The interplay between algebra and geometry in symbolic music information retrieval  Moreno Andreatta Equipe Représentations Musicales IRCAM/CNRS/UPMC http://www.ircam.fr/repmus.html

La sera non è più la tua canzone (by Mario Luzi, Poesie sparse, 1945) 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.

M. Luzi (1914-2005)

Music: M. Andreatta Arrangements and mixage: M. Bergomi & S. Geravini (Perfect Music Production) Mastering: A. Cutolo (Massive Arts Studio, Milan)

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). è http://videotheque.cnrs.fr/

è http://agora2011.ircam.fr

The double movement of a ‘mathemusical’ activity MATHEMATICS  

MUSIC  

Musical   problem  

generalisa9on  

OpenMusic

Music   analysis  

General   theorem   applica9on  

formalisa9on  

Mathema9cal   statement  

Music   theory   Composi9on  

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

Finite Difference Calculus Neo-Riemannian Theory and Spatial Computing

Set Theory, andTransformation Theory

Diatonic Theory and ME-Sets

Block-designs

è AULA 3 dia 22/8

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

ç

Rhythmic Tiling Canons Z-Relation and Homometric Sets

Finite Difference Calculus Neo-Riemannian Theory and Spatial Computing

Set Theory, andTransformation Theory

Diatonic Theory and ME-Sets

Block-designs

è Focus on ATIAM, dia 29/8

Music & mathematics: “prima la musica”!

Pythagora’s monochord, VIe-Ve Century b. C

Mersenne’s Harmonicorum Libri XII, 1648

Euler’s Speculum musicum, 1773

Iannis Xenakis, Musique. Architecture, Tournai, Casterman, 1971, (New, revised edition: Tournai, Casterman, 1976)

Algebra/geometry in Twelve-Tone Music

U

Felix Klein

K

Inversion

Inversion

G

Retrogradation

Retrogradation

Ernst Krenek

Milton Babbitt

“[…] If we represent the permutations G, K, U and KU in a geometric way, we obtain the following figures whose properties are evident for the eyes as well as for the ears. […] If one studies the thoughts by mathematicians and physicists of our time on the new conceptual structures (logical, mathematical, physical, ...) one can mesure, for sure, which big path the musicians have to do before arriving at the stage of a general synthesis.”









KU



Schoenberg, Editions

Main d’Œuvre,

1997. Orig. 1963)



Pierre Barbaud

Combinatorics and axiomatic methods in music

Josef-Mathias Hauer

Marin Mersenne,

Harmonicorum Libri XII, 1648

Combinatorics Axioms Physicists and mathematicians are far in advance of musicians in realizing that their respective sciences do not serve to establish a concept of the universe conforming to an objectively existent nature. As the study of axioms eliminates the idea that axioms are something absolute, conceiving them instead as free propositions of the human mind, just so would this musical theory free us from the concept of major/minor tonality […] as an irrevocable law of nature. Ernst Krenek : Über Neue Musik, 1937 (Engl. Transl. Music here and now, 1939)

Ernst Krenek David Hilbert

Octave reduction and mod 12 congruence 



…

…

…

?

… …

0

do do# ré ré# mi fa fa# sol sol# la la# si do do# re

1

11 si 10 0

1 2

3 4

5

6

7

8

9 10 11 12

9

do

do#

2 ré

la#

ré#

la

8

sol#

mi sol

fa#

7 6

fa 5

3

4

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)

0 1

11 si 10 0

1 2

3 4

5

6

7

8

9 10 11 (12)

9

M. Mesnage

2 ré

0-(2212221)

sol#

ré# mi

sol A. Riotte

do#

la# la

8

do

fa#

7 6

fa 5

3

4

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)

0 1

11 si 10 0

1 2

3 4

5

6

7

8

9 10 11 (12)

9

2 ré

0-(2212221)

sol#

ré# mi

sol

fa#

7 Camille Durutte

do#

la# la

8

do

6

fa 5

3

4

The diatonic bell (P. Audétat & co.)

http://www.cloche-diatonique.ch/

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. 166-179

Musical transpositions are additions... 



…

…

Do maj = {0,2,4,5,7,9,11} +1 Do# maj = {1,3,5,6,8,10,0} …

…

do do# ré ré# mi fa fa# sol sol# la la# si do

0 1

11 si 10 0

1 2

3 4

5

6

7

8

9 10 11 12

9

do#

30°

la#

2 ré ré#

la

8

sol#

mi sol

... or rotations!

do

fa#

7 6

fa 5

3

4

Inversions are subtractions... 



