Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology IIISymbolic Reality III.2 (We Nov 16) Denotators I—definition of a universal concept space

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IIIIII Symbolic RealitySymbolic Reality

III.2 III.2 (We Nov 16) (We Nov 16) Denotators I—definition of a universal concept space Denotators I—definition of a universal concept space and notationsand notations

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Sylvain Auroux: Sylvain Auroux: La sémiotique des encyclopédistesLa sémiotique des encyclopédistes (1979) (1979)

Three encyclopedic caracteristics of general validity:Three encyclopedic caracteristics of general validity:

• unité (unity)unité (unity) grammar of synthetic discourse grammar of synthetic discourse philosophyphilosophy

• intégralité (completeness)intégralité (completeness) all factsall facts dictionarydictionary

• discours (discourse)discours (discourse) encyclopedic orderingencyclopedic ordering representationrepresentation

Jean le Rond D‘AlembertJean le Rond D‘Alembert Denis DiderotDenis Diderot17511751

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ramification typeramification type~ completeness~ completeness

reference ~ unityreference ~ unity

linear ordering ~ discourselinear ordering ~ discourse

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(Kritik der reinen Vernunft, B 324)(Kritik der reinen Vernunft, B 324)

Man kann einen jeden Begriff,Man kann einen jeden Begriff,einen jeden Titel, einen jeden Titel,

darunter viele Erkenntnisse gehören,darunter viele Erkenntnisse gehören,einen logischen Ort nennen.einen logischen Ort nennen.

You may call any concept, You may call any concept, any title (topic) any title (topic)

comprising multiple knowledge,comprising multiple knowledge,a logical site.a logical site.

Immanuel KantImmanuel Kantconcepts are points in concept spaces

concepts are points in concept spaces

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<denotator_name<denotator_name><form_name>(><form_name>(coordinates) coordinates)

<form_name><type>(coordinator)<form_name><type>(coordinator)

FF11

FFnn

DD11

DDs-1s-1

DDss

formform

denotatordenotator

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Simple Forms = Elementary Spaces Simple Forms = Elementary Spaces

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example:example:‘‘Loudness’Loudness’

<denotator_name><denotator_name><form_name><form_name>(coordinates) (coordinates)

example:example:‘‘mezzoforte’mezzoforte’

A = A = STRGSTRG = = set of strings (words) set of strings (words) from a given alphabetfrom a given alphabet

a string of lettersa string of letters

example:example:mfmf

SimpleSimple

Simple 1Simple 1

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example:example:‘‘HiHat-State’HiHat-State’


example:example:‘‘openHiHat’openHiHat’

A = Boole = {NO, YES} (boolean)A = Boole = {NO, YES} (boolean)

boolean valueboolean value

example:example:YESYES

SimpleSimple

Simple 2Simple 2

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example:example:‘‘Pitch’Pitch’


example:example:‘‘thisPitch’thisPitch’

A = integers A = integers Ÿ Ÿ == {...-2,-1,0,1,2,3,...}{...-2,-1,0,1,2,3,...}

integer numberinteger numberfromfrom Ÿ Ÿ

example:example:b-flat ~ 58b-flat ~ 58

SimpleSimple

Simple 3Simple 3

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example:example:‘‘Onset’Onset’


example:example:‘‘myOnset’myOnset’

A = A = real (= decimal) numbersreal (= decimal) numbers — —

real numberreal numberfromfrom — —

example:example:11.2511.25

SimpleSimple

Simple 4Simple 4

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example:example:‘‘Eulerspace’Eulerspace’

example:example:‘‘myEulerpoint’myEulerpoint’

Extend to more general Extend to more general mathematical spaces mathematical spaces MM!!

point inpoint inMM

e.g. Euler pitch e.g. Euler pitch spaces....spaces....


