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Guerino MazzolaGuerino Mazzola
U & ETH ZürichU & ETH Zürich
i2musicsi2musics
[email protected]@mazzola.ch
www.encyclospace.orgwww.encyclospace.org
Guerino MazzolaGuerino Mazzola
U & ETH ZürichU & ETH Zürich
i2musicsi2musics
[email protected]@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
bjec
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
ic o
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
bjec
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
ic o
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
www.encyclospace.orgwww.encyclospace.org