Guerino MazzolaGuerino Mazzola

U & ETH ZürichU & ETH Zürich

i2musicsi2musics

guerino@mazzola.chguerino@mazzola.ch

www.encyclospace.orgwww.encyclospace.org

Guerino MazzolaGuerino Mazzola

U & ETH ZürichU & ETH Zürich

i2musicsi2musics

guerino@mazzola.chguerino@mazzola.ch

www.encyclospace.orgwww.encyclospace.org

Extending Set Theory toExtending Set Theory toHarmonic TopologyHarmonic Topology

and Topos Logicand Topos Logic

1.1. Music objectsMusic objects

2.2. Why topoi?Why topoi?

3.3. LogicLogic

Extending Set Theory toExtending Set Theory toHarmonic TopologyHarmonic Topology

and Topos Logicand Topos Logic

1.1. Music objectsMusic objects

2.2. Why topoi?Why topoi?

3.3. LogicLogic

The address question (ontology):The address question (ontology):What is an elementary musical What is an elementary musical

object?object?

The address question (ontology):The address question (ontology):What is an elementary musical What is an elementary musical

object?object?mus

ic o

bjec

ts

mus

ic o

bjec

ts

xx

ŸŸ12 12 (space of pitch classes)(space of pitch classes)

pp

——EHLD EHLD ——44 (space of note events)(space of note events)

EE

HH

DD

LL

FF

x: x: —— F affine F affinex = ex = ett.g, .g, eett = translation, g = linear = translation, g = linear

mus

ic o

bjec

ts

——00 11

xxA = R-moduleA = R-module

= „address“= „address“A@F = eA@F = eFF.Lin.LinRR(A, F)(A, F)

A = R = A = R = ——

——@F = e@F = eFF.Lin.Lin——((——, F) , F) ªª F F22

R = R = ŸŸ, A = , A = ŸŸ1111, F =, F = ŸŸ1212

A@F = A@F = ŸŸ1111@@ŸŸ1212

S S ŸŸ1111@@ŸŸ1212 ªª ŸŸ12121212

ŸŸ1212

SS

0 11Webern: Op. 28

Dodecaphonic SeriesDodecaphonic Seriesm

usic

obj

ects

Zur Anzeige wird der QuickTime™ Dekompressor „“ benötigt.

gestureH

E

L

score

h

el

Position

Key

E

mus

ic o

bjec

ts

mus

ic o

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ts

Harmony Harmony and and

CounterpointCounterpoint

Grand Grand Unification Unification Perspectives ofPerspectives of

SS(3)(3) TT(3)(3)

k k

A et

et.A

et

modulation modulation SS(3) (3) TT(3) (3) = „cadence + symmetry “= „cadence + symmetry “ modulation modulation SS(3) (3) TT(3) (3) = „cadence + symmetry “= „cadence + symmetry “

mus

ic o

bjec

ts

CC(3)(3) EE bb(3)(3)

MM(3)(3)VVEEbb

VIIVIIEEbb

IIIIEEbb

IIIIIIEEbbVC

IVC

VIIC

IIC

mus

ic o

bjec

ts

Schönberg‘s Modulation DegreesSchönberg‘s Modulation DegreesSchönberg‘s Modulation DegreesSchönberg‘s Modulation Degrees

ŸŸ1212

c

e

g

< f, c< f, c > = {1, c, f, f.c, c.f> = {1, c, f, f.c, c.f , f, f22.c, c.c, c22.f,...}.f,...} ŸŸ1212@@ŸŸ1212

|< e|< e77.3, e.3, e00.0.0 >| = {0, 4, 7}>| = {0, 4, 7}

< f, c< f, c > = {1, c, f, f.c, c.f> = {1, c, f, f.c, c.f , f, f22.c, c.c, c22.f,...}.f,...} ŸŸ1212@@ŸŸ1212

|< e|< e77.3, e.3, e00.0.0 >| = {0, 4, 7}>| = {0, 4, 7}

Circel chords (G. Mazzola, Circel chords (G. Mazzola, Geometrie der TöneGeometrie der Töne) )

ee77.3.3

ee77.3.3

c = 0c = 0f = ef = e77.3.3

{c, f(c), f{c, f(c), f22(c),...} (c),...} = {0, 4, 7} = {c, e, g} = {0, 4, 7} = {c, e, g} = major triad= major triadm

