On two open mathematical problems in music theory: Fuglede spectral conjecture and discrete phase retrieval

Transcription

1 On two open mathematical problems in music theory: uglede spectral conjecture and discrete phase retrieval lgebra Seminar, TU resden, 29 November 2012 Moreno ndreatta Music Representation Team IRM / NRS UMR 9912 / University Paris 6 Moreno.ndreatta@ircam.fr

2 IRM = Institut de Recherche et oordination coustique/musique

3 1999: 4 e orum iderot (Paris, Vienne, Lisbonne), Mathematics and Music (. ssayag, H.. eichtinger, J.. Rodrigues, Springer, 2001) : MaMuPhi Seminar, Penser la musique avec les mathématiques? (ssayag, Mazzola, Nicolas éds., oll. Musique/Sciences, Ircam/elatour, 2006) : International Seminar on MaMuTh (Perspectives in Mathematical and omputational Music Theory (Mazzola, Noll, Luis-Puebla eds, epos, 2004) 2003: The Topos of Music (. Mazzola et al.) : MaMuX Seminar at Ircam : mamuphi Seminar (ns/ircam) 2006: Mathematical Theory of Music (. Jedrzejewski), oll. Musique/Sciences 2007: La vérité du beau dans la musique ( Mazzola), oll. Musique/Sciences 2007: Journal of Mathematics and Music (Taylor & rancis) and SMM 2007: irst MM 2007 (erlin) and Proceedings by Springer : MS Special Session on Mathematical Techniques in Musical nalysis 2009: omputational Music Science (eds:. Mazzola, M. ndreatta, Springer) 2009: MM 2009 (Yale University) and Proceedings by Springer 2010: Mathematics Subject lassification : 0065 Mathematics and music 2011: MM 2011 (Ircam, June 2011) and Proceedings LNS Springer 2013: MM 2013 (Mcill University, anada, June 2013)

4 Modern Mathematics and the process of concepts creation the role of mathematics, which at the beginning was considered as a part of physics, has become thanks to modern mathematics a kind of substitution of philosophy with respect to the creation of concepts (lain onnes).

5 The double movement of a mathemusical activity MTHMTIS Mathema9cal statement generalisa9on eneral theorem formalisa9on OpenMusic applica9on MUSI Musical problem Music analysis Music theory omposi9on

6 Some examples of mathemusical problems M. ndreatta : Mathematica est exercitium musicae, Habilitation Thesis, IRM University of Strasbourg, The construction of Tiling Rhythmic anons - The Z relation and the theory of homometric sets - Set Theory and Transformational Theory - iatonic Theory and Maximally-ven Sets - Periodic sequences and finite difference calculus - lock-designs and algorithmic composition Rhythmic Tiling anons Z-Relation and Homometric Sets inite ifference alculus Set Theory, andtransformation Theory iatonic Theory and M-Sets lock-designs

7 Periodic sequences and finite difference calculus natol Vieru: Zone d oubli for viola (1973) Reducible sequences: k 1 such that k f = 0 Reproducible sequences: k 1 such that k f = f ecomposition theorem (Vuza & ndreatta, Tatra M., 2001) very periodic sequence f can be decomposed in a unique way as a sum + of a reducible sequence and a reproducible sequence. Vuza, M. ndreatta (2001), «On some properties of periodic sequences in natol Vieru's modal theory», Tatra Mountains Mathematical Publications, Vol. 23, p. 1-15

8 lock-designs and algorithmic composition ano plane lock-design (11,6,3) (12,4,3) Jedrzejewski,., M. ndreatta, T. Johnson (2009), «Musical experiences with lock esigns», Second International onference MM 2009, vol. 38, New Haven, p

9 Intervallic structure do do# re re# mi fa fa# sol sol# la la# si do do# re la# si do do# 1 re Z 12 = < T k (T k ) 12 = T 0 > T k : x x+k mod = < T k, I (T k ) 12 =I 2 =T 0, ITI=I(IT) -1 > I : x -x mod 12 la sol# 7 sol Mersenne 1648 fa# 6 fa mi 5 re# 4 3

