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