Algebraic and topological models in computational music analysis!
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1 ! Algebraic and topological models in computational music analysis!! Centro Polifunzionale di Pordenone!! 22 Ottobre 2015!!! Moreno Andreatta! Equipe Représentations Musicales! IRCAM/CNRS/UPMC!
2 IRCAM = Institut de Recherche et Coordination Acoustique/Musique"
3 Acoustics, Signal Processing and Computer Sciences" Computer Science! Informatics mention! PARCOURS MASTER 2!!!!! Parcours multi-mentions du master Sciences et technologies! Université Pierre et Marie Curie! en collaboration avec l Ircam et Télécom ParisTech! (McGill University)! Coordination Ircam! Moreno Andreatta & Cyrielle Fiolet! Engineering Science! Acoustics mention /! (Laboratory of Mechanics and Acoustics / Marseille)!
4 The interplay between algebra and geometry in music " Concerning music, it takes place in time, like algebra. In mathematics, there is this fundamental duality between, on the one hand, geometry which corresponds to the visual arts, an immediate intuition and on the other hand algebra. This is not visual, it has a temporality. This fits in time, it is a computation, something that is very close to the language, and which has its diabolical precision. [...] And one only perceives the development of algebra through music (A. Connes).!
5 The double movement of a mathemusical activity MATHEMATICS& Mathema9cal& statement& generalisa9on& General& theorem& formalisa9on& OpenMusic applica9on& MUSIC& Musical& problem& Music& analysis& Music& theory& Composi9on&
6 Some examples of mathemusical problems M. Andreatta : Mathematica est exercitium musicae, Habilitation Thesis, IRMA University of Strasbourg, The construction of Tiling Rhythmic Canons - The Z relation and the theory of homometric sets - Set Theory and Transformational Theory - Neo-Riemannian Theory, Spatial Computing and FCA - Diatonic Theory and Maximally-Even Sets - Periodic sequences and finite difference calculus - Block-designs and algorithmic composition " Rhythmic Tiling Canons Z-Relation and Homometric Sets Neo-Riemannian Theory and Spatial Computing Finite Difference Calculus Set Theory, andtransformation Theory Diatonic Theory and ME-Sets Block-designs
![A " IC A = [4, 3, 2, 3, 2, 1 ]" ="](/docs-images/92/109779154/images/7-1.jpg "[4, 3, 2, 3, 2, 1 ] = IC A \" Babbitt")
7 A Mathemusical Theorem! "! A" Z-relation IC A =IC A'" homometry A " IC A = [4, 3, 2, 3, 2, 1 ]" =" [4, 3, 2, 3, 2, 1 ] = IC A " Babbitt s Hexachord Theorem:" A hexacord and its complement have the same interval content" (Proofs by Wilcox, Ralph Fox (?), Chemillier, Lewin, Mazzola, Schaub,, Amiot, )"
8 ! Math n Pop Project: formal models for and within popular music!! " Algebraic models Signal-based Music Information Retrieval Computational models Cognitive models Semiotic models Topological models Mathematical models Symbolic Music Information Retrieval Andreatta M. (2014), «Modèles formels dans et pour la musique pop, le jazz et la chanson : introduction et perspectives futures», dans Esthétique & Complexité : Neurosciences, Philosophie et Art, Z. Kapoula, L.-J. Lestocart, J.-P. Allouche éds., éditions du CNRS, 2014
9
10 Octave reduction and mod 12 congruence! " do do# re re# mi fa fa# sol sol# la la# si do do# re 0 11 do si 10 la# do# 1 re la Mersenne" 1648" re# 3 8 sol# 7 sol fa# 6 fa mi 5 4
11 A musical scale as a polygon in a circle! " Do maj = {0,2,4,5,7,9,11} do do# ré ré# mi fa fa# sol sol# la la# si (do) la# si 0 do do# 1 ré (12) 9 la 0-( )" ré# 3 8 sol# mi 4 A. Riotte M. Mesnage 7 sol fa# 6 fa 5
12 Scales and symmetry! " Do maj = {0,2,4,5,7,9,11} do do# ré ré# mi fa fa# sol sol# la la# si (do) la# si 0 do do# 1 ré (12) 9 la 0-( )" ré# 3 8 sol# mi 4 Camille Durutte 7 sol fa# 6 fa 5
13 The diatonic bell (P. Audétat & co.)" Junod, J., Audétat, P., Agon, C., Andreatta, M., «A Generalisation of Diatonicism and the Discrete Fourier Transform as a Mean for Classifying and Characterising Musical Scales», Second International Conference MCM 2009, vol. 38, New Haven, 2009, pp "
14 The pitch-rhythm cognitive isomorphic correspondence" cyclic permutations... J. Pressing, Cognitive isomorphisms between pitch and rhythm in world musics: West Africa, the Balkans and Western tonality, Studies in Music, 17, p !
