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Mathematical Music Theory — Status Quo 2000. Guerino Mazzola U & ETH Zürich Internet Institute for Music Science guerino@mazzola.ch www.encyclospace.org. Time Table The Concept Framework Global Classification Models and Methods Towards Grand Unification . Contents. 1978 1980 1981

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slide1

Mathematical Music Theory

— Status Quo 2000

Guerino Mazzola

U & ETH Zürich

Internet Institute for Music Science

guerino@mazzola.ch

www.encyclospace.org

slide2

Time Table

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

Contents

slide3

1978

1980

1981

1984

1985

1986

1988

1990

1992

1994

1995

1996

1998

1999

2000

2001

Music

Theory

Software

Grants

Status quo

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

slide4

translation

dilinear

  • Mod = category of modules + diaffine morphisms:
  • A = R-module, B = S-module
  • Dilin(A,B) = (l,f) f:A ® B additive, l:R ® S ring homomorphism f(r.a) = l(r).f(a)
  • eb(x) = b+x; translation on B
  • A@B = eB.Dilin(A,B)

Concepts

eb.f: A ® B ® B

slide5

Topos of presheaves over Mod

Mod@ = {F: Mod ®Sets, contravariant}

Example: representable presheaf

@B: @B(A) = A@B

F(A) =: A@F A = address

Concepts

Yoneda Lemma

The functor @: Mod®Mod@ is fully faithfull.

B ~> @B

slide6

K Í B

B

Concepts

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

B @ 0Ÿ@B

K Í A@B

K Í 0Ÿ@B

  • A = Ÿn: sequences (b0,b1,…,bn)
  • A = B: self-addressed tones
  • Need general addresses A

F = W@B

A = 0Ÿ

KÎA @F

F = presheaf over Mod

F = @B

slide7

Functor(F)

Frame(√)

F = Form name

one of four „space types“

a diagramn √ in Mod@

Forms

a monomorphism in Mod@id: Functor(F) >® Frame(√)

Concepts

  • Frame(√)-space for type
  • simple √ = Æ~> @B simple(√) =@B
  • limit √ = Form-Name-Diagram ® Mod@
  • limit(√) = lim(Form-Name-Diagram ® Mod@)
  • colimit √ = Form-Name-Diagram ® Mod@
  • colimit(√) = colim(Form-Name-Diagram ® Mod@)
  • power √ = Form-Name F ~> Functor(F)
  • power(√) =WFunctor(F)
slide8

A

address A

K

Functor(F)

Frame(√)

Denotators

D = denotator name

Concepts

KÎA @ Functor(F)

„A-valued point“

Form F

slide9

Satellites

AnchorNote

MakroNote

Onset

Pitch

Loudness

Duration

STRG

Ÿ

MakroNote

  • Ornaments
  • Schenker Analysis

Concepts

slide10

Java Classes for

Modules,

Forms, and Denotators

L

L

S

S

RUBATO®

Concepts

slide11

Fr

F2

x3

xn

F1

x2

x1

Galois Theory

Form Theory

Defining equation

Defining diagram

Concepts

fS(X) = 0

id √(F)

Field S

Form System

Mariana Montiel Hernandez, UNAM

slide12

objects

a = affine morphism

f, h = natural transformations

morphisms

specify „address change“ a

Category Loc of local compositions

Type = PowerF ~> Functor(F) = G

Classification

local composition K Î A@WG

K Í @A´G

generalizes K Í A@G „objective“ local compositions

K Í @A´G

@a´h

f/a

L Í @B´H

slide13

Embedding

functor

Trace

functor

ObLoc

Loc

Classification

ObLocA

LocA

  • Theorem
  • Loc is finitely complete (while ObLoc is not!)
  • On ObLocA andLocA Embedding and Trace
  • are an adjoint pair:
  • ObLocA(Embedding(K),L) @ LocA(K,Trace(L))
slide14

K

A@Gi◊ Ki

@A´Gi◊ Ki

KtÍ A@Gt

KtÍ @A´Gt

Kit

Kti

local isomorphism/A

Classification

slide15

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

slide16

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 ~> GF(KI) of global affine functions is a sheaf.
slide17

res

KI

ADn*

Have universal construction of a „resolution of KI“

res:ADn*® KI

It is determined only by the KIaddress A and the

nerve n* of the covering atlas I.

Classification

slide18

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
  • res*F(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 SƒRA with (*).

Classification

slide19

Applications of classification:

  • String Quartet Theory: Why four strings?
  • Composition: Generic compositional material
  • Performance Theory: Why deformation?

Classification

slide20

Mazzola

Mazzola/Noll

Noll

Ferretti

Nestke

Noll

  • 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

slide21

What is a mathematical model of a musical phenomenon?

Precise Concept Framework

Instance specification

Formal process restatement

Proof of structure theorems

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!

slide22

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?
slide23

II

III

IV

V

VI

VII

I

Models

slide24

g

graviton

gluon

W+

electromagnetic

force

strong force

weak force

gravitation

quantum = set of

pitch classes = M

S(3)

T(3)

force = symmetry between

S(3) and T(3)

k

k

Models

slide25

IVC

IIEb

VIIEb

M(3)

IIC

VEb

VIIC

IIIEb

VC

Eb(3)

C(3)

Models

slide26

Ÿ12[e]

ƒ1

e e.2.5

Ÿ12 @Ÿ3 x Ÿ4

Unification

K = Ÿ12 +e.{0,3,4,7,8,9} = consonances

D = Ÿ12 +e.{1,2,5,6,10,11} = dissonances

slide27

Rules of CounterpointFollowing J.J. Fux

C/D Symmetry inHuman Depth-EEG

Extension to ExoticInterval Dichotomies

Unification

slide28

0

Ÿ12 @Ÿ12

X = { }

Trans(X,X)

Ÿ12@ 0@Ÿ12

slide29

ƒ1

Z12[e]

Z12

Trans(D, T) = Trans(K,K)|ƒe

Z12 [e] @ Z12 [e]

Z12 @ Z12

ƒe

D = C-dominant triad

T = C-tonic triad

K

slide30

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