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Z12-StoryA MatheMusical Tutorial
Thomas Noll
Escola Superior de Musica de Catalunya, BarcelonaDepartament de Teoria i Composició
Basic Concepts
Semigroup, Monoid, GroupGroup Action, Monoid Action
Semigroup
Concatenation is associative:
A family of abstract „acts“.Any two acts can be concatenatedand yield another act of this family.
Semigroup
Concatenation is associative: =
A family of abstract „acts“.Any two acts can be concatenatedand yield another act of this family.
Monoid
Neutral Element: =
=
A Semigroup.
Neutral Element: =
=
A Semigroup.
Monoid
Neutral Element: =
=
Monoid
A Semigroup.
Every acthas an inverse:
=
Group
A Monoid.
=
= =
γf
γgf = γg γf
γg
X
X
X
Group Action on a Set X
gf
γ: G x X X
gf
Monoid Action on a Set X
X
XX
nm
µ: M x X X
µm
µnm= µn µm
nm µn
Groups and Group Actions
Musical Examples
T1
Example: Cyclic group of order 12
T2
T3
T4
T5T6
T7
T8
T9
T10
T11
T0
T0T1
T4
T5T6
T7
T8
T9 T3
T10T2
T11
Caley-Graph for C12
Action of C12 on Z12
Abstract Acts Concrete Acts of Tone Transposition
T1
Abstract Concatenation Concrete Concatenation
T1
T2
Example: Melodic Progressions as Acts of TranspositionA. Schönberg: Five Pieces for Orchestra Op. 16/III: Farben
Bars 1- 9
Bars 32- 38
Example: Chord Relations as Acts of TranspositionA. Schönberg: Five Pieces for Orchestra Op. 16/III: Farben
Bars 1- 9 T-1
Two Actions of C12 („T - Actions“)
C12 x Z12 Z12
Transposition of Tones („pitch classes“)
C12 x Subsets(Z12) Subsets(Z12)
Transposition of Chords („pitch class sets“)
Musically Interesting: Common Tone Relations
bar 1 bar 26 bar 44
T4 T8
3 commontones
Diatonic Scale (Balzano 1980):Solidarity between Fifth Transposition and Chromatic Alteration
T7
T1
6 commontones
Minimalvoiceleading
=
T0
T0 I
T1
T1I
T4
T4I T5
T5I
T6
T6I
T7
T7I
T8
T8I
T9 T9 IT3 I T3
T10
T10 IT2I T2
T11
T11 I
Caley-Graph for D12
C-G-
E+
E- B+
B-
F#+
F#-
C#+
C#-
Ab+Ab-
Eb+ Eb-A- A+
Bb+
Bb-D-F+
F-
D12-Actions onthe 24 Triads:
T/I-Action
C+G+
D+
C-F-
E+
Ab- B+
C#-
F#+
F#-
C#+
B
Ab+E-
Eb+ A-Eb- A+
Bb+
D-Bb-F+
G-
D12-Actions onthe 24 Triads :S/W-Action
C+G+
D+
C-
E+
E-
A-A+
C+G+
D+
Neo-RiemannianTransformationsAsAlternativeGenerators
P R
LParallelLeading-ToneRelative
C
E
Bb
F G
D
A
BF#C#
G#
Eb
(Neo-)Riemannian Operation P = „Parallel“
Hugo Riemann
R. Cohn: „The overdetermined triad“: Voice leading parsimony
C
E
Bb
F G
D
A
BF#C#
G#
Eb
(Neo-)Riemannian Operation R = „Relative“
Hugo Riemann
C
E
Bb
F G
D
A
BF#C#
G#
Eb
(Neo-)Riemannian Operation L = „Leading-Tone“
Hugo Riemann
C
E
Bb
F G
D
A
BF#C#
G#
Eb
Hexatonic System <L, P>
C
E
Bb
F G
D
A
BF#C#
G#
Eb
Octatonic System <R, P>
Hexatonic System (Cohn 1996):Solidarity between Transformation and Voiceleading Parsimony
P P PL L L
T7
T1
=Remember:
T1 T1 T1T-1T-1T-1
Decomposition,Stabilizers,
Conjugationetc.