…

…

Do maj = {0,4,7} La min = {0,4,9} …

…

do do# ré ré# mi fa fa# sol sol# la la# si do

0 1

11 si 10 0

1 2

3 4

5

6

7

8

9 10 11 12

9

I4(x)=4-x

do

do# ré

la#

ré#

la sol#

I4

... or axial symmetries!

2

8

mi sol

fa#

7 6

fa 5

3

4

Inversions are subtractions... 



…

…

Do maj = {0,4,7} Do min = {0,3,7} …

…

do do# ré ré# mi fa fa# sol sol# la la# si do

0 1

11 si

10

I7 0

1 2

3 4

5

6

7

8

9 10 11 12

do

do#

2 ré

la#

ré#

la

9

8

sol#

mi sol

... or axial symmetries!

I7(x)=7-x

fa#

7 6

fa 5

3

4

Inversions are subtractions... 



…

…

Do maj = {0,4,7} Mi min = {4,7,11} …

…

do do# ré ré# mi fa fa# sol sol# la la# si do

I11 si

0

1 2

3 4

5

6

7

8

9 10 11 12

do

do#

2 ré

la#

ré#

la

9

8

sol#

mi sol

... or axial symmetries!

0 1

11 10

I11(x)=11-x

fa#

7 6

fa 5

3

4

The equal tempered space is a cyclic group



…

…

…

The generators of the cyclic group of order 12 are the transpositions T1 , T5 , T7 et T11

…

do do# ré ré# mi fa fa# sol sol# la la# si do do# ré

0

1 2

3 4

Z12 = < T1 | (T1

)12

5

6

= T0 >

7

8

9 10 11 12

Z12

The equal tempered space is a cyclic group



…

…

…

The generators of the cyclic group of order 12 are the transpositions T1 , T5 , T7 et T11

…

do do# ré ré# mi fa fa# sol sol# la la# si do do# ré

0

1 2

3 4

)12

5

6

7

Z12 = < T1 | (T1 = T0 > = = < T5 | (T5)12 = T0 >

8

9 10 11 12

Z12

The equal tempered space is a cyclic group



…

…

…

The generators of the cyclic group of order 12 are the transpositions T1 , T5 , T7 et T11

…

do do# ré ré# mi fa fa# sol sol# la la# si do do# ré

0

1 2

3 4

)12

5

6

7

Z12 = < T1 | (T1 = T0 > = = < T5 | (T5)12 = T0 > = = < T7 | (T7)12 = T0 >

8

9 10 11 12

Z12

The equal tempered space is a cyclic group



…

…

…

The generators of the cyclic group of order 12 are the transpositions T1 , T5 , T7 et T11

…

do do# ré ré# mi fa fa# sol sol# la la# si do do# ré

0

1 2

3 4

)12

5

6

7

Z12 = < T1 | (T1 = T0 > = = < T5 | (T5)12 = T0 > = = < T7 | (T7)12 = T0 > = = < T11 | (T11)12 = T0 >

8

9 10 11 12

Z12

From the circular representation to the Tonnetz

P R L

Axis of minor thirds

Speculum Musicum (Euler, 1773)

Axe de tierces mineures

?

The Tonnetz construction

(animation by Gilles Baroin)

http://www.mathemusic.net

Building Chord Complexes 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(Zn)

… C

E

Intervallic structure

major/minor triads

F

B

KTI[3,4,5]

G

F#

B♭

A

C#

Classifying Chord Complexes L. Bigo, Représentation symboliques musicales et calcul spatial, PhD, Ircam / LACL, 2013

•  Complexes enumeration in the chromatic system

C

E

KTI[3,4,5]

B

[Cohn – 1997]

A

F

C#

G

F#

B♭

KTI[2,3,3,4]

[Gollin - 1998]

KT[2,2,3]

[Mazzola – 2002]











…

?

