Algebra and Geometry in Music and Musicology

Transcription

1 Algebra and Geometry in Music and Musicology Séminaire PMMH / ESPCI 9 Juin 2017 Moreno Andreatta Music Representations Team IRCAM / CNRS UMR 9912 / UPMC, Paris IRMA & GREAM, University of Strasbourg

2 The musical and scientific research at IRCAM...

3 ... at the interface between contemporary and pop music MusiqueLab 2 Orig. OMAX (computer-aided impro) Var. 1 Var. 2

4 The galaxy of geometrical models at the service of music

5 Bach s enigmatic canons and geometry Do

6 My end is my beginning (but twisted!)

7 [min ]

8 The galaxy of geometrical models at the service of music

9 The galaxy of geometrical models at the service of music

10 Mersenne and the birth of (musical) combinatorics Marin Mersenne, Harmonicorum Libri XII, 1648 Marin Mersenne

11 The circular representation of the pitch space Marin Mersenne la # la sol # sol si 11 7 do 0 6 fa # 1 2 fa Harmonicorum Libri XII, 1648 re mi dodo# re re# mi fa fa# sol sol# la la# si 5 do # 3 4 re # do è DEMO

12 The Pitch/Rhythm Isomorphism Bembé 12 The problem of concentric circles? Abadja or Bembé Dinner Table Problem

13 ME-sets and the Ising antiferromagnetic spin-1/2 model Jack Douthett & Richard Krantz, Energy extremes and spin configurations for the one-dimensional antiferromagnetic Ising model with arbitrary-range interaction, J. Math. Phys. 37 (7), July 1996 Jack Douthett & Richard Krantz, Maximally even sets and configurations: common threads in mathematics, physics, and music, J. Comb. Optm. 14, /2-1/2 ferromagnetic anti-ferromagnetic

14 African-cuban ME-rhythms El cinquillo El trecillo 8 8

15 The Discrete Fourier Transform in symbolic MIR Hexachord Theorem Homometric structures IC A = [4, 3, 2, 3, 2, 1 ] = [4, 3, 2, 3, 2, 1 ] = IC A E. Amiot : «Une preuve élégante du théorème de Babbitt par transformée de Fourier discrète», Quadrature, 61, 2006.

16 The Tonnetz (or honeycomb hexagonal tiling) Fifths axis Major thirds axis Speculum Musicum (Euler, 1773)

17 From music analysis to the sonifications of complex systems

18 Can you hear the shape of a tiling? Honeycomb Tonnetz Grain boundaries Disordered M. Potier, De la sonification à la «musification» de systèmes complexes, Master MPRI, Université Paris Diderot, 6 septembre 2012 (codirection avec Wiebke Drenckhan, LPS-CNRS-université Paris Sud)

19 Can you hear the transition towards the scaling state? Average cell area Easy to see Difficult to see Scaling state: <A> ~ t 1 Number n of neighbours of cell Iteration: Iteration (time)

20 Can you hear the transition towards the scaling state? Average cell area Easy to see Difficult to see Scaling state: <A> ~ t 1

21 Major thirds axis L P R R P L The Tonnetz (or honeycomb hexagonal tiling) Fifths axis Gilles Baroin Major thirds axis Speculum Musicum (Euler, 1773)

22 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 dodo# re re# mi fa fa# sol sol# la la# si do la# si 0 do do# 30 1 re la 0-(435) re# 3...or rotations! 8 sol# 7 sol fa# 6 fa mi 5 4

23 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 dodo# re re# mi fa fa# sol sol# la la# si do la# si 0 do do# 1 re la 1-(435) re# 3...or rotations! 8 sol# 7 sol fa# 6 fa mi 5 4

24 Musical inversions are differences... I : x -x mod 12 Do maj = {0,4,7} La min = {0,4,9} I 4 (x)=4-x dodo# 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

25 Musical inversions are differences... I : x -x mod 12 Do maj = {0,4,7} Do min = {0,3,7} I 7 (x)=7-x dodo# re re# mi fa fa# sol sol# la la# si do I la# si 0 do 1 do# re la re# 3... or axial symmetries! 8 sol# 7 sol fa# 6 fa mi 5 4

26 Musical inversions are differences... I : x -x mod 12 Do maj = {0,4,7} Mi min = {4,7,11} I 11 (x)=11-x dodo# 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

27 The Tonnetz, its symmetries and its topological structure P R L Minor third axis transposition Axe de tierces mineures R as relative? P as parallel L = Leading Tone

28 The Tonnetz, its symmetries and its topological structure P R L Minor third axis transposition Axe de tierces mineures R as relative P as parallel L = Leading Tone

29 The Tonnetz, its symmetries and its topological structure R P L Minor third axis transposition Axe de tierces mineures R as relative P as parallel è Source: Wikipedia L = Leading Tone

30 Symmetries in Paolo Conte s Madeleine La b Re b Si b Mi b Si Mi Re b Fa # Re Sol Mi La Re La b Re b Do Mi b Almost total covering of the major-chords space!

31 è Gilles Baroin

32 A zig-zag harmonic progression L R è Source :

33 è Gilles Baroin

34 Reading Beethoven backwards time

35 The collection of 124 Hamiltonian Cycles! G. Albini & S. Antonini, University of Pavia, 2008

36 Hamiltonian Cycles with inner periodicities L P L P L R... 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, in Poesie sparse) min 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. R L P Music: M. Andreatta Arrangement and mix: M. Bergomi & S. Geravini (Perfect Music Production) Mastering: A. Cutolo (Massive Arts Studio, Milan) 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.

37 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

38 From the OuLiPo to the OuMuPo (Ouvroir de Musique Potentielle) Valentin Villenave Mike Solomon Jean-François Piette Martin Granger Joseph Boisseau Moreno Andreatta Tom Johnson è

39 The geometric character of musical logic 0,5 Johann Sebastian Bach - BWV 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] Johann Sebastian Bach - BWV 328 K[2,3,7] K[2,4,6] random chords K[2,5,5] K[3,3,6] K[3,4,5] K[4,4,4] 0,5 Claude Debussy - Voiles 2-compactness 0, compactness 0,5 0,25 Schönberg - Pierrot Lunaire - Parodie 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 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] Schönberg - Pierrot Lunaire - Parodie random chords

40 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 P(Z n ) C E Intervallic structure major/minor triads B F A G K TI [3,4,5] C# F# B

41 The musical style...is the space! 3 2

42 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 Du rer with his Treatise of human proportions (Lévi-Strauss, conversation with Eribon, 1988).

43 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

44 Spatial music analysis via Hexachord è

45 Keeping the space...but changing the trajectory! è

46 Keeping the space...but changing the trajectory! Rotation (autour du do)

47 Rotational symmetry applied to traditional Brazilian music

48 Thank you for your attention!