0
9
6
8
5
2
4
1
10
3 117
0 1 20
1
2
3
Z3
Z4
Z3 x Z4Direct Product
0 1
2
3
0
123
3
0
2
1
0 1
2
0
12
01
2
2
1
0
0 1
2
3
4
56
7
11
8
10
9
a = t mod 3 b = t mod 4
t = 4 a + 9 b mod 12
a b
t
Full Symmetry Group of affine Transformations of Z12
Z12*Z12
Translations Kleinian Four-Group
1 5 7 115 1 11 7
7 11 1 5
11 7 5 1
(t, u) (s, v) = (t + u s, u v)
Sym(Z12) =
5,0 5,6 5,9 5,3
5,4 5,10 5,1 5,7
5,8 5,5 5,11 5,2
1,0 1,6 1,9 1,3
1,4 1,10 1,1 1,7
1,8 1,5 1,11 1,2
7,0 7,6 7,9 7,3
7,4 7,10 7,1 7,7
7,8 7,5 7,11 7,2
11,0 11,6 11,9 11,3
11,4 11,10 11,1 11,7
11,8 11,5 11,11 11,2
Stabilizer of an Element x of X:Subgroup H of all g with γg(x) = x
γ: G x X X
5,0 5,6 5,9 5,3
5,4 5,10 5,1 5,7
5,8 5,5 5,11 5,2
1,0 1,6 1,9 1,3
1,4 1,10 1,1 1,7
1,8 1,5 1,11 1,2
7,0 7,6 7,9 7,3
7,4 7,10 7,1 7,7
7,8 7,5 7,11 7,2
11,0 11,6 11,9 11,3
11,4 11,10 11,1 11,7
11,8 11,5 11,11 11,2
Internal Symmetries of Chords and ScalesExample: Octatonic Strategies in Bartok (c.f. Cohn 1991)Mikrokosmos Book IV: From the Island of Bali
Internal Symmetries of Chords and ScalesExample: Octatonic Strategies in Bartok (c.f. Cohn 1991)Sonata for two Pianos and Percussion
Internal Symmetries of the DiatonicRiemann 1878, Balzano 1980, Mazzola 1985, Cohn 1997
T0
T0 I
T7
T7I
T5
T5 I
Bb+ = {10, 2, 5}
G- = {2, 10, 7}
F+ = {5, 9, 0}Eb+ = {3, 7, 10}
D- = {9, 5, 2} C- = {7, 3, 0}
The Balzano - Family Zk(k+1)
2
8
5
11
2
6
0
9
3
6
2
8
5
11
2
10
4
1
7
10
I
V
IV
ii
vi
iii
18
6
2
10
14
3
11
7
15
19
13
1
17
5
9
8
16
12
0
4
I
V
VI
iv
viii
iii
18
6
2
10
14
18 3 8 13 18
ix
II
3
13
8
18
23
9
19
14
24
29
21
1
26
6
11
15
25
20
0
5V
IX
I
x
iii
vii
27
7
2
12
17
vi
VIII
3
13
8
18
23
28 4 10 16 22 28
3 9 15 21 27 3
ii
IV
Monoidsand Monoid Actions
e
fg
fg
fg
gg fg
fg
gf fffg
ggg fgg
fg
gfg ffg
fg
ggf fgf
fg
gff ffffg fgfgfgfgfgfgfg
Caley-Graph for a Free Monoid <f, g>
Example: Rhythms with two durational Values: 2, 3c.f. Chemillier & Truchet (2004)
e
bf
ff
f2
fag
gg
g2
g f
fff = f
g
ggg = g
cf g
ffg = ggf
g
g
g
gfx = gf
fff
gfx = gf
Caley-Graph for the Triadic Monoid
1
Caley-Graph for Z2*
0
0 1
0 0 0
1 0 1
*
What is a Z2* - Action ?µ: Z2
* x X X
Answer: (1) µ1 is the identical map idX of X.
X
(2) µ0 is a projection to a subset Y of X.
