Exploring the mathemusical dynamics: some aspects of a musically driven mathematical practice

Exploring the mathemusical dynamics: some aspects of a musically driven mathematical practice C E B F A Istituto Veneto di Lettere Scienze e Arti G F# B C# March 31, 2017 Moreno Andreatta Music Representations Team IRCAM/CNRS/UPMC & IRMA/USIAS Strasbourg http://www.ircam.fr/repmus.html

The double movement of a mathemusical activity MATHEMATICS Mathematical statement generalisation General theorem formalisation OpenMusic application MUSIC Musical problem Music analysis Music theory Composition

The double movement of a mathemusical activity MATHEMATICS Mathematical statement generalisation General theorem OpenMusic formalisation application MUSIC Musical problem Music analysis Music theory Composition

The double movement of a mathemusical activity MATHEMATICS Mathematical statement generalisation General theorem formalisation OpenMusic application MUSIC Musical problem Music analysis Music theory Composition

Some examples of mathemusical problems M. Andreatta : Mathematica est exercitium musicae, Habilitation Thesis, IRMA University of Strasbourg, 2010 - 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

The interplay between algebra and geometry in music è http://agora2011.ircam.fr 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). è http://videotheque.cnrs.fr/

The galaxy of geometrical models at the service of music

The galaxy of geometrical models at the service of music

The circular representation of the pitch space Marin Mersenne la # la 10 9 8 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 0 1 2 3 4 5 6 7 8 9 10 11 12

The circular representation of the pitch space Marin Mersenne la # la sol # 10 9 8 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 0 1 2 3 4 5 6 7 8 9 10 11 12

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 10 11 la# si 0 do do# 30 1 re 2 0 1 2 3 4 5 6 7 8 9 10 11 12 9 la 0-(435) re# 3...or rotations! 8 sol# 7 sol fa# 6 fa mi 5 4

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 10 11 la# si 0 do do# 1 re 2 0 1 2 3 4 5 6 7 8 9 10 11 12 9 la 1-(435) re# 3...or rotations! 8 sol# 7 sol fa# 6 fa mi 5 4

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 10 11 la# si 0 do do# 1 re 2 R 0 1 2 3 4 5 6 7 8 9 10 11 12 9 la re# 3... or axial symmetries! I 4 8 sol# 7 sol fa# 6 fa mi 5 4

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 7 11 10 la# si 0 do 1 do# re 2 0 1 2 3 4 5 6 7 8 9 10 11 12 9 la re# 3 P... or axial symmetries! 8 sol# 7 sol fa# 6 fa mi 5 4

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 10 11 la# si I 11 0 do do# 1 re 2 0 1 2 3 4 5 6 7 8 9 10 11 12 9 la re# 3... or axial symmetries! 8 sol# 7 sol fa# 6 fa L mi 5 4

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

Minor thirds axis The Tonnetz (or honeycomb hexagonal tiling)

Minor thirds axis The Tonnetz (or honeycomb hexagonal tiling)

Minor thirds axis The Tonnetz (or honeycomb hexagonal tiling)

The Tonnetz (or honeycomb hexagonal tiling) Gilles Baroin Minor thirds axis

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

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

Harmonic progressions as spatial trajectories L R è Source : https://upload.wikimedia.org/wikipedia/commons/6/67/tonnetztorus.gif

Hamiltonian cycles and song writing G. Albini et S. Antonini, «Hamiltonian Cycles in the Topological Dual of the Tonnetz», MCM 2009, LNCS Springer

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 (1863-1938)

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) http://www.mathemusic.net min. 1 02 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.

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

From the OuLiPo to the OuMuPo (ouvroir de musique potentielle) http://oumupo.org/ Valentin Villenave Mike Solomon Jean-François Piette Martin Granger Joseph Boisseau Moreno Andreatta Tom Johnson

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

The geometric character of musical logic 0,5 Johann Sebastian Bach - BWV 328 2 3 3 4 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] 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 0,25 0 2-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

Spatial music analysis via Hexachord èhttp://www.lacl.fr/~lbigo/hexachord

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