Classifying Chord Complexes L. Bigo, Représentation symboliques musicales et calcul spatial, PhD, Ircam / LACL, 2013

•  Complexes enumeration in the chromatic system

C

E

KTI[3,4,5]

B

[Cohn – 1997]

F

C#

G

F#

KTI[2,3,3,4]

[Gollin - 1998]

KT[2,2,3]

[Mazzola – 2002]











…

A

B♭

Spatial symmetries in pop music

Guy Capuzzo, "Neo-Riemannian Theory and the Analysis of Pop-Rock Music", Music Theory Spectrum 26(2), 177-199, 2004

d

Db

f

Bb

Shake the disease - 1985

(Depeche Mode) – min. 2’17’’

d

Bb

f

Db

Trajectories and harmonic progressions in the Tonnetz

Guy Capuzzo, "Neo-Riemannian Theory and the Analysis of Pop-Rock Music", Music Theory Spectrum 26(2), 177-199, 2004

d

RP

f

L

L

Db

RP

Bb

Shake the

disease - 1985

(Depeche Mode)

Sequence: RPLRPL

RP RP

L P R P L

RP R

L min. 0’33’’ Hexachord (by Louis Bigo, 2013)

LR

P

RL R

LR

LR

P

PRP

Symmetries in Frank Zappa’s music

[Guy Capuzzo, Music Theory Spectrum, 2004]

« Easy Meat » - 1981 (Frank Zappa)

min. 1’44’’ – 2’39’’

Symmetries in Frank Zappa’s music: the generating cell

Guy Capuzzo, "Neo-Riemannian Theory and the Analysis of Pop-Rock Music", Music Theory Spectrum 26(2), 177-199, 2004

L

P' PP'

« Easy Meat » - 1981 (Frank Zappa)

P'

L P

P'

P'

P'

PP'

Symmetries in Zappa’s music: the P’ transformation

• 

Guy Capuzzo, "Neo-Riemannian Theory and the Analysis of Pop-Rock Music", Music Theory Spectrum 26(2), p. 177-199, 2004

L

P' PP'

« Easy Meat » - 1981 (Frank Zappa)

P'

0

P

1

11

P'

si 10

do

do#

2 ré

la#

P' ré#

la

9

P' 8

sol#

mi sol

fa#

7 6

fa 5

3

4

The generating cell and its spatial transformations

Guy Capuzzo, "Neo-Riemannian Theory and the Analysis of Pop-Rock Music", Music Theory Spectrum 26(2), 177-199, 2004

L L

P' PP'

L

P' PP'

P' PP'

L

P' PP'

« Easy Meat » - 1981 (Frank Zappa)

L

P'

PP'

T-3 L

P'

PP'

L

P'

PP'

L

P'

PP'

T-3 T-3

The trajectory of the harmonic progression

Fa lam Lab Sol Ré fa#m Fa Mi Si

la#m Ré Réb Lab dom Si Sib

!

è

http://www.mathemusic.net

Symmetries in Paolo Conte’s Madeleine Lab→Réb/Fa→Sib7→Mib7/Réb

S. La Via, Poesia per musica e musica per poesia. Dai trovatori a Paolo Conte, Carocci, 2006

Lab→Réb/Fa→Sib7→Mib7/Réb Si/Ré#→Mi→Do#→Fa# Ré/La→Sol→Mi7→La7 Ré→Lab7→Réb→Do7→Mib

Madeleine’s spatial trajectory Lab Réb Sib

Mib Si

Mi

Réb Fa# Ré

Sol Mi

La

Ré

Lab Réb Do Mib

! http://www.mathemusic.net

Partial covering of the Tonnetz Lab→Réb/Fa→Sib7→Mib7/Réb

Missing major chord

S. La Via, Poesia per musica e musica per poesia. Dai trovatori a Paolo Conte, Carocci, 2006

Lab→Réb/Fa→Sib7→Mib7/Réb Si/Ré#→Mi→Do#→Fa# Ré/La→Sol→Mi7→La7 Ré→Lab7→Réb→Do7→Mib

? (« chanson ouverte », based on a poetry by Livio Andeatta)

T-3

T-1

1

T[1,3,4] 4

3

Notation: C = Do minor C# = Do# minor . . . B = Si minor

Notation : C = Do minor C# = Do# minor ...

http://www.mathemusic.net

T T

The “T” operator (as “trick”) T T T

Notation: C = Do minor C# = Do# minor ... http://www.mathemusic.net

Extract of the 2nd movement of the Symphony No. 9 (L. van Beethoven)

C#

D B♭

F#

E# B

G E♭

B

G# E

C G♭

C F C#

A

B♭

b

Enumeration of Hamiltonian Cycles in the Tonnetz

!