15 Musical transpositions are additions...! " T k : x x+k mod 12 Do maj = {0,2,4,5,7,9,11} Do# maj = {1,3,5,6,8,10,0} +1 do do# re re# mi fa fa# sol sol# la la# si do la# si 0 do 30 do# 1 re la 0-(435)" re# 3 8 sol# mi 4...or rotations!" 7 sol fa# 6 fa 5
16 Musical transpositions are additions...! " T k : x x+k mod 12 Do maj = {0,2,4,5,7,9,11} Do# maj = {1,3,5,6,8,10,0} +1 do do# re re# mi fa fa# sol sol# la la# si do la# si 0 do do# 1 re la 1-(435)" re# 3 8 sol# mi 4...or rotations!" 7 sol fa# 6 fa 5
17 Musical inversions are differences...! " I : x -x mod 12 Do maj = {0,4,7} La min = {0,4,9} I 4 (x)=4-x do do# re re# mi fa fa# sol sol# la la# si do la# si 0 do do# 1 re la re# 3... or axial symmetries!" I 4 8 sol# 7 sol fa# 6 fa mi 5 4
18 Musical inversions are differences...! " I : x -x mod 12 Do maj = {0,4,7} Do min = {0,3,7} I 7 (x)=7-x do do# re re# mi fa fa# sol sol# la la# si do I la# si 0 do do# 1 re la re# 3... or axial symmetries!" 8 sol# 7 sol fa# 6 fa mi 5 4
19 Musical inversions are differences...! " I : x -x mod 12 Do maj = {0,4,7} Mi min = {4,7,11} I 11 (x)=11-x do do# re re# mi fa fa# sol sol# la la# si do la# si I 11 0 do do# 1 re la re# 3... or axial symmetries!" 8 sol# 7 sol fa# 6 fa mi 5 4
20 Composing Transposition and Inversion operators" I: x 12-x T k : x k+x {0, 5, 9, 11} T 11 I: x 11-x {11, 6, 3, 0}
21 «Entités formelles pour l analyse musicale» " Marcel Mesnage (1998)" A. Schoenberg : Klavierstück Op. 33a, 1929 T 3 T 1 I T 1 I
22 Equivalence relation between musical structures" Transposition Transposition and/or inversion T 3 {0, 4, 7} = 3+{0, 4, 7} = {3, 7, 10} T 3 I{0, 4, 7} = 3+{0, -4, -7} = {3, 11, 8} Multiplication (or affine transformation) M 5 {0, 4, 7} = 5 {0, 4, 7} = {0, 8, 11}
23 Equivalence classes of chords (up to a group action) Z 12 = < T k (T k ) 12 = T 0 > D 12 = < T k, I (T k ) 12 =I 2 =T 0, ITI=I(IT) -1 > Aff = { f f (x)=ax+b, a (Z 12 )*, b Z 12 } Cyclic group Dihedral group Paradigmatic architecture Affine group
24 Group actions and the classification of musical structures" Cyclic group Dihedral group Zalewski / Vieru / Halsey & Hewitt" Forte/ Rahn Carter" Morris / Mazzola Estrada" 77" 158" 224" 352" F. Klein" W. Burnside" Affine group Symmetric group G. Polya" 1" 2" 3" 4" 5" 6" 7" 8" 9" 10" 11" 12" 1" 6" 19" 43" 66" 80" 66" 43" 19" 6" 1" 1" 1" 6" 12" 29" 38" 50" 38" 29" 12" 6" 1" 1" 1" 5" 9" 21" 25" 34" 25" 21" 9" 5" 1" 1" 1! 6! 12! 15! 12! 11! 7! 5! 3! 2! 1! 1!!
