Transcript Extending Set Theory to Harmonic Topology and Topos Logic 1. Music objects Guerino Mazzola U & ETH Zürich i2musics [email protected] www.encyclospace.org 2.

Extending Set Theory to
Harmonic Topology
and Topos Logic
1. Music objects
Guerino Mazzola
U & ETH Zürich
i2musics
[email protected]
www.encyclospace.org
2. Why topoi?
3. Logic
music objects
The address question (ontology):
What is an elementary musical
object?
music objects
H
—EHLD  —4 (space of note events)
x
D
L
E
p
Ÿ12 (space of pitch classes)
music objects
F
x
—
0
1
x: —  F affine
x = et.g,
et = translation, g = linear
A = R-module
= „address“
A@F = eF.LinR(A, F)
A=R=—
—@F = eF.Lin—(—, F) ª F2
music objects
Dodecaphonic Series
Ÿ12
S
Webern: Op. 28
0
R = Ÿ, A = Ÿ11, F = Ÿ12
A@F = Ÿ11@Ÿ12
S  Ÿ11@Ÿ12 ª Ÿ1212
11
music objects
Key
E
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Position
gesture
H
h
score
E
L
l
e
music objects
Grand
Unification
Perspectives of
Harmony
and
Counterpoint
et
music objects
A
et.A
S(3)
T(3)
k
et
k
modulation S(3)  T(3) = „cadence + symmetry “
IVC
IIEb
music objects
VIIEb
IIC
M(3)
VC
VIIC
VEb
IIIEb
Schönberg‘s Modulation Degrees
C(3)
E b(3)
music objects
Circel chords (G. Mazzola, Geometrie der Töne)
c
c=0
f = e7.3
{c, f(c), f2(c),...}
= {0, 4, 7} = {c, e, g}
= major triad
Ÿ12
e7.3
e
g
e7.3
< f, c > = {1, c, f, f.c, c.f , f2.c, c2.f,...}  Ÿ12@Ÿ12
|< e7.3, e0.0 >| = {0, 4, 7}
music objects
Modeling Riemann Harmony (Th. Noll, PhD Thesis)
Dominant Triad {g, h, d}
Tonic Triad {c, e, g}
Dt
Tc
f
Trans(Dt,Tc) = < f:Dt Tc >  Ÿ12@Ÿ12
„relative consonances“
music objects
Ÿ12  Ÿ3  Ÿ4
11
0
1
2
10
3
9
8
z ~> (z mod 3, -z mod4)
4.u+3.v <~ (u,v)
4
7
6
5
8
11
4
3
7
0
5
6
2
9
1
10
Ÿ12
music objects
Ÿ12[e]= Ÿ12[X]/(X2)
c+e.d
c
c+e.Ÿ12
music objects
Ke = Ÿ12+e.{0,3,4,7,8,9} = consonances
ee.2.5
De = Ÿ12+e.{1, 2, 5, 6, 10, 11} = dissonances
music objects
Parallels of fifths are always forbidden
music objects
K e, D e
ƒe
Ÿ12
add.ch
Ÿ12[e]
Trans(Dt,Tc) = Trans(Ke,Ke)|ƒe
Ÿ12@Ÿ12
Ÿ12 [e]@Ÿ12 [e]
ƒe
Trans(Dt,Tc)
add.ch
Trans(Ke,Ke)
music objects
Prize for parametrization addresses:
Parametrized objects need
parametric evaluation!
address A
space F
music objects
K  —@F
F = —EH ª —2
H
E
f10: 0—  —: 0 ~> 10
music objects
series
S  Ÿ11@Ÿ12
Ÿ12
S
Webern: Op. 28
0
11
More general: set of k sequences of pitch classes of length t+1
K = {S1,S2,...,Sk}
This is a „polyphonic“ local composition
Ÿ12
K  Ÿt@Ÿ12
S1
Sk
music objects
Ÿ12
S1
Sk
s ≤ t, define affine map f: Ÿs  Ÿt
e0 ~> ei(0)
e1 ~> ei(1)
.................
es ~> ei(s)
f@K
Ÿ12
S1.f
Sk.f
es
e0
e1
Ÿs
music objects
Gegenstand der Untersuchungen sind aber nicht
die Töne selbst, denn deren Beschaffenheit spielt
gar keine Rolle, sondern die
Verknüpfungen und Verbindungen
der Töne untereinander.