!

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

Aprile, a Hamiltonian « decadent » song Do←dom←Sol#←fam←Fa←lam←La←fa#m←Fa#←sibm←Do#←do#m La

mim→Sol→sim→Ré→rém→Sib→solm→Mib→mibm→Si→sol#m→Mi

G. D’Annunzio (1863-1938)

!

!

!

!

Do←dom←Sol#←fam←Fa←lam←La←fa#m←Fa#←sibm←Do#←do#m La

mim→Sol→sim→Ré→rém→Sib→solm→Mib→mibm→Si→sol#m→Mi Do→mim→Mi→sol#m→Si→ré#m→Re#→dom→Lab→fam→Do#→do#m lam←Fa←rém←Ré←sim←Sol←solm←Sib←sibm←Fa#←fa#m←La Mi←mim←Do←lam←Fa←fam←Reb←sibm←Fa#←mibm←Mib←dom La

do#m→La→fa#m→Ré→rém→Sib→solm→Sol→sim→Si→sol#m→Sol# è Hexachord (by Louis Bigo, 2013)

M. Andreatta, « Math’n pop : symétries et cycles hamiltoniens en chanson », Tangente

http://www.mathemusic.net

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) 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.

R L

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.

P

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).

è Hexachord (by Louis Bigo, 2013)

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,

Italo Calvino Raymond Queneau

Il castello dei destini incrociati, 1969

Analyzing harmonic progressions as paths in a generic Tonnetz

?

• 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.

è Hexachord (by Louis Bigo, 2013)

The spatial character of the « musical style » Beethoven, 2nd mouvement of the 9e Symphony

T[3,4,7] Babbitt, Semi-Simple Variations

T[1,2,9]

The geometric space as a parameter of style

Thelonious Monk, Brilliant Corners

Chick Corea, Eternal Child

Bill Evans, Turn Out the Stars

è Hexachord

The permutohedron as a parameter of style

Julio Estrada

L. Van Beethoven, Quatuor n° 17

1

2

3

4

5

6

7

8

9

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

5

1

1

1

6

12

15

12

11

7

2

1

1

21

25

34

25

21

9

5

3

10

11

12

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)

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)

Music analysis as a path in a permutohedron

B. Bartok, Quartet n° 4 (3d movement)

J. Estrada, “The intervallic thought”, Joint course ATIAM/Cursus , 20th November 2012 è http://ressources.ircam.fr/archiprod.html!

A. Schoenberg, Six pieces op. 19

The permutohedron of 77 possible partitions of 12

W. Reckziegel, “Musikanalyse und Wissenschaft”, Studia Musicologica 9(1-2), 1967, 163-186

The permutohedron as a lattice of formal concepts

1+1+1+1+1

1+1+1+2

1+1+3

1+2+2

1+4

2+3

5

•  T. Schlemmer, M. Andreatta, « Using Formal Concept Analysis to represent Chroma Systems », MCM 2013, McGill Univ., Springer, LNCS.

A concept lattice for the diatonic scale

Rudolf Wille

[R. Wille & R. Wille-Henning, « Towards a Semantology of Music », ICCS 2007, Springer, 2007]

A concept lattice for the diatonic scale

CM

Em

mi

X

X

sol

X

X

({mi, sol},{CM, Em})

[R. Wille & R. Wille-Henning, « Towards a Semantology of Music », ICCS 2007, Springer, 2007]

A concept lattice for the diatonic scale

CM

Em

FAm

FM

mi

X

X

sol

X

X

la

X

X

do

X

X

({la, do},{FM, Am})

[R. Wille & R. Wille-Henning, « Towards a Semantology of Music », ICCS 2007, Springer, 2007]

From binary relations to formal context

Attributs

A = extension of the concept (A,B) B = intension of the concept (A,B)

Objects

B A X

X

X

X

X

X

X

X

X

X

X

X

[R. Wille, « Restructuring Lattice Theory: An approach based on Hierarchies of Concepts », I. Rival (ed.), Ordered Sets, 1982]

