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

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology Sylvain Auroux: La sémiotique des encyclopédistes (1979) Three encyclopedic caracteristics of general validity: unité (unity) grammar of synthetic discourse philosophy unité (unity) grammar of synthetic discourse philosophy intégralité (completeness) all facts dictionary intégralité (completeness) all facts dictionary discours (discourse) encyclopedic ordering representation discours (discourse) encyclopedic ordering representation Jean le Rond D‘Alembert Jean le Rond D‘Alembert Denis Diderot Denis Diderot 1751

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology ramification type ~ completeness reference ~ unity linear ordering ~ discourse

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology (Kritik der reinen Vernunft, B 324) Man kann einen jeden Begriff, einen jeden Titel, darunter viele Erkenntnisse gehören, einen logischen Ort nennen. You may call any concept, any title (topic) comprising multiple knowledge, a logical site. Immanuel Kant concepts are points in concept spaces

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology (coordinates) (coordinates) <form_name><type>(coordinator) F1F1F1F1 FnFnFnFn D1D1D1D1 D s-1 DsDsDsDs form denotator

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology Simple Forms = Elementary Spaces

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology <form_name><type>(coordinator) example:‘Loudness’ (coordinates) (coordinates) example:‘mezzoforte’ A = STRG = set of strings (words) from a given alphabet a string of letters example:mf Simple Simple 1

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology <form_name><type>(coordinator) example:‘HiHat-State’ (coordinates) (coordinates) example:‘openHiHat’ A = Boole = {NO, YES} (boolean) boolean value example:YES Simple Simple 2

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology <form_name><type>(coordinator) example:‘Pitch’ (coordinates) (coordinates) example:‘thisPitch’ A = integers Ÿ = {...-2,-1,0,1,2,3,...} integer number from Ÿ example: b-flat ~ 58 Simple Simple 3

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology <form_name><type>(coordinator) example:‘Onset’ (coordinates) (coordinates) example:‘myOnset’ A = real (= decimal) numbers — real number from — example:11.25 Simple Simple 4

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology example:‘Eulerspace’ example:‘myEulerpoint’ Extend to more general mathematical spaces M! point in M e.g. Euler pitch spaces.... <form_name><type>(coordinator)<denotator_name><form_name>(coordinates) Simple octave fifth third Simple +

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology <form_name><type>(coordinator) A module M over a ring R (e.g., a real vector space) Simple Examples: M =  — 3 space for space music description M =  — 3 space for space music description M = – 3 pitch space o.log(2) + f.log(3) + t.log(5) M = – 3 pitch space o.log(2) + f.log(3) + t.log(5) M = Ÿ 12, Ÿ 3, Ÿ 4 for pitch classes M = Ÿ 12, Ÿ 3, Ÿ 4 for pitch classes M = Ÿ  Ÿ 365  Ÿ 24  Ÿ 60  Ÿ 60  Ÿ 28 (y:d:h:m:s:fr) for time M = Ÿ  Ÿ 365  Ÿ 24  Ÿ 60  Ÿ 60  Ÿ 28 (y:d:h:m:s:fr) for time M = ¬, Polynomials R[X] etc. for sound, analysis, etc. M = ¬, Polynomials R[X] etc. for sound, analysis, etc. ( Ÿ 12 ) ( Ÿ 12 )

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology

Rubato Composer: 1.Check the construction of rings and modules! 2.Check general simple forms! 3.Check construction of forms! 4.Check construction of denotators!

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology Compound Forms = Recursive Spaces

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology spaces/formsspaces/forms product/limitproduct/limitunion/colimitunion/colimit collections/powers ets exist three compound space types:

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology <form_name><type>(coordinator) example:‘Note’ (coordinates) (coordinates) example:‘myNote’ n denotators from F 1, F 1,... F n example (n=2): (‘myOnset’,’thisPitch’) Limit sequence F 1, F 2,... F n of n forms

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology example:‘Interval’ example:‘myInterval’ n denotators, plus arrow conditions example: (‘note 1 ’,’on’,’note 2 ’) Note Onset Note Note Onset Note <form_name><type>(coordinator)<denotator_name><form_name>(coordinates) Limit extend to diagram of n forms + functions F1F1F1F1 FnFnFnFn FiFiFiFi K-nets (networks!)

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology Db} J1J1J1J1 J2J2J2J2 J3J3J3J3 J4J4J4J4 Klumpenhouwer (hyper)networks

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology Ÿ 12 T4T4T4T4 T2T2T2T2 T 5.-1 T     Ÿ 12 T4T4T4T4 T2T2T2T2 T 5.-1 T limit

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology Rubato Composer: 1.Check the construction of limits!