10 Paradigmatic architecture yclic group ihedral group orte/ Rahn arter Morris / Mazzola strada Klein W. urnside ffine group Symmetric group. Halsey &. Hewitt, ine gruppentheoretische Methode in der Musik-theorie, Jahr. der t. Math.-Vereinigung, 80, Reiner, numeration in Music Theory, mer. Math. Month. 92:51-54, 1985 H. ripertinger, numeration in Musical Theory, eiträge zur lektr. Musik,1,1992 R.. Read, ombinatorial problems in the theory of music, iscrete Mathematics 1997 H. ripertinger, numeration of mosaics, iscrete Mathematics, 1999 H. ripertinger, numeration of non-isomorphic canons, Tatra Mt. Math. Publ., Polya

11 Z-relation homometry I = [4, 3, 2, 3, 2, 1 ] = [4, 3, 2, 3, 2, 1 ] = I abbitt s Hexachord Theorem: hexacord and its complement have the same interval content (Proofs by Wilcox, Ralph ox (?), hemillier, Lewin, Mazzola, Schaub,, miot, )

12 [Thomas Noll 2004] ρ = <L,R L 2 = (LR) 12 =1 LRL=L(LR) -1 > uler : Speculum musicum, 1773 ρ Sym ( 12 ) 12 Sym (ρ) 12 = <I, T I 2 = T 12 =1 ITI=I(IT) -1 > [Hook, Science, 2006] rans., iore T., and Satyendra R. "Musical ctions of ihedral roups." The merican Mathematical Monthly, Vol. 116 (2009), No. 6:

13 L. igo, J.-L. iavitto,. Spicher, "uilding Topological Spaces for Musical Objects", MM 2011 L. igo, Ph (ongoing), Ircam / University of Paris 12 # # #

14 xtract of the 2 nd movement of the Symphony No. 9 (L. van eethoven) # b # # b # # b # # b # b # # b

15 xtract of the 2 nd movement of the Symphony No. 9 (L. van eethoven) # b # # b # # b # # b # b # # b

16 xtract of the 2 nd movement of the Symphony No. 9 (L. van eethoven) # b # # b # # b # # b # b # # b

17 xtract of the 2 nd movement of the Symphony No. 9 (L. van eethoven) # b # # b # # b # # b # b # # b

18 xtract of the 2 nd movement of the Symphony No. 9 (L. van eethoven) # b # # b # # b # # b # b # # b

19 xtract of the 2 nd movement of the Symphony No. 9 (L. van eethoven) # b # # b # # b # # b # b # # b

20 xtract of the 2 nd movement of the Symphony No. 9 (L. van eethoven) # b # # b # # b # # b # b # # b

21 xtract of the 2 nd movement of the Symphony No. 9 (L. van eethoven) # b # # b # # b # # b # b # # b

22 xtract of the 2 nd movement of the Symphony No. 9 (L. van eethoven) # b # # b # # b # # b # b # # b

23 xtract of the 2 nd movement of the Symphony No. 9 (L. van eethoven) # b # # b # # b # # b # b # # b

24 xtract of the 2 nd movement of the Symphony No. 9 (L. van eethoven) # b # # b # # b # # b # b # # b

25 xtract of the 2 nd movement of the Symphony No. 9 (L. van eethoven) # b # # b # # b # # b # b # # b

26 xtract of the 2 nd movement of the Symphony No. 9 (L. van eethoven) # b # # b # # b # # b # b # # b

27

28 l cinquillo l trecillo

29 embé? Jack outhett & Richard Krantz, nergy extremes and spin configurations for the one-dimensional antiferromagnetic Ising model with arbitrary-range interaction, J. Math. Phys. 37 (7), July 1996!