usic

obj

ects

Trans(Dt,Tc) = < f:Trans(Dt,Tc) = < f:DtDt TcTc > > ŸŸ1212@@ŸŸ1212 Trans(Dt,Tc) = < f:Trans(Dt,Tc) = < f:DtDt TcTc > > ŸŸ1212@@ŸŸ1212

f

DtDt

Dominant Triad {g, h, d}Dominant Triad {g, h, d}

TcTc

Tonic Triad {c, e, g}Tonic Triad {c, e, g}

Modeling Riemann Harmony (Th. Noll, Modeling Riemann Harmony (Th. Noll, PhD ThesisPhD Thesis))m

usic

obj

ects

„„relative consonances“relative consonances“

ŸŸ12 12 ŸŸ3 3 ŸŸ44

z ~> (z mod 3, -z mod4)z ~> (z mod 3, -z mod4)4.u+3.v <~ (u,v)4.u+3.v <~ (u,v)

11

10

8

1

2

34

567

9

0

0 12

3

4

567

8

9

1011

mus

ic o

bjec

ts

ŸŸ12 12 ŸŸ1212[[]= ]= ŸŸ1212[X]/(X[X]/(X22))

c+c+..ŸŸ1212

ccc+c+.d.d

mus

ic o

bjec

ts

= = ŸŸ1212++ = consonances = consonances

DD = = ŸŸ1212++{1, 2, 5, 6, 10, 11} = dissonances{1, 2, 5, 6, 10, 11} = dissonances

ee.2.2.5.5

mus

ic o

bjec

ts

Parallels of fifths are always forbiddenParallels of fifths are always forbiddenParallels of fifths are always forbiddenParallels of fifths are always forbidden

mus

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bjec

ts

ŸŸ1212 ŸŸ1212[[]]

ŸŸ1212@@ŸŸ1212 ŸŸ12 12 [[]]@@ŸŸ12 12 [[]]

Trans(Dt,Tc) = Trans(KTrans(Dt,Tc) = Trans(K,K,K)|)|ƒƒ

ƒƒƒƒ

ƒƒ

add.chadd.ch add.chadd.ch

Trans(Dt,Tc)Trans(Dt,Tc) Trans(Trans(KK,K,K))

KK, D, Dm

usic

obj

ects

space Fspace F

Prize for parametrization addresses:Prize for parametrization addresses:Parametrized objects need Parametrized objects need

parametric evaluation!parametric evaluation!

Prize for parametrization addresses:Prize for parametrization addresses:Parametrized objects need Parametrized objects need

parametric evaluation!parametric evaluation!

mus

ic o

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ts

address Aaddress A

f0: 0—— ——: 0 ~> 0f1: 0—— ——: 0 ~> 1

EE

HHF = F = ——EHEH ªª ——22 KK——@F@F

mus

ic o

bjec

ts

ŸŸ1212

series series S S ŸŸ1111@@ŸŸ1212

More general: set of k sequences of pitch classes of length t+1

K = {S1,S2,...,Sk}

This is a „polyphonic“ local composition K ŸŸtt@@ŸŸ1212

S1

Sk

mus

ic o

bjec

ts ŸŸ1212

SS

0 11Webern: Op. 28

s ≤ t, define affine map f: ŸŸs s ŸŸtt

e0 ~> ei(0)

e1 ~> ei(1)

.................es ~> ei(s)

S1

Sk

ŸŸ1212

e0e1

es

ŸŸss

S1.f

Sk.f

ŸŸ1212

f@K

mus

ic o

bjec

ts

mus

ic o

bjec

tsGegenstand der Untersuchungen sind aber nichtGegenstand der Untersuchungen sind aber nichtdie Töne selbst, denn deren Beschaffenheit spieltdie Töne selbst, denn deren Beschaffenheit spielt

gar keine Rolle, sondern diegar keine Rolle, sondern dieVerknüpfungen und VerbindungenVerknüpfungen und Verbindungen

der Töne untereinander.der Töne untereinander.