<denotator_name><form_name>(coordinates) <denotator_name><form_name>(coordinates)

SimpleSimple

octave

fifth

third

Simple +Simple +

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A module A module MM over a ring over a ring RR (e.g., a (e.g., a real vector space)real vector space)

SimpleSimple

Examples:Examples:

• M = M = ——33 space for space music description space for space music description• M = M = ––3 3 pitch space o.log(2) + f.log(3) + t.log(5)pitch space o.log(2) + f.log(3) + t.log(5)• M = M = ŸŸ1212,, ŸŸ33,, ŸŸ4 4 for pitch classesfor pitch classes

• M =M = ŸŸ ŸŸ365 365 ŸŸ24 24 ŸŸ60 60 ŸŸ60 60 ŸŸ28 28 (y:d:h:m:s:fr) for time(y:d:h:m:s:fr) for time

• M = M = ¬¬,, Polynomials R[X] etc. for sound, analysis, etc.Polynomials R[X] etc. for sound, analysis, etc.

<PitchClass><Simple>(<PitchClass><Simple>(ŸŸ1212))

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Compound Forms = Recursive SpacesCompound Forms = Recursive Spaces

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spaces/formsspaces/formsspaces/formsspaces/forms

product/limitproduct/limitproduct/limitproduct/limit union/colimitunion/colimitunion/colimitunion/colimit collections/collections/powersetspowersets

collections/collections/powersetspowersets

exist three compound space types:exist three compound space types:exist three compound space types:exist three compound space types:

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example:example:‘‘Note’Note’


example:example:‘‘myNote’myNote’

n denotators fromn denotators fromFF11, , FF11,... ,... FFnn

example (n=2):example (n=2): ((‘myOnset’‘myOnset’,,’thisPitch’’thisPitch’))

LimitLimit

sequence sequence FF11, , FF22,... ,... FFnn

of n formsof n forms

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example:example:‘‘Interval’Interval’

example:example:‘‘myInterval’myInterval’

n denotators, plus arrow n denotators, plus arrow conditions conditions example: example: (‘note(‘note11’,’on’,’note’,’on’,’note22’)’)

Note Onset NoteNote Onset Note


<denotator_name><form_name>(coordinates) <denotator_name><form_name>(coordinates)

LimitLimit

extend to diagram ofextend to diagram ofn forms + functionsn forms + functions

FF11

FFnnFFii

K-nets (networks!)K-nets (networks!)

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Db}

JJ11 JJ22 JJ33 JJ44

Klumpenhouwer (hyper)networksKlumpenhouwer (hyper)networks

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ŸŸ1212

ŸŸ1212

ŸŸ1212

TT44

TT22

TT55.-1.-1 TT1111.-1.-1

33 77

22 44

ŸŸ1212

ŸŸ1212

ŸŸ1212

ŸŸ1212

TT44

TT22

TT55.-1.-1 TT1111.-1.-1limit

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example:example:‘‘Orchestra’Orchestra’


example:example:‘‘mySelection’mySelection’

one denotator for one denotator for i-th form i-th form FFii

example:example:Select a note from Select a note from celestacelesta

ColimitColimit

sequence sequence FF11, , FF22,... ,... FFnn

of n formsof n forms

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Example:Example:‘‘Motif’Motif’


example:example:‘‘thisMotif’thisMotif’

one form one form FF

A set of A set of denotators denotators of form of form FF

example:example:{n{n11,n,n22,n,n33,n,n44,n,n55}} F = NoteF = Note

PowersetPowersetPowersetPowerset

Power 1Power 1

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Example:Example:‘‘Chord’Chord’


example:example:‘‘thisChord’thisChord’

one form one form FF

A set of A set of denotators denotators of form of form FF

example:example:{p{p11,p,p22,p,p33}} F = PitchClassF = PitchClass

PowersetPowersetPowersetPowerset

Power 2Power 2

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ColimitColimit

diagram ofdiagram ofn formsn forms

FF11

FFnnFFii

Gluing together spaces

Gluing together spaces

of musical objects!of musical objects!

Idea: take union of all FIdea: take union of all Fii and identify corresponding points and identify corresponding points

under the given maps.under the given maps.

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TTnn{c{c11,c,c22,...,c,...,ckk} =} = {n+c{n+c11, n+c, n+c22,..., n+c,..., n+ckk} mod 12} mod 12

(transposition by n semitones)(transposition by n semitones)

Result = set of n-transposition chord classes!Result = set of n-transposition chord classes!

ChordChordD D ==

TTnn

BTW: What would the Limit of BTW: What would the Limit of DD be?be?