30 Rhythmic Tiling anons as group factorisations <Z n <Z n Z n <Z n Z n Z n periodic subset Z n =

31 [ ]

32

33 Tiling Rhythmic anons as a mathémusical problem Minkowski/Hajós Problem ( ) In any simple lattice tiling of the n-dimensional uclidean space by unit cubes, some pair of cubes must share a complete (n-1)-dimensional face Vieru s problem and Vuza s formalization (PNM, 1991) Vuza anon is a factorization of a cyclic group in a direct sum of two nonperiodic subsets Z/nZ =R S Link between Minkowski problem and Vuza anons (ndreatta, Master diss. 1996) Hajós groups (good groups) Z/nZ with n {p α, p α q, pqr, p 2 q 2, p 2 qr, pqrs} where p, q, r, s, are distinct prime numbers Non-Hajós group (bad groups) (Sloane s sequence ) M. ndreatta, onstructing and ormalizing Tiling Rhythmic anons : Historical Survey of a Mathemusical Problem, Perspectives of New Music, Special Issue, 49(1-2), 2011

34 rom Minkowski onjecture to Hajós groups Minkowski onjecture (1907) In any simple lattice tiling of the n-dimensional uclidean space by unit cubes, some pair of cubes must share a complete (n-1)- dimensional face Hajós Theorem (1941) iven a finite abelian group and given n elements a 1, a 2,, a n of. If is a direct product of cyclic subsets 1,.., n where with q i > 0 for all i=1, 2,, n, then k is a group for a given k Hajós groups (good groups) S. Stein, S. Szabó: lgebra and Tiling, arus Math. Mon Rédei 1947 (p, p) Hajós 1950 Z Z/nZ with n= p α e rujin 1953 (p α, q) (p, q, r) Sands 1957 (p 2, q 2 ) (p 2, q, r) (p, q, r, s) Sands 1959 (2 2, 2 2 ) (3 2, 3) (2 n, 2) Sands 1962 (p, 3, 3) (p, 2 2, 2) (p, 2, 2, 2, 2) (p 2, 2, 2, 2) (p 3, 2, 2) (p, q, 2, 2) Sands 1964 Q Z+Z/pZ Q+Z/pZ

35 Weak versions of Minkowski onjecture Minkowski onjecture (1907) In any simple lattice tiling of the n-dimensional uclidean space by unit cubes, some pair of cubes must share a complete (n-1)- dimensional face The four conditions of Minkowski onjecture [1] The cubes are translates of each other [2] The translation vectors form a lattice [3] The interiors of the cubes are disjoint [4] ach point of the space which is not on the boundary of any cube is contained in exactly one cube S. Szabó: ube Tilings as contribution of lgebra to eomery, eiträge zur lgebra und eometrie, 34(1), 1993 Keller onjecture (1930) = Minkowski [2] True for n 6 (Perron, 1940) alse for n 8 (Lagarias et Shor, 1992; Mackey, 2000) Open for n=7 S. Stein, S. Szabó: lgebra and Tiling, arus Math. Mon urtwängler onjecture (1936) = Minkowski [3, 4] + new k-fold tiling condition : [4 ] ach point not lying on the boudary of any cube is contained in exactly k cubes True for n 2 (Hajos 1941) alse in general, since for every k > 1 and every n > 2 there is a k-fold tiling of n-dimensional space by cubes with no shared faces (Szabó 1982).

36 Minkowski onjecture (1907) In any simple lattice tiling of the n-dimensional uclidean space by unit cubes, some pair of cubes must share a complete (n-1)- dimensional face Hajós quasi-periodic onjecture S. Stein, S. Szabó: lgebra and Tiling, arus Math. Mon Hajos quasi-periodic onjecture (1950) very factorisation of a finite abelian group = + is quasiperiodic i.e. is decomposable in subsets 1,, k and there exists a periodic subset {g 1,, g k } of such that + i =g i alse for Z 5 Z 25 (Sands, 1974) True for Z p Z p (Steinberger, 2008) Open in general