Bach‘s „Art of Fugue“ (1924)Bach‘s „Art of Fugue“ (1924)

Wolfgang Graeser (1908—1928)

Need recursive combination of constructions such asNeed recursive combination of constructions such as

„„sequences of sets of sets of curves of sets of chordssequences of sets of sets of curves of sets of chords“, “, etc. etc.

This leads to the theory of This leads to the theory of denotatorsdenotators, which we omit here., which we omit here.

Need recursive combination of constructions such asNeed recursive combination of constructions such as

„„sequences of sets of sets of curves of sets of chordssequences of sets of sets of curves of sets of chords“, “, etc. etc.

This leads to the theory of This leads to the theory of denotatorsdenotators, which we omit here., which we omit here.

mus

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bjec

ts

Eine kontrapunktische Form ist eine Eine kontrapunktische Form ist eine Menge von Mengen von Menge von Mengen von

Mengen (von Tönen)Mengen (von Tönen)

Bach‘s „Art of Fugue“ (1924)Bach‘s „Art of Fugue“ (1924)

Wolfgang Graeser

ModMod@@

F:F: Mod Mod Sets Setspresheavespresheaves

have all these have all these propertiesproperties

SetsSetscartesian products Xcartesian products X YYdisjoint sums X disjoint sums X YYpowersets Xpowersets XYY

characteristic maps characteristic maps X X no „algebra“no „algebra“

ModModdirect products Adirect products A≈≈B etc.B etc.

has „algebra“has „algebra“no powersetsno powersets

no characteristic mapsno characteristic maps

why

topo

i?

Yoneda LemmaYoneda Lemma

The functorial mapThe functorial map @: Mod @: Mod ModMod@ @ is is fully faithfullfully faithfull..M M @M = Hom(?,M) @M = Hom(?,M)

M@F ≈ Hom(@M,F)M@F ≈ Hom(@M,F)

Yoneda LemmaYoneda Lemma

The functorial mapThe functorial map @: Mod @: Mod ModMod@ @ is is fully faithfullfully faithfull..M M @M = Hom(?,M) @M = Hom(?,M)

M@F ≈ Hom(@M,F)M@F ≈ Hom(@M,F)

ModMod@@

@Mod@Mod@Mod@Mod ModModModMod

why

topo

i?

Const.Const.Const.Const.

SetsSetsSetsSets

Functorial Local CompositionsFunctorial Local Compositions

Are left with two important problems for Are left with two important problems for local compositions K local compositions K A@FA@F::

• The definition of a The definition of a general evaluation proceduregeneral evaluation procedure;;• There are no general There are no general fiber productsfiber products for local compositions. for local compositions.

Are left with two important problems for Are left with two important problems for local compositions K local compositions K A@FA@F::

• The definition of a The definition of a general evaluation proceduregeneral evaluation procedure;;• There are no general There are no general fiber productsfiber products for local compositions. for local compositions.

Solution:Solution:

A@A@F F = {= {subfunctors a subfunctors a @A@A FF} „generalized } „generalized sieves“sieves“

Kˆ Kˆ @A @A FF

X@Kˆ = {(f:X X@Kˆ = {(f:X A, k.f), k A, k.f), k K} K} X@A X@A X@FX@F

Kˆˇ = IdKˆˇ = IdAA@Kˆ@Kˆ = K= KKˆˇ = IdKˆˇ = IdAA@Kˆ@Kˆ = K= K

why

topo

i?

Classical logic: Classical logic: F = 0 = zero moduleF = 0 = zero module

subsets dsubsets d 0@F = 0@0 0@F = 0@0 = {0}= {0}

Have two values: Have two values: d = d = 0@0 0@0 = = TT, “true”, “true”d = d = ˆ̂ = =F F = = ˘˘TT, “false” , “false”

Fuzzy logic: Fuzzy logic: F = S = F = S = ——//ŸŸ = = circle groupcircle group

subsets subsets dd = = [0, e[ [0, e[ 0@F = 0@S 0@F = 0@S

This logic is known as the This logic is known as the Gödel algebra,Gödel algebra,in fact a Heyting algebra defined by thein fact a Heyting algebra defined by thetopology of these subsets.topology of these subsets.

logi

c

e

S

Hugo Riemann: Hugo Riemann: Logik ist in der Funktionstheorie Logik ist in der Funktionstheorie ein fundamentaler, aber dunkler Begriff.ein fundamentaler, aber dunkler Begriff.