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology <form_name><type>(coordinator) example:‘Orchestra’ (coordinates) (coordinates) example:‘mySelection’ one denotator for i-th form F i one denotator for i-th form F i example: Select a note from celesta Colimit sequence F 1, F 2,... F n of n forms

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology <form_name><type>(coordinator) Example:‘Motif’ (coordinates) (coordinates) example:‘thisMotif’ one form F A set of denotators of form F example: {n 1,n 2,n 3,n 4,n 5 } F = Note PowersetPowerset Power 1

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology <form_name><type>(coordinator) Example:‘Chord’ (coordinates) (coordinates) example:‘thisChord’ one form F A set of denotators of form F example: {p 1,p 2,p 3 } F = PitchClass PowersetPowerset Power 2

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology Rubato Composer: Check the construction of powerset denotators!

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology <form_name><type>(coordinator)Colimit diagram of n forms F1F1F1F1 FnFnFnFn FiFiFiFi Gluing together spaces of musical objects! Idea: take union of all F i and identify corresponding points under the given maps.

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology T n {c 1,c 2,...,c k } = {n+c 1, n+c 2,..., n+c k } mod 12 (transposition by n semitones) Result = set of n-transposition chord classes! Chord D = TnTnTnTn BTW: What would the Limit of D be? <form_name><type>(coordinator)Colimit F1F1F1F1 FnFnFnFn FiFiFiFi

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology — — OnsetOnsetLoudnessLoudnessDurationDurationPitchPitch NoteNote STRG Ÿ Note form

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology GeneralNoteGeneralNote — — OnsetOnsetLoudnessLoudnessDurationDurationPitchPitch NoteNote STRG Ÿ— — DurationDuration OnsetOnset PausePause GeneralNote form GeneralNote form

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology FM-Synthesis FM-Synthesis

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology NodeNode FM-ObjectFM-Object — — AmplitudeAmplitudePhasePhaseFrequencyFrequency FM-Synthesis FM-Synthesis — SupportSupport ModulatorModulator FM-ObjectFM-Object

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology NodeNode FM-ObjectFM-Object — — AmplitudeAmplitudePhasePhaseFrequencyFrequency FM-Synthesis FM-Synthesis— SupportSupport FM-ObjectFM-Object

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology ? Schenker Analysis GTTM Composition EmbellishmentsEmbellishments hierarchies!

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology macroscoremacroscore nodenode macroscoremacroscore scorescore NoteNote Flatten Nodify — — STRG ŸNoteNoteonsetonsetloudnessloudnessdurationdurationpitchpitchvoicevoice Ÿ

The denoteX notation for forms and denotators

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology 1.Forms Name:.TYPE(Coordinator); Name = word (string)Name = word (string) TYPE = one of the following: - Simple - Limit - Colimit - PowersetTYPE = one of the following: - Simple - Limit - Colimit - Powerset Coordinator = one of the following: - TYPE = Simple: STRING, Boole, Ÿ, — - TYPE = Limit, Colimit: A sequence F 1,... F n of form names - TYPE = Powerset: One form name FCoordinator = one of the following: - TYPE = Simple: STRING, Boole, Ÿ, — - TYPE = Limit, Colimit: A sequence F 1,... F n of form names - TYPE = Powerset: One form name F

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology 2.Denotators Name = word (string)Name = word (string) FORM = name of a defined formFORM = name of a defined form Coordinates = x, which looks as follows: - FORM:.Simple(F), then x is an element of F (STRING, Boole, Ÿ, — )Coordinates = x, which looks as follows: - FORM:.Simple(F), then x is an element of F (STRING, Boole, Ÿ, — ) - FORM:.Powerset(F), then x = {x 1, x 2, x 3,... x k } x i = F-denotators, only names x i : - FORM:.Limit(F 1,... F n ), then x = (x 1, x 2, x 3,... x n ) x i = F i -denotators, i = 1,...n - FORM:.Colimit(F 1,... F n ), then x = denotator of one F i (i>x, only names x:)

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology Exercise: A FM form and a denotator for this function: f(t) = sin(2  5t+3)+cos(t -sin(6  t+sin(  t+89)))A FM form and a denotator for this function: f(t) = sin(2  5t+3)+cos(t -sin(6  t+sin(  t+89)))NodeNode FM-ObjectFM-Object —— AmplitudeAmplitudePhasePhaseFrequencyFrequency — SupportSupportFM-ObjectFM-Object