ColimitColimit

FF11

FFnnFFii

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—— ——

OnsetOnsetOnsetOnset LoudnessLoudnessLoudnessLoudness DurationDurationDurationDurationPitchPitchPitchPitch

NoteNoteNoteNote

STRGSTRGŸŸ

Note formNote formNote formNote form

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GeneralNoteGeneralNoteGeneralNoteGeneralNote

—— ——

OnsetOnsetOnsetOnset LoudnessLoudnessLoudnessLoudness DurationDurationDurationDurationPitchPitchPitchPitch

NoteNoteNoteNote

STRGSTRGŸŸ—— ——

DurationDurationDurationDurationOnsetOnsetOnsetOnset

PausePausePausePause

GeneralNote formGeneralNote form

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FM-SynthesisFM-Synthesis

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NodeNodeNodeNode

FM-ObjectFM-ObjectFM-ObjectFM-Object

—— ——

AmplitudeAmplitudeAmplitudeAmplitude PhasePhasePhasePhaseFrequencyFrequencyFrequencyFrequency


——

SupportSupportSupportSupport ModulatorModulatorModulatorModulator


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NodeNodeNodeNode


—— ——



——

SupportSupportSupportSupport


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??

Schenker AnalysisSchenker AnalysisGTTMGTTM

CompositionComposition

Embellishments Embellishments Embellishments Embellishments

hierarchies!hierarchies!

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macroscoremacroscoremacroscoremacroscore

nodenodenodenode

macroscoremacroscoremacroscoremacroscorescorescorescorescore

NoteNoteNoteNote FlattenFlatten

NodifyNodify

—— ——STRGSTRGŸŸ

NoteNoteNoteNote

onsetonsetonsetonset loudnessloudnessloudnessloudness durationdurationdurationdurationpitchpitchpitchpitch voicevoicevoicevoice

ŸŸ

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The denoteX notation for forms and denotators The denoteX notation for forms and denotators

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1.1. FormsForms

Name:.TYPE(Coordinator);Name:.TYPE(Coordinator);

• Name = word (string)Name = word (string)

• TYPE = one of the following:TYPE = one of the following:- Simple- Simple- Limit- Limit- Colimit- Colimit- Powerset- Powerset

• Coordinator = one of the following:Coordinator = one of the following:- TYPE = Simple: STRING, Boole, - TYPE = Simple: STRING, Boole, ŸŸ, , —— - TYPE = Limit, Colimit: A sequence F- TYPE = Limit, Colimit: A sequence F11,... F,... Fnn of form names of form names

- TYPE = Powerset: One form name F- TYPE = Powerset: One form name F

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2.2. DenotatorsDenotators

Name:@FORM(Coordinates);Name:@FORM(Coordinates);

• Name = word (string)Name = word (string)

• FORM = name of a defined formFORM = name of a defined form

• Coordinates = x, which looks as follows:Coordinates = x, which looks as follows:- FORM:.Simple(F), then x is an element of F - FORM:.Simple(F), then x is an element of F

(STRING, Boole, (STRING, Boole, ŸŸ, , ——))

- FORM:.Powerset(F), then x = {x- FORM:.Powerset(F), then x = {x11,, xx22,, xx33,... x,... xkk}}

xxi i = F-denotators, = F-denotators, only names only names

xxii::

- FORM:.Limit(F- FORM:.Limit(F11,... F,... Fnn), then x = (x), then x = (x11, x, x22, x, x33,... x,... xnn))

x xi i = F= Fii-denotators, i = 1,...n-denotators, i = 1,...n

- FORM:.Colimit(F- FORM:.Colimit(F11,... F,... Fnn), then x = denotator of one F), then x = denotator of one Fi i

(i>x, (i>x, only names x:only names x:) )

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Exercise:Exercise:

• A FM form and a denotator for this function:A FM form and a denotator for this function:

f(t) = -12.5 sin(2f(t) = -12.5 sin(25t+3)+cos(t -sin(65t+3)+cos(t -sin(6t+sin(t+sin(t+89))) t+89)))

NodeNodeNodeNode


—— ——


——

SupportSupportSupportSupport


Documents

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology IIISymbolic Reality III.2 (We Nov 16) Denotators I—definition of a universal concept space