Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology IIISymbolic Reality III.2 (We Nov...
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Transcript of Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology IIISymbolic Reality III.2 (We Nov...
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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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<form_name><type>(coordinator)<form_name><type>(coordinator)
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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<form_name><type>(coordinator)<form_name><type>(coordinator)
example:example:‘‘HiHat-State’HiHat-State’
<denotator_name><denotator_name><form_name><form_name>(coordinates) (coordinates)
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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<form_name><type>(coordinator)<form_name><type>(coordinator)
example:example:‘‘Pitch’Pitch’
<denotator_name><denotator_name><form_name><form_name>(coordinates) (coordinates)
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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<form_name><type>(coordinator)<form_name><type>(coordinator)
example:example:‘‘Onset’Onset’
<denotator_name><denotator_name><form_name><form_name>(coordinates) (coordinates)
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....
<form_name><type>(coordinator)<form_name><type>(coordinator)
<denotator_name><form_name>(coordinates) <denotator_name><form_name>(coordinates)
SimpleSimple
octave
fifth
third
Simple +Simple +
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<form_name><type>(coordinator)<form_name><type>(coordinator)
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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<form_name><type>(coordinator)<form_name><type>(coordinator)
example:example:‘‘Note’Note’
<denotator_name><denotator_name><form_name><form_name>(coordinates) (coordinates)
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
<form_name><type>(coordinator)<form_name><type>(coordinator)
<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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<form_name><type>(coordinator)<form_name><type>(coordinator)
example:example:‘‘Orchestra’Orchestra’
<denotator_name><denotator_name><form_name><form_name>(coordinates) (coordinates)
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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<form_name><type>(coordinator)<form_name><type>(coordinator)
Example:Example:‘‘Motif’Motif’
<denotator_name><denotator_name><form_name><form_name>(coordinates) (coordinates)
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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<form_name><type>(coordinator)<form_name><type>(coordinator)
Example:Example:‘‘Chord’Chord’
<denotator_name><denotator_name><form_name><form_name>(coordinates) (coordinates)
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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<form_name><type>(coordinator)<form_name><type>(coordinator)
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?
<form_name><type>(coordinator)<form_name><type>(coordinator)
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
FM-SynthesisFM-Synthesis
——
SupportSupportSupportSupport ModulatorModulatorModulatorModulator
FM-ObjectFM-ObjectFM-ObjectFM-Object
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NodeNodeNodeNode
FM-ObjectFM-ObjectFM-ObjectFM-Object
—— ——
AmplitudeAmplitudeAmplitudeAmplitude PhasePhasePhasePhaseFrequencyFrequencyFrequencyFrequency
FM-SynthesisFM-Synthesis
——
SupportSupportSupportSupport
FM-ObjectFM-ObjectFM-ObjectFM-Object
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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
FM-ObjectFM-ObjectFM-ObjectFM-Object
—— ——
AmplitudeAmplitudeAmplitudeAmplitude PhasePhasePhasePhaseFrequencyFrequencyFrequencyFrequency
——
SupportSupportSupportSupport
FM-ObjectFM-ObjectFM-ObjectFM-Object