37 uglede Spectral onjecture subset of the n- dimensional uclidean space tiles by translation iff it is spectral. (J. unc. nal. 16, 1974) alse in dim. n 3 (Tao, Kolountzakis, Matolcsi, arkas and Mora) Open in dim. 1 et 2 n=1 xample: ={0,1,6,7} Λ={0,1/12,1/2,7/12} since exp(πi), exp(πi/6), exp(-πi/6), exp(5πi/6), exp(-5πi/6) are the roots of the associated polynomial (X)=1 + X + X 6 + X 7 Z n =

38 Minkowski/Hajos Problem and uglede onjecture Minkowski/Hajos Problem uglede Spectral onjecture? In any simple lattice tiling of the n-dimensional uclidean space by unit cubes, some pair of cubes must share a complete (n-1)-dimensional face subset of the n-dimensional uclidean space tiles by translation iff it is spectral R. Tijdeman: ecomposition of the Integers as a direct sum of two subsets, Number Theory, ambridge University Press, The fundamental Lemma: tiles Z n => p tiles Z n when <p,n>=1. oven &. Meyerowitz: Tiling the integers with translates of one finite set, J. lgebra, 212, pp , 1999 T1 + T2 => tile Tile => T1 I. Laba : The spectral set conjecture and multiplicative properties of roots of polynomials, J. Lond Math Soc, 2002 T1 + T2 => spectral T2 => spectral spectral => T1

39 Polynomial Representations of Tiling anons Z n 1 + X + X X n-1 ={0,5,6,7} (X)=1 + X 5 + X 6 + X 7 ={0,4,8} (X)=1 + X 4 + X 8 Δ 12 = 1 + X + +X 11 = (X) (X) mod X 12-1

40 Using cyclotomic polynomials to build tiling canons Φ 1 (X) = X - 1 Φ 2 (X) = 1 + X Φ 3 (X) = 1 + X + X 2 Φ 4 (X) = 1 + X 2 Φ 5 (X) = 1 + X + X 2 + X 3 + X 4 Φ 6 (X) = 1 - X + X 2? {0,1} {0,1,2} {0,2} {0,1,2,3,4}? Δ n = 1 + X + X X n-1 = Φ d (X) Δ 4 = 1 + X + X 2 + X 3 = Φ 2 (X) Φ 4 (X) d n d 1

41 ood and bad decompositions Δ n = 1 + X + X X n-1 = Φ d (X) d n d 1 Φ 2 (X) = 1 + X Φ 3 (X) = 1 + X + X 2 Φ 4 (X) = 1 + X 2 Φ 6 (X) = 1 - X + X 2 Δ 12 = 1 + X + +X 11 = Φ 2 Φ 3 Φ 4 Φ 6 Φ 12 (X) = Φ 2 Φ 3 Φ 6 Φ 12 = 1 + X + X 4 + X 5 + X 8 + X 9 (X) = Φ 4 = 1 + X 2 S = {0, 2} Φ 2 (X) = 1 + X Φ 3 (X) = 1 + X + X 2 oven & Meyerowitz conditions R = {0, 1, 4, 5, 8, 9} (T1) (1) = 6 = Φ 2 (1) Φ 3 (1) = 2 3 (T2) Φ 2 (X) et Φ 3 (X) Φ 2 3 (X)

42 oven and Meyerowitz onditions. oven &. Meyerowitz : Tiling the integers with translates of one finite set, J. lgebra, 212, pp , 1999

43 bad decomposition Δ n = 1 + X + X X n-1 = Φ d (X) d n d 1 Φ 2 (X) = 1 + X Φ 3 (X) = 1 + X + X 2 Φ 4 (X) = 1 + X 2 Φ 6 (X) = 1 - X + X 2 Δ 12 = 1 + X + +X 11 = Φ 2 Φ 3 Φ 4 Φ 6 Φ 12 (X) = Φ 2 Φ 3 Φ 12 = 1 +2X +X 2 X 3 X 4 + X 5 + 2X 6 + X 7 (X) = Φ 4 Φ 6 = 1 X +2X 2 X 3 +X 4 oven & Meyerowitz conditions Φ 2 (X) = 1 + X S =? Φ 3 (X) = 1 + X + X 2 R =? (T1) (1) = 7 Φ 2 (1) Φ 3 (1) = 2 3 (T2) Φ 2 (X) et Φ 3 (X) Φ 2 3 (X)

