Kein Folientitel

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Penser la musique dans la logique fonctorielle des
topoi
Guerino Mazzola U ETH Zürich Internet
Institute for Music Science guerino_at_mazzola.ch www
.encyclospace.org
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• Topographie musicale
• Stratification sémiotique
• Models mathématiques
• Espaces de concepts
• Fiction et facticité
• Vérité et beauté

programme
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topographie
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topographie
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topographie
T(E) (dvE/dE)-1 q /sec
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topographie
Tonal modulation G major E b major
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Hjelmslev Stratification
Expression
Signification
Content
stratification
connotation
motivation
meta system
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Stratification on the mental level
denotator layer
forms
topoi
stratification
score layer
classical sheaves
interpretation layer
performance fields
differential geometry
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What is a mathematical model of a musical
phenomenon?
Mathematics
Music

• Field of Concepts
• Material Selection
• Process Type
• Grown rules for process
• construction and
• analysis

models
Deduction of rules from structure theorems
Why this material, these rules, relations?
Generalization!
Anthropic Principle!
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Arnold Schönberg Harmonielehre (1911)
Old Tonality Neutral Degrees (IC, VIC)
Modulation Degrees (IIF, IVF, VIIF)
New Tonality Cadence Degrees (IIF VF)
models

• What is the considered set of tonalities?
• What is a degree?
• What is a cadence?
• What is the modulation mechanism?
• How do these structures determine the modulation
  degrees?

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models
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models
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models
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Denotators
D denotator name
concepts
K Î A _at_ Functor(F) A-valued point
Form F
DA.F(K)
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F Form name one of four space types a
name diagram v in Mod_at_
Forms
a monomorphism in Mod_at_ id Functor(F) gt
Frame(v)
concepts

• Frame(v)-space for type
• simple v gt _at_B simple(v) _at_B
• limit v Form-Name-Diagram Mod_at_
• limit(v) lim(Form-Name-Diagram Mod_at_)
• colimit v Form-Name-Diagram Mod_at_
• colimit(v) colim(Form-Name-Diagram Mod_at_)
• power v Form-Name F gt Functor(F)
• power(v) WFunctor(F)

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MakroNote

• Ornaments
• Schenker Analysis

concepts
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Galois Theory
Form Theory
Defining equation
Defining diagram
concepts
fS(X) 0
id v(F)
Field S
Form System
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RUBATO
concepts
Os X
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(Textual) Predicates
Exi
facticité
Sig
D/Exi
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What is facticity of predicate Ex at denotator D?
D/Ex A.TRUTH(F)(d) TRUTH(F) space of
subsets of space F of truth values d Î W
F(A) d Í _at_A x F
facticité
The coordinate d of a truth denotator D/Ex is a
sieve in A x F.
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Special case 1 I 0-module Then F _at_0
final object 1 in Mod_at_ d Î W1(A) W (A)
A 0 W (0) Hom(1, W) set of
topos-theoretic truth values
Sub(_at_0) Special values d ˆ F ? T, d
_at_0 T
facticité
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Special case 2 I /Ÿ S circle group, F
_at_S. d Î W F(A) means this Take again
special address A 0, i.e., d Í _at_S In
particular, if d d_at_, d 0,e Í S an
interval, we have fuzzy logic defined by the
truth quantity e in the closed unit interval.
facticité
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Summary

• The truth denotators D/Ex
• associated with a predicate Ex
• are local compositions
• at address A
• and in the truth space F.
• They generalize and unify
• the topos-theoretic and fuzzy logic values, and
• classical objects of music theory.

facticité
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Classification of Predicate Constructions

• 1. Arbitrary/Atomic Predicates
• Mathematical
• Musical (Primavista)
• Deictic (Shifters)
• 2. Motivated/Compound Predicates
• Logical
• Geometric

facticité
D/Ex
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beauté
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TON C, F, A , D , G , C , F, B, E, A, D,
G val T,S, D, t, s, d S /Ÿ F _at_S, A
0
beauté
RieMD,d(Chord(222)) d d_at_, d 0,e Í S
T,v D ,d
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TON C, F, A , D , G , C , F, B, E, A, D,
G Val T,S, D, t, s, d, T,S, D, t, s,
d F Chords(Ÿ12) TRUTH(F) sets of chords
in F
beauté
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 basic open set in
the extension topology