Formal Concept Analysis: the double history

Algebraic structures Order structures

•  M. Barbut, « Note sur l’algèbre des techniques d’analyse hiérarchique », in B. Matalon (éd.), L’analyse hiérarchique, Paris, Gauthier-Villars, 1965. •  M. Barbut, B. Monjardet, Ordre et Classification. Algèbre et Combinatoire, en deux tomes, 1970 •  M. Barbut, L. Frey, « Techniques ordinales en analyse des données », Tome I, Algèbre et Combinatoire des Méthodes Mathématiques en Sciences de l’Homme, Paris, Hachette, 1971. •  B. Leclerc, B. Monjardet, « Structures d’ordres et sciences sociales », Mathématiques et sciences humaines, 193, 2011, 77-97

Topological structures

•  R. Wille, « Mathematische Sprache in der Musiktheorie », in B. Fuchssteiner, U. Kulisch, D. Laugwitz, R. Liedl (Hrsg.): Jahrbuch Überblicke Mathematik. B.I.Wissenschaftsverlag, Mannheim, 1980, p. 167-184. •  R. Wille, « Restructuring Lattice Theory: An approach based on Hierarchies of Concepts », I. Rival (ed.), Ordered Sets, 1982 •  R. Wille, « Sur la fusion des contextes individuals », Mathématiques et sciences humaines, tome 85, 1984. •  B. Ganter & R. Wille, Formal Concept Analysis: Mathematical Foundations, Springer, Berlin, 1998

Formal Concept Analysis: the common root

Garrett Birkhoff (1911-1996)

•  M. Barbut, « Note sur l’algèbre des techniques d’analyse hiérarchique », in B. Matalon (éd.), L’analyse hiérarchique, Paris, Gauthier-Villars, 1965. •  M. Barbut, B. Monjardet, Ordre et Classification. Algèbre et Combinatoire, en deux tomes, 1970 •  M. Barbut, L. Frey, « Techniques ordinales en analyse des données », Tome I, Algèbre et Combinatoire des Méthodes Mathématiques en Sciences de l’Homme, Paris, Hachette, 1971. •  B. Leclerc, B. Monjardet, « Structures d’ordres et sciences sociales », Mathématiques et sciences humaines, 193, 2011, 77-97

•  R. Wille, « Mathematische Sprache in der Musiktheorie », in B. Fuchssteiner, U. Kulisch, D. Laugwitz, R. Liedl (Hrsg.): Jahrbuch Überblicke Mathematik. B.I.Wissenschaftsverlag, Mannheim, 1980, p. 167-184. •  R. Wille, « Restructuring Lattice Theory: An approach based on Hierarchies of Concepts », I. Rival (ed.), Ordered Sets, 1982 •  R. Wille, « Sur la fusion des contextes individuals », Mathématiques et sciences humaines, tome 85, 1984. •  B. Ganter & R. Wille, Formal Concept Analysis: Mathematical Foundations, Springer, Berlin, 1998

FCA as the « restructuration » of lattice theory

FCA and the generalized just tuning Tonnetz

Mutabor (Darmstadt 1980)

è http://www.math.tu-dresden.de/~mutabor/

Formal Concept Analysis and topology: the Q-analysis

A

B

C

D

X1

1

0

0

1

X2

1

1

1

0

X3

0

0

1

1

X4

1

1

1

1

duality

Concept lattice vs simplicial complex

Lattice

Lattice complex

Simplicial complex

Concept lattice vs simplicial complex

Simplicial complex

Lattice

Lattice complex

tion n retrac

io eformat Strong d

Conclusions: •  The concept lattice alone cannot be fully reconstructed from the simplicial complex •  The simplicial complex cannot be fully determined from the concept lattice alone •  The concept lattice alone allows to determine the homotopy type of the simplicial complex

The simplicial complex of a Chopin’s Prelude

è Hexachord (by Louis Bigo, 2013)

Towards a topological signature of a musical piece A structural approach in Music Information Retrieval The simplices and their self-assembly

The score

Score reduction

The simplicial complex generated by the piece

Topological signature?

A specific trajectory in the complex

A trajectory realized in different support spaces 3

2

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 1917. 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).

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

Musically interesting Trajectory Transformations The “M” transformation

T[2,3,7]

T[3,4,7] M M

T[3,4,7]

M

Musically interesting Trajectory Transformations Automorphism of the support space

inversion do minor chord

Fa major chord

Rotation (autour du do)

T[3,4,7] Beatles, Hey Jude (orig. version)

T[3,4,7] Beatles, Hey Jude (transformed version)

èHexachord



Thank you for your attention!

Hexachord (by Louis Bigo, 2013)