Guerino Mazzola

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• Extending Set Theory to
• Harmonic Topology
• and Topos Logic
• Music objects
• Why topoi?
• Logic

• Guerino Mazzola
• U ETH Zürich
• i2musics
• guerino_at_mazzola.ch
• www.encyclospace.org

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The address question (ontology) What is an
elementary musical object?
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A R-module address A_at_F eF.LinR(A, F) A
R _at_F eF.Lin(, F) ª F2
x ? F affine x et.g, et translation, g
linear
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Dodecaphonic Series
music objects
R Ÿ, A Ÿ11, F Ÿ12 A_at_F Ÿ11_at_Ÿ12 S Î
Ÿ11_at_Ÿ12 ª Ÿ1212
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modulation S(3) T(3) cadence symmetry
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Schönbergs Modulation Degrees
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Circel chords (G. Mazzola, Geometrie der Töne)
c 0f e7.3
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c, f(c), f2(c),... 0, 4, 7 c, e, g
major triad
lt f, c gt 1, c, f, f.c, c.f , f2.c, c2.f,... ?
Ÿ12_at_Ÿ12 lt e7.3, e0.0 gt 0, 4, 7
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Modeling Riemann Harmony (Th. Noll, PhD Thesis)
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Trans(Dt,Tc) lt fDt Tc gt ? Ÿ12_at_Ÿ12
relative consonances
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Ÿ12 ? Ÿ3 ? Ÿ4 z gt (z mod 3, -z
mod4) 4.u3.v lt (u,v)
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Ke Ÿ12e.0, 3, 4, 7, 8, 9 consonances
De Ÿ12e.1, 2, 5, 6, 10, 11 dissonances
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Parallels of fifths are always forbidden
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Ke, De
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Trans(Dt,Tc)
Trans(Ke,Ke)
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Prize for parametrization addresses Parametrized
objects need parametric evaluation!
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series S Î Ÿ11_at_Ÿ12
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More general set of k sequences of pitch classes
of length t1 K S1,S2,...,Sk This is a
polyphonic local composition K ? Ÿt_at_Ÿ12
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s t, define affine map f Ÿs ? Ÿt e0 gt
ei(0) e1 gt ei(1) ................. es gt
ei(s)
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Gegenstand der Untersuchungen sind aber nichtdie
Töne selbst, denn deren Beschaffenheit spieltgar
keine Rolle, sondern dieVerknüpfungen und
Verbindungender Töne untereinander. Bachs Art
of Fugue (1924)
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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.
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Eine kontrapunktische Form ist eine Menge von
Mengen von Mengen (von Tönen)Bachs Art of
Fugue (1924)
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Sets cartesian products X ? Y disjoint sums X È
Y powersets XY characteristic maps c X ? 2 no
algebra
why topoi?
Mod direct products AB etc. has algebra no
powersets no characteristic maps
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Yoneda Lemma The functorial map _at_ Mod Mod_at_
is fully faithfull. M gt _at_M
Hom(?,M) M_at_F Hom(_at_M,F)
why topoi?
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Functorial Local Compositions

• Are left with two important problems for local
  compositions K ? A_at_F
• The definition of a general evaluation
  procedure
• There are no general fiber products for local
  compositions.

why topoi?
Solution A_at_WF subfunctors a ? _at_A ? F
generalized sieves Kˆ ? _at_A ? F X_at_Kˆ (fX
A, k.f), k Î K ? X_at_A ? X_at_F
Kˆ? IdA_at_Kˆ K
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Hugo Riemann Logik ist in der Funktionstheorie
ein fundamentaler, aber dunkler Begriff.
Classical logic F 0 zero module subsets d
? 0_at_F 0_at_0 0 Have two values d 0_at_0
T, true d ˆ F ? T, false Fuzzy
logic F S /Ÿ circle group subsets d
0, e ? 0_at_F 0_at_S This logic is known as the
Gödel algebra,in fact a Heyting algebra defined
by thetopology of these subsets.
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Have natural generalization! d ? 0_at_0 d 0, e
? 0_at_S F any space (functor)A any address d
? A_at_F objective local compositiond ? _at_A ?
F functorial local composition In this context,
local compositions d are structurally legitimate
supports of logical values and their combinations
(conjunction, disjunction, implication, negation).
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The functorial change K gt Kˆ has dramatic
consequences for the global theory!
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A 0Ÿ
A Ÿ12
X ? Ÿ12 gt X End(X) ? Ÿ12_at_Ÿ12
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logic
ToM, ch. 25
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logic
e0.4.e11.0 e11.3.e11.0 e8.0
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Extension Topology
logic
Fix a space functor F, End(F) set of
endomorphisms of F, and an address A.
ExTopA(F) A_at_WF a ? _at_A ? F Extension
topology on ExTopA(F) Subsets M ? End(F),
Basic open sets ExtA(M) a, M ? End(a)
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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
logic
ExTopB(F)
ExTopA(F)
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Naturality of Heyting Logic of Open Sets
logic
U?V U?V U?V U?V U?V ? W ? U ? V W ?U
(-U)o
.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))
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TON C, F, Bb , Eb , Ab , Db , Gb, B, E, A, D,
G Val T, S, D, t, s, d, T,S, D, t, s,
d F Chords(Ÿ12) W_at_Ÿ12 TRUTH(F) sets of
chords in F
logic
RieNT,v(Chord) dChord.Ext0(MT,v) MT,v
monoid of all endomorphisms of prototypical
triadic chords Ext0(MT,v) chords
invariant under MT,v
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Birkhäuser 20021368 pages, hardcover incl.
CD-ROM 128. / CHF 188. ISBN
3-7643-5731-2 English
www.encyclospace.org