25 Permutations are partitions..." DIA= (2,2,1,2,2,2,1) DIA E = (1,1,2,2,2,2,2) do do# ré ré# mi fa fa# sol sol# la la# si (do) la# si 0 do 1 do# ré (12) 9 la 0-( )" ré# 3... mathematically speaking!" 8 sol# 7 sol fa# 6 fa mi 5 4
26 The permutohedron as a combinatorial space" Julio Estrada, Théorie de la composition : discontinuum continuum, université de Strasbourg II, 1994 Julio Estrada" DIA E = (1,1,2,2,2,2,2) J. Estrada"
27 The permutohedron as a musical conceptual space" L. Van Beethoven, Quatuor n 17
28 The permutohedron as a musical conceptual space" B. Bartok, Quartet n 4 (3 d movement) A. Schoenberg, Six pieces op. 19
29 Towards an OpenMusic computational model "
30 The permutohedron as a lattice of formal concepts" T. Schlemmer, M. Andreatta, «Using Formal Concept Analysis to represent Chroma Systems», MCM 2013, McGill Univ., Springer, LNCS.
31 Permutohedron and Tonnetz: a structural inclusion" R P L R P L (3 5 4) (5 3 4) (5 4 3) (4 5 3) (4 3 5) (3 4 5)
32 P R L The Tonnetz as a topological structure! Axis of minor thirds Axe de tierces mineures Speculum Musicum (Euler, 1773)"?
33 P R L The Tonnetz as a topological structure: a torus" Axis of minor thirds Axe de tierces mineures Speculum Musicum (Euler, 1773)" Torus
34 P R L The Tonnetz and its symmetries " Axis of minor thirds Axe de tierces mineures
35 Symmetries in Frank Zappa s music" Fa la m La b Sol Ré fa# m Fa Mi Si la# m Ré Ré b La b do m Si Si b «Easy Meat» (Frank Zappa)" min !
36 Symmetries in Paolo Conte s Madeleine Lab Réb/Fa Sib 7 Mib 7 /Réb S. La Via, Poesia per musica e musica per poesia. Dai trovatori a Paolo Conte, Carocci, 2006 Lab Réb/Fa Sib 7 Mib 7 /Réb Si/Ré# Mi Do# Fa# Ré/La Sol Mi 7 La 7 Ré Lab 7 Réb Do 7 Mib
37 Symmetries in Paolo Conte s Madeleine Lab Réb/Fa Sib 7 Mib 7 /Réb S. La Via, Poesia per musica e musica per poesia. Dai trovatori a Paolo Conte, Carocci, 2006 Lab Réb/Fa Sib 7 Mib 7 /Réb Si/Ré# Mi Do# Fa# Ré/La Sol Mi 7 La 7 Ré Lab 7 Réb Do 7 Mib
38 Symmetries in Paolo Conte s Madeleine Lab Réb/Fa Sib 7 Mib 7 /Réb S. La Via, Poesia per musica e musica per poesia. Dai trovatori a Paolo Conte, Carocci, 2006 Lab Réb/Fa Sib 7 Mib 7 /Réb Si/Ré# Mi Do# Fa# Ré/La Sol Mi 7 La 7 Ré Lab 7 Réb Do 7 Mib
39 Symmetries in Paolo Conte s Madeleine Lab Réb/Fa Sib 7 Mib 7 /Réb S. La Via, Poesia per musica e musica per poesia. Dai trovatori a Paolo Conte, Carocci, 2006 Lab Réb/Fa Sib 7 Mib 7 /Réb Si/Ré# Mi Do# Fa# Ré/La Sol Mi 7 La 7 Ré Lab 7 Réb Do 7 Mib
40 Madeleine s spatial trajectory La b Ré b Si b Mi b Si Mi Ré b Fa # Ré Sol Mi La Ré La b Ré b Do Mi b!
41 Partial covering of the Tonnetz Lab Réb/Fa Sib 7 Mib 7 /Réb Missing major chord S. La Via, Poesia per musica e musica per poesia. Dai trovatori a Paolo Conte, Carocci, 2006 Lab Réb/Fa Sib 7 Mib 7 /Réb Si/Ré# Mi Do# Fa# Ré/La Sol Mi 7 La 7 Ré Lab 7 Réb Do 7 Mib
42 Aprile, a Hamiltonian «decadent» song Do do m Sol# fa m Fa la m La fa# m Fa# sib m Do# do# m La mi m Sol si m Ré ré m Sib sol m Mib mib m Si sol# m Mi G. D Annunzio ( )!