Bach‘s „Art of Fugue“ (1924)
Wolfgang Graeser (1908—1928)
music objects
Need recursive combination of constructions such as
„sequences of sets of sets of curves of sets of chords“,
etc.
This leads to the theory of denotators, which we omit here.
Eine kontrapunktische Form ist eine
Menge von Mengen von
Mengen (von Tönen)
Bach‘s „Art of Fugue“ (1924)
Wolfgang Graeser
why topoi?
Sets
cartesian products X  Y
disjoint sums X  Y
powersets XY
characteristic maps c:X  2
no „algebra“
Mod@
F: Mod  Sets
presheaves
have all these
properties
Mod
direct products A≈B etc.
has „algebra“
no powersets
no characteristic maps
why topoi?
Yoneda Lemma
The functorial map @: Mod  Mod@ is fully faithfull.
M ~> @M = Hom(?,M)
M@F ≈ Hom(@M,F)
Mod@
Const.
Sets
@Mod
Mod
why topoi?
Functorial Local Compositions
Are left with two important problems for
local compositions K  A@F:
• The definition of a general evaluation procedure;
• There are no general fiber products for local compositions.
Solution:
A@WF = {subfunctors a  @A  F}
„generalized sieves“
Kˆ  @A  F
X@Kˆ = {(f:X A, k.f), k  K}  X@A  X@F
Kˆˇ = IdA@Kˆ = K
logic
Hugo Riemann: Logik ist in der Funktionstheorie
ein fundamentaler, aber dunkler Begriff.
Classical logic: F = 0 = zero module
subsets d  0@F = 0@0 = {0}
Have two values:
d = 0@0 = T, “true”
d =  = F =  T, “false”
Fuzzy logic: F = S = —/Ÿ = circle group
subsets d = [0, e[  0@F = 0@S
This logic is known as the Gödel algebra,
in fact a Heyting algebra defined by the
topology of these subsets.
e
S
logic
Have natural generalization!
d  0@0
d = [0, e[  0@S
F = any space (functor)
A = any address
d  A@F
d  @A  F
objective local composition
functorial local composition
In this context, local compositions d are structurally
legitimate supports of logical values and their
combinations (conjunction, disjunction, implication,
negation).
logic
The „functorial“ change K ~> Kˆ has dramatic consequences
for the global theory!
II
A = 0Ÿ
I
II
III
IV
V
VI
V
VII
VI
I
VII
A = Ÿ12
IV
X  Ÿ12
~>
III
X* = End*(X)  Ÿ12@Ÿ12
logic
IIII*
VI
VI*
VV*
IV*
IV
II*
VII*
VII
ToM, ch. 25
III*
III
I*
I*  II* = 
Ÿ12@Ÿ12
II*
logic
@Ÿ12
I*
f@I*ˆ
e0.4
I*ˆ
II*
f@I*ˆf@II*ˆ
e11.3
II*ˆ
f@II*ˆ
@Ÿ12
1Ÿ12
f = e11.0: Ÿ12  Ÿ12
e0.4.e11.0 = e11.3.e11.0 = e8.0
e8.0
logic
Extension Topology
Fix a space functor F,
End(F) = set of endomorphisms of F,
and an address A.
ExTopA(F) = A@WF = {a  @A  F}
Extension topology on ExTopA(F):
Subsets M  End(F),
Basic open sets: ExtA(M) = {a, M  End(a)}
logic
Naturality of Extension Topologies
Proposition: Fix a space functor F two addresses A, B,
and a retraction a: A  B. Then we have this
continuous map:
.a
ExTopB(F)
a
ExTopA(F)
@B  F
@a  IdF
a.a
@A  F
logic
Naturality of Heyting Logic of Open Sets
.a
ExTopB(F)
UV
UV
UV
U
Proposition:
OpenB(F)
ExTopA(F)
= UV
= UV
=  WUV W
= (-U)o
.a-1
OpenA(F)
is a logical homomorphism
.a-1 (UV)  (.a-1 (U) .a-1 (V))
.a-1 (UV  (.a-1 (U)  .a-1 (V))
.a-1 (UV)  (.a-1 (U)  .a-1 (V))
Birkhäuser 2002
1368 pages, hardcover
incl. CD-ROM
€ 128.– / CHF 188.–
ISBN 3-7643-5731-2
English
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