Y
e
bf
ff
f2
fag
gg
g2
g f
fff = f
g
ggg = g
cf g
ffg = ggf
g
g
g
gfx = gf
fff
gfx = gf
Caley-Graph for the Triadic Monoid
C
E
Bb
F G
D
A
BF#C#
G#
Eb
C
E
Bb
F G
D
A
BF#C#
G#
Eb
Generators of a Triadic Action
ff2 g2g
ba
ce
˙ =
Concatenation Table
ff2 g2g
ba
ce
˙ =
Concatenation Table
ff2 g2g
ba
ce
˙ =
Concatenation Table
e
bf
ff
f2
fag
gg
g2
g fg
cf g
g
g
gfff
Cosieves B: mB ⊂ B for all Elements m of M
e
bf
ff
f2
fag
gg
g2
g fg
cf g
g
g
gfff
Cosieves B: mB ⊂ B for all Elements m of M
e
bf
ff
f2
fag
gg
g2
g fg
cf g
g
g
gfff
Cosieves B: mB ⊂ B for all Elements m of M
e
bf
ff
f2
fag
gg
g2
g fg
cf g
g
g
gfff
Cosieves B: mB ⊂ B for all Elements m of M
e
bf
ff
f2
fag
gg
g2
g fg
cf g
g
g
gfff
Cosieves B: mB ⊂ B for all Elements m of M
e
bf
ff
f2
fag
gg
g2
g fg
cf g
g
g
gfff
Cosieves B: mB ⊂ B for all Elements m of M
ff2 g2g
ba
c
e
g2g
ba
c
ff2 g2g
ba
c
ff2
ba
c ba
c
RL
P
T
C
ø
Action of M on Sieves
Go to the blackgoard :-)
From Monoids to Categories
with a neutral Element.=
=
A Monoid is Semigroup
Recall:
From Monoids to Categories
In a Category arrows are sorted:A
B B
CA
C
From Monoids to Categories
In a Category arrows are sorted:A
B
C
A
A
B
BCodomain
Domain
B
AIdentities = „Objects“
From Monoids to Categories
In a Category arrows are sorted:A
B
C
A
B =A
B
A
B=
A
B
Monoid Action on a Set X
X
XX
nm
µ: M x X X
µm
µnm= µn µm
nm µn
Functor
F(B)nm
F: C Sets
F(m)
F(nm) = F(n) F(m)
nm
A
B
C
F(C)F(A)
F(n)
Intuitionistic Logics of ProjectingIs not a sub-object of this Projection
Are and Elements of ?
No.No, but....
Z(π)
Z(τ)
Z(σ)
ø
ø{ } {σ} {σ, π}
τ
σ
{ } {τ}
π
Z(π)
Z(τ)
Z(σ)
Ω(π)
Ω(τ)
Ω(σ)
Play-ActionAbstractDiagram Evaluation-Action
C
E
Bb
F G
D
A
HF#C#
G#
Eb
Perspectival Action
Tones are mapped to tones Pipes are mapped to keys
Z(π)
Z(τ)
Z(σ)
Play Action
ff2 g2g
ba
c
e
g2g
ba
c
ff2 g2g
ba
c
ff2
ba
c ba
c
RL
P
T
C
ø
T
T
R
Characteristic Function of {0, 4, 1}
χ{0,4,1}(t) =
C T
C
R
LRR
L
P
P
P
PP
CL
T
R
ø
L
T
R
øL
T
R
øL
T
R
ø
C
C
C
jP
jL jR
jC
as UpgradeOperations
Lawvere-TierneyTopologies
TT
T
T
T
T
R
C T
C
R
LR
R
L
P R
RR
R
C
C
L
L
jP χ{0,4,1}
TT
T
T
TT
T
TT
T
R
C T
C
R
LR
R
L
P
L
L
L
L
jR χ{0,4,1}
TT
T
T
T
T
T
T
R
C T
C
R
LR
R
L
P R
RR
R
R
R
jL χ{0,4,1}
T
T
T
C R
L
R
CP
R
L
R
TR
R
T
TR
R
T
C
L
C
L
jP χ{0,4,10}
T
T
T
C R
L
R
CP
R
L
R
TT
T
T
TT
T
T
L
L
L
L
jR χ{0,4,10}
T
T
T
C R
L
R
CP
R
L
R
TR
R
T
TR
R
T
R
T
R
T
jL χ{0,4,10}
Fifth - Transposition through Chromatic Alteration
Z20 = Z4(4+1) Z30 = Z5(5+1)
PerspectiveτF
PerspectiveτG
τF(x) = 3x+7 mod 12 τF(x) = 8x+4 mod 12
Triadic Action on Z12
Upgrades
PerspectiveτF
PerspectiveτG
τF(x) = 4x+9 mod 20 τF(x) = 15x+5 mod 20
Triadic Action on Z20
Upgrades
PerspectiveτF
PerspectiveτG
τF(x) = 5x+11 mod 30 τF(x) = 24x+6 mod 30
Triadic Action on Z30
Upgrades