Hugo Riemann: Hugo Riemann: Logik ist in der Funktionstheorie Logik ist in der Funktionstheorie ein fundamentaler, aber dunkler Begriff.ein fundamentaler, aber dunkler Begriff.

logi

cHave natural generalization!Have natural generalization!

dd 0@0 0@0 dd = = [0, e[ [0, e[ 0@S 0@S

F = any space (functor)F = any space (functor)A = any addressA = any address

d d A@F A@F objective local compositionobjective local compositiond d @A @A F F functorial local compositionfunctorial local composition

In this context, local compositions In this context, local compositions dd are are structurally legitimate supports of logical values structurally legitimate supports of logical values and their combinations (conjunction, disjunction, and their combinations (conjunction, disjunction, implication, negation).implication, negation).

The „functorial“ change K ~> Kˆ has dramatic consequences for the global theory!

The „functorial“ change K ~> Kˆ has dramatic consequences for the global theory!

I

IV

IIVIV

IIIVII

I IV VII III VI VIIA = 0ŸŸ

A = ŸŸ1212 X ŸŸ1212 ~> X* = End*(X) ŸŸ1212@@ŸŸ1212

logi

c

I

IV

II

VIV

III

VII

I*

IV*

II*

VI*V*

III*

VII*

II*

I* ŸŸ1212@@ŸŸ1212

I* II* =

ToM, ch. 25ToM, ch. 25

logi

c

I*ˆ̂

II*ˆ̂

@@ŸŸ1212

@@ŸŸ1212I* I*

11ŸŸ1212

II* II*

f = ef = e1111.0: .0: ŸŸ1212 ŸŸ1212

f@II*ˆ̂

ee00.4.4

ee1111.3.3

ee88.0.0

ee00.4.4..ee1111.0 = .0 = ee1111.3.3..ee1111.0 = .0 = ee88.0.0 ee00.4.4..ee1111.0 = .0 = ee1111.3.3..ee1111.0 = .0 = ee88.0.0

f@I*ˆ̂

f@I*ˆ̂f@II*ˆ̂

logi

c

Extension TopologyExtension Topology

Fix a space functor F, Fix a space functor F, End(F) = set of endomorphisms of F,End(F) = set of endomorphisms of F,and an address A.and an address A.

ExTopExTopAA(F)(F) = A@ = A@F F = {a = {a @A @A F} F}

EExtension topologyxtension topology on ExTop on ExTopAA(F): (F):

Subsets M Subsets M End(F), End(F),

Basic open sets: Basic open sets: ExtExtAA(M) = {a, M (M) = {a, M End(a)} End(a)}

logi

c

Naturality of Extension TopologiesNaturality of Extension Topologies

PropositionProposition: Fix a space functor F two addresses A, : Fix a space functor F two addresses A, B, and a retraction B, and a retraction : A : A B. Then we have this B. Then we have this continuous map:continuous map:

ExTopExTopBB(F)(F) ExTopExTopAA(F)(F)

@B @B F F

@A @A F F

@@ Id IdFF

aa

aa

logi

c

Naturality of Heyting Logic of Open SetsNaturality of Heyting Logic of Open Sets

ExTopExTopBB(F)(F) ExTopExTopAA(F)(F)

UUV V = U= UVVUUV V = U= UVV

UUV V = = W W U U V V W WU U = (-U)= (-U)oo

(U(UV) V) ( ( (U) (U) (V)) (V)) (U (UV V ( ( (U) (U) (V)) (V)) (U(UV) V) ( ( (U) (U) (V)) (V))

(U(UV) V) ( ( (U) (U) (V)) (V)) (U (UV V ( ( (U) (U) (V)) (V)) (U(UV) V) ( ( (U) (U) (V)) (V))

logi

c

OpenOpenBB(F)(F) OpenOpenAA(F)(F)

is a logical homomorphismis a logical homomorphism

PropositionProposition::

Birkhäuser 2002Birkhäuser 20021368 pages, hardcover 1368 pages, hardcover incl. CD-ROMincl. CD-ROM€ € 128.– / CHF 188.–128.– / CHF 188.–ISBN 3-7643-5731-2ISBN 3-7643-5731-2EnglishEnglish

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