43 M. Andreatta, «Math n pop : symétries et cycles hamiltoniens en chanson», Tangente
44 Hamiltonian Cycles with inner periodicities L P L P L R LPLPLR... P L P L R L... L P L R L P... P L R L P L... L R L P L P... R L P L P L... La sera non è più la tua canzone (Mario Luzi, 1945, tratto da Poesie sparse) R L P La sera non è più la tua canzone, è questa roccia d ombra traforata dai lumi e dalle voci senza fine, la quiete d una cosa già pensata. Ah questa luce viva e chiara viene solo da te, sei tu così vicina al vero d una cosa conosciuta, per nome hai una parola ch è passata nell intimo del cuore e s è perduta. Caduto è più che un segno della vita, riposi, dal viaggio sei tornata dentro di te, sei scesa in questa pura sostanza così tua, così romita nel silenzio dell essere, (compiuta). L aria tace ed il tempo dietro a te si leva come un arida montagna dove vaga il tuo spirito e si perde, un vento raro scivola e ristagna.
45 Extract of the 2 nd movement of the Symphony No. 9 (L. van Beethoven)" C# F# B B E G D G C B E A E# G# C# C F B b
46 time Math n Pop R L
47 The use of constraints in arts" OuLiPo (Ouvroir de " Littérature Potentielle)" Georges Perrec Cent mille milliards de poèmes, 1961 La vie mode d emploi, Raymond Queneau Italo Calvino Il castello dei destini incrociati, 1969
48 The Tonnetz as a simplicial complex L. Bigo, Représentation symboliques musicales et calcul spatial, PhD, Ircam / LACL, 2013 Assembling chords related by some equivalence relation" Transposition/inversion: Dihedral group action on )" C! E! Intervallic structure major/minor triads" B! F! A G! K TI [3,4,5]" C# F#! B!
![- 1998]" Classifying Chord Complexes L.](/docs-images/92/109779154/images/49-2.jpg "Bigo, Représentation symboliques musicales et")
49 Complexes enumeration in the chromatic system" K TI [3,4,5]" [Cohn 1997]" K TI [2,3,3,4]" [Gollin ]" Classifying Chord Complexes L. Bigo, Représentation symboliques musicales et calcul spatial, PhD, Ircam / LACL, 2013 B! G! C! F#! F! E! B! A! C#! K T [2,2,3]" [Mazzola 2002]" " " " " "
![Analyzing harmonic progressions as paths in a generic Tonnetz? T[1,2,9] T[3,4,7] L. Bigo, M. Andreatta, J.-L. Giavitto, O. Michel, A.](/docs-images/92/109779154/images/50-4.jpg "Spicher, «Computation and Visualization of Musical Structures in Chord-based Simplicial Complexes», MCM 2013, McGill University, Springer, LNCS. Bigo L.")
50 Analyzing harmonic progressions as paths in a generic Tonnetz? T[1,2,9] T[3,4,7] L. Bigo, M. Andreatta, J.-L. Giavitto, O. Michel, A. Spicher, «Computation and Visualization of Musical Structures in Chord-based Simplicial Complexes», MCM 2013, McGill University, Springer, LNCS. Bigo L., D. Ghisi, A. Spicher, M. Andreatta (2014), Proceedings ICMC SMC 2014, Sept. 2014, Athens (revised and enlarged version forthcoming in Computer Music Journal, 39(3), 2015) Bigo L., M. Andreatta, «Musical analysis with simplicial chord spaces», in D. Meredith (ed.), Computational Music Analysis, Springer (in press)
51 Analyzing harmonic progressions as paths in Hexachord demo
![K[1,1,10] K[1,2,9] K[1,3,8] K[1,4,7] K[1,5,6] K[2,2,8] K[2,3,7] K[2,4,6] K[2,5,5] K[3,3,6] K[3,4,5] K[4,4,4] Johann Sebastian Bach - BWV 328](/docs-images/92/109779154/images/52-3.jpg "random chords 0,5 Claude Debussy - Voiles 0,5 Schönberg - Pierrot Lunaire - Parodie 2-compactness 0,25 2-compactness 0,25 0 0 K[1,1,10]")
