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Exploring the “mathemusical” dynamics: some aspects of a musically driven
mathematical practice
Istituto Veneto di Lettere Scienze e Arti
March 31, 2017
Moreno AndreattaMusic Representations Team
IRCAM/CNRS/UPMC & IRMA/USIAS Strasbourghttp://www.ircam.fr/repmus.html
CE
G
B
B�F#
C#
AF
MUSIC
MATHEMATICS
Musicalproblem
Mathematicalstatementform
alisa
tion
Generaltheorem
generalisation
Musicanalysis
Musictheory
Composition
OpenMusic
application
The double movement of a ‘mathemusical’ activity
MUSIC
MATHEMATICS
Musicalproblem
Mathematicalstatement
Generaltheorem
generalisation
Musicanalysis
Musictheory
Composition
OpenMusic
application
The double movement of a ‘mathemusical’ activity
form
alisa
tion
MUSIC
MATHEMATICS
Musicalproblem
Mathematicalstatement
Generaltheorem
generalisation
Musicanalysis
Musictheory
Composition
OpenMusic
application
form
alisa
tion
The double movement of a ‘mathemusical’ activity
Set Theory, andTransformation Theory
Finite Difference Calculus
Diatonic Theory and ME-Sets Block-designs
Rhythmic TilingCanons
Z-Relation and Homometric Sets
Some examples of ‘mathemusical’ problems
Neo-Riemannian Theoryand Spatial Computing
- 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
M. Andreatta : Mathematica est exercitium musicae, Habilitation Thesis, IRMA University of Strasbourg, 2010
The interplay between algebra and geometry in music
�Concerning music, it takes place in time, like algebra. In mathematics, there is this fundamental dualitybetween, on the one hand, geometry – whichcorresponds 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 whichhas its diabolical precision. [...] And one only perceivesthe development of algebra through music� (A. Connes).
è http://videotheque.cnrs.fr/
è http://agora2011.ircam.fr
The galaxy of geometrical models at the service of music
The galaxy of geometrical models at the service of music
do…
do# re# fa# sol# la#re mi fa sol la si
0 1 2 3 4 5 6 7 8 9 10 11 12
do…
Marin Mersenne
Harmonicorum Libri XII, 1648
The circular representation of the pitch space
do
re
mi
fasol
la
si0 1
2
3
45
678
9
1011
do#
re#
fa#
sol#
la#
do…
do# re# fa# sol# la#re mi fa sol la si
0 1 2 3 4 5 6 7 8 9 10 11 12
do…
Marin Mersenne
Harmonicorum Libri XII, 1648
do
re
mi
fasol
la
si0 1
2
3
45
678
9
1011
do#
re#
fa#
sol#
la#
The circular representation of the pitch space
do
… …
…do# re# fa# sol# la#re mi fa sol la si
0 1 2 3 4 5 6 7 8 9 10 11 12
do… 0
1
2
3
4
56
7
8
9
10
11
sol#
la#
mi
do
re
do#
re#
fa# fasol
la
si
Musical transpositions are additions...
Do maj = {0,2,4,5,7,9,11}Do# maj = {1,3,5,6,8,10,0}
...or rotations !
+1
30�
0-(435)
Tk : x ® x+k mod 12
do
… …
…do# re# fa# sol# la#re mi fa sol la si
0 1 2 3 4 5 6 7 8 9 10 11 12
do… 0
1
2
3
4
56
7
8
9
10
11
sol#
la#
mi
do
re
do#
re#
fa# fasol
la
si
Musical transpositions are additions...
Do maj = {0,2,4,5,7,9,11}Do# maj = {1,3,5,6,8,10,0}
...or rotations !
+1
1-(435)
Tk : x ® x+k mod 12
do
… …
…do# re# fa# sol# la#re mi fa sol la si
0 1 2 3 4 5 6 7 8 9 10 11 12
do… 0
1
2
3
4
56
7
8
9
10
11
sol#
la#
mi
do
re
do#
re#
fa# fasol
la
si
Musical inversions are differences...
Do maj = {0,4,7}La min = {0,4,9}
... or axial symmetries!I4
I4(x)=4-x
I : x ® -x mod 12
R
do
… …
…do# re# fa# sol# la#re mi fa sol la si
0 1 2 3 4 5 6 7 8 9 10 11 12
do…
01
2
3
4
56
7
8
9
10
11
sol#
la#
mi
do
re
do#
re#
fa# fasol
la
si
Musical inversions are differences...
