, pp ; 30 2, pp ; 31 1, pp Wild, J., Tessellating the chromatic, Perspectives of New Music, Yust, J., Schubert s harmonic language and Fourier phase space, JMT 59, pp (2015). 97. Yust, J., Restoring the Structural Status of Keys through DFT Phase Space, to appear in Proceedings of ICMM, Puerto Vallarta, Springer, Yust, J., Applications of DFT to the theory of twentieth-century harmony, Proceedings of MCM, London, 2015, Springer, pp Yust, J., Analysis of Twentieth-Century Music Using the Fourier Transform, Music Theory Society of New York State, Binghamton, April Yust, J., Special Collections: Renewing Set Theory, to appear in JMT, 2016.

6 Index k-homometric, 48 k-homometric, 46 k-homometry, 47 algorithms, 9, 22, 51, 60, 62, 63, 69, 75, 76, 80, 83, 88, 140, 144 all-interval, 17, 27, 41, 122, 123, 126 atonal, 57, 117, 122, 164, 166, 168 Bach, VII, 61, 142, 145 Bartok, VII, 114, 116 Berg, 57, 89, 121, 122, 168, 180 multiplication d accords, 3, 10, 71, 72 Callender, 4, 111, 135, 136, 138, 155 Chopin, VII, 3, 10, 71, 101, 164 circulant matrix, 1, 18 22, 25, 26, 31 34, 36, 40, 49, 53 56, 124 convolution, VIII, 1 3, 7 10, 13, 16, 18, 20, 24, 25, 29, 31, 32, 49, 63, 71, 72, 112, 179 Debussy, 14, 100, 116 diatonic, 5, 24, 25, 57, 92, 94 96, 100, 101, 107, , 121, 122, 132, , 148, 159, 160, 162, 163, diatonicity, 111, 114, 117, , 142, 143, 149, 169 dihedral group, XI, 11, 16, 41, 48, 49, 64, 70, 100, 154, 155, 170, 171 Dirac distribution, 3, 63, 127 distributions, 1 3, 6 9, 20, 24, 27, 28, 30 33, 40, 46, 47, 49, 59, , 135, 139, 140, 158, 171, 172 eigenvalues, 1, 19, 21, 23, 31, 33 35, 37, 38, 40, 49 Euler, 35, 83, 96 Fuglede, 60, 78, 79, 88, 126 generators, 40, 41, 92, 94 99, 101, 107, 111, 112, 132, 141 Guidonian hexachord, 92, 111, 113, 121, 160 hexachordal, 10, 17, 26 28, 31, 41, 43, 49, 114, , 140, 176 Homometry, 48 homometry, V, VI, 11, 16, 18, 27, 28, 30 34, 39 41, 43, 45, 49, 64, 105, 127, 131, 135, 154, 160, 176 interval content, 16, 17, 42, 123, 124, 126 interval function, 9, 14 16, 18, 20, 21, 25, 44, 45, 51, 52, 82 invariant, 11, 12, 36, 37, 47, 59, 64, 66, 69, 80, 100, 103, 123, 128, 131, 161, 164, 173, 175, 176 isometry, V, 6, 7, 16, 30, 41, 64, 110, 121, 145 isomorphism, 4, 19, 20, 22, 24, 32, 34, 36, 39, 41, 83, 85, 124, 142, 172 Kolountzakis, 62, 80, 83, 88 Krumhansl, 96, 164 Lewin, V, VI, 15, 17, 18, 20, 29, 31, 51 54, 58, 59, 71, 82, 138, 179 Ó Springer International Publishing Switzerland 2016 E. Amiot, Music Through Fourier Space, Computational Music Science, DOI /

7 206 Index Matolcsi, 87 Mazzola, 7, 135, 151, 153 oversampling, 12, 13, 71, 73, 139 Parseval, 7, 25, 30, 44, 125, 131 periodic, V, VI, 1, 4, 11, 14, 16, 25, 62, 67 69, 71, 72, 74, 83, 88, 91 94, 96, , 107, 111, 131, 150, 153, 180 permutation, 8, 11, 12, 19, 41, 56, 86, 93, 122 phase, VI, VII, 12, 18, 27, 28, 30, 44, 46, 51, 58, 64, 82, 94, 95, 118, 119, 146, 150, 153, 157, , , 175, polynomial, characteristic, 23 25, 58 polynomial, cyclotomic, 24, 51, 58, 66, 80, 88 Quinn, V, VII, 53, 91, 101, , 111, 113, 135, 138, 153, 159, 179, 180 repetition, 12, 13, 31, 61, 62, 68 71, 73, 105, 107, 109, 151, 175 saliency, V, VI, VIII, 12, 91, 111, 113, 114, 116, 127, 142, 149, 153, 154, 159, 160 saturation, 93 singular, 15, 22, 27, 31, 33, 44, 49, 53 57, 71, 89, 188 spectral unit, IX, 27, 30 34, 39, 40 Stravinsky, 117, 160, 162, 168 Szabó, 62, 75, 83, 86 tango, 1, 14, 101, 109, 132 tonal, 112, 114, 142, 166, 168 totient function, 35, 83, 96, 97 Tymoczko, 100, 112, 118, 135, 138, 139, 145, 146, 148, 149, 165 Vuza, VI, 62, 68, 70, 72, 74 77, 79, 83, Yust, VII, 12, 57, 58, 71, 94, 111, 114, 116, 117, 157, , 163, 164, 166, 167, 169, 170, 172, 173, 175, 176, 180