44 &M onditions, tiling and spectrality &M (1999) Laba (2002) miot (2009) T1 + T2 => tiling tiling => T1 tiling Z n where n has at most two prime factors => T1 + T2 T1 + T2 => spectral T2 => spectral spectral => T1 Tiling of a Hajos group => T2. This means that if tiles Z n without being spectral => is the rhythm of a Vuza anon. miot, New perspectives on rhythmic canons and the spectral conjecture, Special Issue Tiling Problems in Music, Journal of Mathematics and Music, July 2009, 3(2). Is condition T2 mandatory for tiling?

45 Vuza anons of period n n=p 1 p 2 n 1 n 2 n 3 <p 1 n 1, p 2 n 2 >=1 n 3 > n=72 Tijdeman s undamental Lemma : R tiles Z n => ar tiles Z n <a,n>=1 onclusion: There are only two «types» of Vuza anons of period 72 (up to an affine transformation)

46 Tiling and iscrete ourier Transform (. Lewin, JMT, 1958)

47 short (and elegant) proof of abbitt s Theorem hexacord and its complement have the same interval content I = [4, 3, 2, 3, 2, 1 ] = [4, 3, 2, 3, 2, 1 ] = I. miot : «Une preuve élégante du théorème de abbitt par transformée de ourier discrète», Quadrature, 61, 2006.

48 Z-relation, homometry and phase retrieval problem " Two sets are Z-related if they have the same module of the T Z-relation homometry Mandereau J.,. hisi,. miot, M. ndreatta,. gon, (2011), «Z-relation and homometry in musical distributions», Journal of Mathematics and Music, vol. 5, n 2, p Mandereau J,. hisi,. miot, M. ndreatta,. gon (2011), «iscrete phase retrieval in musical structures», Journal of Mathematics and Music, vol. 5, n 2, p

49 Paradigmatic classification and homometry Is there a (non-trivial) group action whose orbits are the equivalence classes of homometric sets? Is there an enumeration formula for homometric sets? John Mandereau, Étude des ensembles homométriques et leur application en théorie mathématique de la musique et en composition assistée par ordinateur, Master Thesis, TIM, Ircam/Université Paris 6, juin 2009 [ ]

50 High-order interval content: Lewin s mv k vector aniele hisi, "rom Z-relation to homometry: an introduction and some musical examples", Séminaire MaMuX, 10 décembre 2010 [

51 High-order Z-relation and k-homometric nesting Z 18 = {0, 1, 2, 3, 5, 6, 7, 9, 13} = {0, 1, 4, 5, 6, 7, 8, 10, 12} mv 3 ()= mv 3 () 3-homometry ( 2-homometry) Where does this nesting stop?

52 The extended phase retrieval and related open problems " The reconstruction index R(n) is the minimum integer k for which there exist no k-homometric sets in Z n (i.e. mv 3 provides enough information to reconstruct the set) Z 36 = {0, 1, 2, 3, 4, 5, 7, 10, 12, 15, 19, 20, 22, 23, 24, 25, 27, 28} = {0, 1, 2, 3, 4, 5, 6, 9, 14, 17, 18, 19, 21, 22, 24, 26, 27, 29} mv 4 ()= mv 4 () 4-homometry

53 Homometry and Tiling Rhythmic anons. miot, «New Perspectives on Rhythmic anons and the Spectral onjecture», JMM, 2009.

54 Homometry and Tiling Rhythmic anons. miot, «New Perspectives on Rhythmic anons and the Spectral onjecture», JMM, 2009.

55 Homometry and Tiling Rhythmic anons. miot, «New Perspectives on Rhythmic anons and the Spectral onjecture», JMM, 2009.

56 Thank you for your attention!