Guerino Mazzola

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Mathematical Music Theory Status Quo 2000

• Guerino Mazzola
• U ETH Zürich
• Internet Institute for Music Science
• guerino_at_mazzola.ch
• www.encyclospace.org

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• Time Table
• The Concept Framework
• Global Classification
• Models and Methods
• Towards Grand Unification

Contents
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Theory
Grants
Software
Kelvin Null
Akroasis
Gruppentheore-tische Methode in der Musik
M(2,Z)\Z2 Karajan
Time
Gruppen und Kategorien in der Musik
Depth-EEG for Consonances and Dissonances
Presto
Synthesis
Geometrie der Töne
Immaculate Concept
RUBATO Project
Morphologie abendländischer Harmonik
Kuriose Geschichte
RUBATO NeXT
KiT-MaMuTh Project
Mac OS X
Kunst der Fuge
Topos of Music
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• Mod category of modules diaffine morphisms
• A R-module, B S-module
• Dilin(A,B) (l,f) fA B additive,
  lR S ring homomorphism f(r.a)
  l(r).f(a)
• eb(x) bx translation on B
• A_at_B eB.Dilin(A,B)

Concepts
eb.f A B B
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Topos of presheaves over Mod Mod_at_ F Mod
Sets, contravariant Example representable
presheaf _at_B _at_B(A) A_at_B F(A) A_at_F A
address
Concepts
Yoneda Lemma The functor _at_ Mod Mod_at_ is
fully faithfull. B gt _at_B
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Concepts

• Database Management Systems
• require recursively stable object types!
• k Î B
• K Î 2B no module!
• Need more general spaces F

B _at_ 0_at_B
K Í 0_at_B
K Í A _at_B

• A n sequences (b0,b1,,bn)
• A B self-addressed tones
• Need general addresses A

F W_at_B A 0
K Î A _at_F F presheaf over Mod
F _at_B
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F Form name one of four space types a
diagramn v in Mod_at_
Forms
a monomorphism in Mod_at_ id Functor(F) gt
Frame(v)
Concepts

• Frame(v)-space for type
• simple v Ægt _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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Denotators
D denotator name
Concepts
K Î A _at_ Functor(F) A-valued point
Form F
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MakroNote

• Ornaments
• Schenker Analysis

Concepts
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RUBATO
Concepts
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Galois Theory
Form Theory
Defining equation
Defining diagram
Concepts
fS(X) 0
id v(F)
Field S
Form System
Mariana Montiel Hernandez, UNAM
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Category Loc of local compositions Type
PowerF gt Functor(F) G
Classification
local composition K Î A_at_WG K Í _at_A G
generalizes K Í A_at_G objective local
compositions
K Í _at_A G
_at_a h
f/a
L Í _at_B H
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ObLoc
Loc
Classification
ObLocA
LocA

• Theorem
• Loc is finitely complete (while ObLoc is not!)
• On ObLocA and LocA Embedding and Trace
• are an adjoint pair
• ObLocA(Embedding(K),L) _at_ LocA(K,Trace(L))

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Classification
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Category Gl of global compositions Objects KI
functor K which is covered by a finite
atlas I (Ki) of local compositions in LocA
at address A Morphisms KI at address A
LJ at address B f/a KI LJ f
natural transformation, a A B address
change f induces local morphisms fij/a on the
charts
Classification
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Have Grothendieck topology Cov(Gl) on
Gl Covering families (fi/ai KIi LJ)i are
finite, generating families.
Classification

• Theorem
• Cov(Gl) is subcanonical
• The presheaf GF KI gt GF(KI) of global affine
  functions is a sheaf.

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Have universal construction of a resolution of
KI res ADn KI It is determined only by
the KI address A and the nerve n of the covering
atlas I.
Classification
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• Theorem (global addressed geometric
  classification)
• Let A locally free of finie rank over
  commutative ring R
• Consider the objective global compositions KI at
  A with ()
• locally free chart modules R.Ki
• the function modules GF(Ki) are projective
• (i) Then KI can be reconstructed from the
  coefficient system of
• retracted functions
• resF(KI) Í F(ADn)
• (ii) There is a subscheme Jn of a projective
  R-scheme of finite type whose points w Spec(S)
  Jn parametrize the isomorphism classes of
  objective global compositions at address SRA
  with ().

Classification
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• Applications of classification
• String Quartet Theory Why four strings?
• Composition Generic compositional material
• Performance Theory Why deformation?

Classification
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• There are models for these musicological topics
• Tonal modulation in well-tempered and
  just intonation and general scales
• Classical Fuxian counterpoint rules
• Harmonic function theory
• String quartet theory
• Performance theory
• Melody and motive theory
• Metrical and rhythmical structures
• Canons
• Large forms (e.g. sonata scheme)
• Enharmonic identification

Models
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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!
Anthropomorphic 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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12 _at_ 3 x 4
Unification
K 12 e.0,3,4,7,8,9 consonances
D 12 e.1,2,5,6,10,11 dissonances
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Unification
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12 _at_ 0 _at_ 12
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D C-dominant triad T C-tonic triad
K
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The Topos of Music Geometric Logic of Concepts,
Theory, and Performance
in collaboration with Moreno Andreatta, Jan
Beran, Chantal Buteau, Karlheinz Essl, Roberto
Ferretti, Anja Fleischer, Harald Fripertinger,
Jörg Garbers, Stefan Göller, Werner Hemmert,
Mariana Montiel, Andreas Nestke, Thomas Noll,
Joachim Stange-Elbe, Oliver Zahorka
www.encylospace.org