![K[1,2,9] K[1,3,8] K[1,4,7] K[1,5,6] K[2,2,8] K[2,3,7] K[2,4,6] K[2,5,5] K[3,3,6] K[3,4,5] K[4,4,4] K[1,1,10] K[1,2,9] K[1,3,8] K[1,4,7]](/docs-images/92/109779154/images/52-4.jpg "K[1,5,6] K[2,2,8] K[2,3,7] K[2,4,6] K[2,5,5] K[3,3,6] K[3,4,5] K[4,4,4] Claude Debussy - Voiles random chords Schönberg - Pierrot Lunaire -")
52 The spatial character of the «musical style» Bigo L., M. Andreatta, «Musical analysis with simplicial chord spaces», in D. Meredith (ed.), Computational Music Analysis, Springer (in press) C 0,5 Johann Sebastian Bach - BWV 328 E G B 2-compactness 0,25 T[2,3,7] T[3,4,5] 0 K[1,1,10] K[1,2,9] K[1,3,8] K[1,4,7] K[1,5,6] K[2,2,8] K[2,3,7] K[2,4,6] K[2,5,5] K[3,3,6] K[3,4,5] K[4,4,4] Johann Sebastian Bach - BWV 328 random chords 0,5 Claude Debussy - Voiles 0,5 Schönberg - Pierrot Lunaire - Parodie 2-compactness 0,25 2-compactness 0, K[1,1,10] K[1,2,9] K[1,3,8] K[1,4,7] K[1,5,6] K[2,2,8] K[2,3,7] K[2,4,6] K[2,5,5] K[3,3,6] K[3,4,5] K[4,4,4] K[1,1,10] K[1,2,9] K[1,3,8] K[1,4,7] K[1,5,6] K[2,2,8] K[2,3,7] K[2,4,6] K[2,5,5] K[3,3,6] K[3,4,5] K[4,4,4] Claude Debussy - Voiles random chords Schönberg - Pierrot Lunaire - Parodie random chords
53 The spatial character of the «musical style» Beethoven, 2 nd mouvement of the 9 e Symphony T[3,4,7] Babbitt, Semi-Simple Variations T[1,2,9]
54 The «shape» of space distributions in jazz standards Thelonious Monk, Brilliant Corners Chick Corea, Eternal Child Bill Evans, Turn Out the Stars
55 Beatles natural Space and stylistic embeddings The Beatles, Hey Jude
56 A trajectory realized in different support spaces 3 2
57 The morphological vs the mathematical genealogy of the structuralism" [The notion of transformation] comes from a work which played for me a very important role and which I have read during the war in the United States : On Growth and Form, in two volumes, by D'Arcy Wentworth Thompson, originally published in The author (...) proposes an interpretation of the visible transformations between the species (animals and vegetables) within a same gender. This was fascinating, in particular because I was quickly realizing that this perspective had a long tradition: behind Thompson, there was Goethe s botany and behind Goethe, Albert Dürer with his Treatise of human proportions (Lévi-Strauss, conversation with Eribon, 1988).
58 Musically interesting Trajectory Transformations Isomorphism from a support space to a different one Beatles, Hey Jude T[3,4,7]?????? T[2,3,7] L. Bigo, Représentation symboliques musicales et calcul spatial, PhD, Ircam / LACL, 2013??
59 Musically interesting Trajectory Transformations Automorphism of the support space F major chord inversion C minor chord Rotation (autour du do) Beatles, Hey Jude (orig. version) T[3,4,7] Beatles, Hey Jude (transformed version) T[3,4,7]!Hexachord "
60 Musically interesting Trajectory Transformations The M transformation T[3,4,7] T[2,3,7] M M M T[3,4,7]
61 Signal/Symbolic articulation in MIR = Ionian mode Locrian mode M. Bergomi, Dynamics and Algebraic Topology Tools for Music in the Symbolic/Signal interaction domain, ongoing PhD! Towards a geometric dynamic modeling of a musical piece? SPACE MUSIC! Towards a topological signature of a musical piece??
62 Acotto E. et M. Andreatta (2012), «Between Mind and Mathematics. Different Kinds of Computational Representations of Music», Mathematics and Social Sciences, n 199, 2012(3), p Neurosciences and Tonnetz
63 Thank you for your attention!" Hexachord (by Louis Bigo, 2013)
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