Do maj = {0,4,7}Do min = {0,3,7}
... or axial symmetries!
I7
I7(x)=7-x
I : x ® -x mod 12
P
do
… …
…do# re# fa# sol# la#re mi fa sol la si
0 1 2 3 4 5 6 7 8 9 10 11 12
do…
01
2
3
4
56
7
8
9
10
11
sol#
la#
mi
do
re
do#
re#
fa# fasol
la
si
Musical inversions are differences...
Do maj = {0,4,7}Mi min = {4,7,11}
... or axial symmetries!
I11
I11(x)=11-x
I : x ® -x mod 12
L
Majorthirds axis
Speculum Musicum(Euler, 1773)
Maj
or th
irds
axis
The Tonnetz
(or ‘honeycomb’ hexagonal tiling)R
R
P
P
L
L
FifthsaxisR
P
L
The Tonnetz(or ‘honeycomb’ hexagonal tiling)
Minor thirdsaxis
The Tonnetz(or ‘honeycomb’ hexagonal tiling)
Minor thirdsaxis
The Tonnetz(or ‘honeycomb’ hexagonal tiling)
Minor thirdsaxis
Gilles Baroin
The Tonnetz(or ‘honeycomb’ hexagonal tiling)
Minor thirdsaxis
Axe de tierces mineures
The Tonnetz, its symmetries and its topological structure P
R LMinorthird axis
?
transposition
L = Leading Tone
P as parallel
R as relative
Axe de tierces mineures
PR L
L = Leading Tone
P as parallel
R as relative
transposition
è Source: Wikipedia
Minorthird axis
The Tonnetz, its symmetries and its topological structure
Harmonic progressions as spatial trajectories
RL
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
LaDo¬dom¬Sol#¬fam¬Fa¬lam¬La¬fa#m¬Fa#¬sibm¬Do#¬do#m
mim®Sol®sim®Ré®rém®Sib®solm®Mib®mibm®Si®sol#m®Mi
G. D’Annunzio (1863-1938)
Hamiltonian Cycles with inner periodicitiesL 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 ...
RL
P
La sera non è più la tua canzone (Mario Luzi, 1945, in Poesie sparse)
La sera non è più la tua canzone,è questa roccia d’ombra traforatadai lumi e dalle voci senza fine,la quiete d’una cosa già pensata.
Ah questa luce viva e chiara vienesolo da te, sei tu così vicinaal vero d’una cosa conosciuta,per nome hai una parola ch’è passatanell’intimo del cuore e s’è perduta.
Caduto è più che un segno della vita,riposi, dal viaggio sei tornatadentro di te, sei scesa in questa purasostanza così tua, così romitanel silenzio dell’essere, (compiuta).
L’aria tace ed il tempo dietro a tesi leva come un’arida montagnadove vaga il tuo spirito e si perde,un vento raro scivola e ristagna.
http://www.mathemusic.netmin. 1’02’’
Music: M. AndreattaArrangement and mix: M. Bergomi & S. Geravini(Perfect Music Production)Mastering: A. Cutolo (Massive Arts Studio, Milan)
The use of constraints in arts
Raymond QueneauIl castello dei destiniincrociati, 1969
Cent mille milliards de poèmes, 1961
Italo Calvino
La vie mode d’emploi,
Georges Perrec
OuLiPo (Ouvroir de Littérature Potentielle)
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
http://oumupo.org/
The musical style...is the space!3 2
The geometric character of musical logicJohann Sebastian Bach - BWV 328
Johann Sebastian Bach - BWV 328 random chords
K[1,1,1
0]
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
]0
0,25
0,5
2-co
mpa
ctne
ss
Schönberg - Pierrot Lunaire - Parodie
Schönberg - Pierrot Lunaire - Parodie random chords
K[1,1,1
0]
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
]0
0,25
0,5
2-co
mpa
ctne
ss
Claude Debussy - Voiles
Claude Debussy - Voiles random chords
K[1,1,1
0]
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
]0
0,25
0,5
2-co
mpa
ctne
ss
C
E
G
B
T[2,3,7] T[3,4,5]
2 33 4
Spatial music analysis via Hexachord
èhttp://www.lacl.fr/~lbigo/hexachord
Thank you for your attention!
