Guerino Mazzola Spring 2018 Music Informatics Seminar Denotatorsdefinition

Guerino Mazzola (Spring 2018 ©): Music Informatics Seminar Denotators—definition of a universal concept space and notations

Guerino Mazzola (Spring 2018 ©): Music Informatics Seminar Jean le Rond D‘Alembert 1751 Denis Diderot Sylvain Auroux: La sémiotique des encyclopédistes (1979) Three encyclopedic caracteristics of general validity: • unité (unity) grammar of synthetic discourse philosophy • intégralité (completeness) all facts dictionary • discours (discourse) representation encyclopedic ordering

Guerino Mazzola (Spring 2018 ©): Music Informatics Seminar ramification type ~ completeness linear ordering ~ discourse reference ~ unity

Guerino Mazzola (Spring 2018 ©): Music Informatics Seminar (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. con ce pt s are Immanuel Kant poi nt s i n co nc ept sp ac es

Guerino Mazzola (Spring 2018 ©): Music Informatics Seminar <denotator_name><form_name>(coordinates) denotator form D 1 <form_name><type>(coordinator) F 1 Ds-1 Fn Ds

Guerino Mazzola (Spring 2018 ©): Music Informatics Seminar Simple Forms = Elementary Spaces

Guerino Mazzola (Spring 2018 ©): Music Informatics Seminar <form_name><type>(coordinator) Simple 1 A = STRG = set of strings (words) from a given alphabet example: ‘Loudness’ Simple <denotator_name><form_name>(coordinates) a string of letters example: ‘mezzoforte’ example: mf

Guerino Mazzola (Spring 2018 ©): Music Informatics Seminar <form_name><type>(coordinator) Simple 2 A = Boole = {NO, YES} (boolean) example: ‘Hi. Hat-State’ Simple <denotator_name><form_name>(coordinates) boolean value example: ‘open. Hi. Hat’ example: YES

Guerino Mazzola (Spring 2018 ©): Music Informatics Seminar <form_name><type>(coordinator) Simple 3 A = integers Ÿ = {. . . -2, -1, 0, 1, 2, 3, . . . } example: ‘Pitch’ Simple <denotator_name><form_name>(coordinates) integer number from Ÿ example: ‘this. Pitch’ example: b-flat ~ 58

Guerino Mazzola (Spring 2018 ©): Music Informatics Seminar <form_name><type>(coordinator) Simple 4 A = real (= decimal) numbers — example: ‘Onset’ Simple <denotator_name><form_name>(coordinates) real number from — example: ‘my. Onset’ example: 11. 25

Simple + Guerino Mazzola (Spring 2018 ©): Music Informatics Seminar <form_name><type>(coordinator) Extend to more general mathematical spaces M! example: ‘Eulerspace’ Simple <denotator_name><form_name>(coordinates) third point in M example: ‘my. Eulerpoint’ e. g. Euler pitch spaces. . octave fifth

Guerino Mazzola (Spring 2018 ©): Music Informatics Seminar <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 pitch space o. log(2) + f. log(3) + t. log(5) M = Ÿ 12, Ÿ 3, Ÿ 4 for pitch classes M = Ÿ Ÿ 365 Ÿ 24 Ÿ 60 Ÿ 28 (y: d: h: m: s: fr) for time M = ¬, Polynomials R[X] etc. for sound, analysis, etc. <Pitch. Class><Simple>(Ÿ 12)

Guerino Mazzola (Spring 2018 ©): Music Informatics Seminar Compound Forms = Recursive Spaces

Guerino Mazzola (Spring 2018 ©): Music Informatics Seminar spaces/forms exist three compound space types: product/limit union/colimit collections/powers ets

Guerino Mazzola (Spring 2018 ©): Music Informatics Seminar <form_name><type>(coordinator) sequence F 1, F 2, . . . F n of n forms example: ‘Note’ Limit <denotator_name><form_name>(coordinates) n denotators from F 1, . . . Fn example: ‘my. Note’ example (n=2): (‘my. Onset’, ’this. Pitch’)

Guerino Mazzola (Spring 2018 ©): Music Informatics Seminar <form_name><type>(coordinator) F 1 extend to diagram of n forms + functions example: ‘Interval’ Limit !) s k r o etw n ( s t e K-n <denotator_name><form_name>(coordinates) example: ‘my. Interval’ Fi Fn n denotators, plus arrow conditions example: (‘note 1’, ’on’, ’note 2’) Note Onset Note

Guerino Mazzola (Spring 2018 ©): Music Informatics Seminar Klumpenhouwer (hyper)networks J 1 J 2 J 3 J 4 Db}

Guerino Mazzola (Spring 2018 ©): Music Informatics Seminar T 4 T 5. -1 2 Ÿ 12 7 Ÿ 12 T 11. -1 Ÿ 12 T 2 Ÿ 12 3 4 Ÿ 12 limit T 4 T 5. -1 Ÿ 12 T 11. -1 T 2 Ÿ 12

Guerino Mazzola (Spring 2018 ©): Music Informatics Seminar <form_name><type>(coordinator) sequence F 1, F 2, . . . F n of n forms example: ‘Orchestra’ Colimit <denotator_name><form_name>(coordinates) one denotator for i-th form F i example: ‘my. Selection’ example: Select a note from celesta

Guerino Mazzola (Spring 2018 ©): Music Informatics Seminar <form_name><type>(coordinator) Power 1 one form F Example: ‘Motif’ Powerset <denotator_name><form_name>(coordinates) example: ‘this. Motif’ A set of denotators of form F example: {n 1, n 2, n 3, n 4, n 5} F = Note

Guerino Mazzola (Spring 2018 ©): Music Informatics Seminar <form_name><type>(coordinator) Power 2 one form F Example: ‘Chord’ Powerset <denotator_name><form_name>(coordinates) example: ‘this. Chord’ A set of denotators of form F example: {p 1, p 2, p 3} F = Pitch. Class

Guerino Mazzola (Spring 2018 ©): Music Informatics Seminar <form_name><type>(coordinator) F 1 diagram of n forms Fi Fn Colimit Idea: take union of all Fi and identify corresponding points s e c under the given maps. a p s er h t e g o t Gl ui ng l objects! ica s u m f o

Guerino Mazzola (Spring 2018 ©): Music Informatics Seminar <form_name><type>(coordinator) Colimit F 1 Fi Tn Fn D= Chord Tn{c 1, c 2, . . . , ck} = {n+c 1, n+c 2, . . . , n+ck} mod 12 (transposition by n semitones) Result = set of n-transposition chord classes! BTW: What would the Limit of D be?

Guerino Mazzola (Spring 2018 ©): Music Informatics Seminar Note form Note Onset — Pitch Loudness Ÿ STRG Duration —

Guerino Mazzola (Spring 2018 ©): Music Informatics Seminar General. Note form General. Note Pause Onset — Note Duration — Onset — Pitch Loudness Ÿ STRG Duration —

Guerino Mazzola (Spring 2018 ©): Music Informatics Seminar FM-Synthesis

Guerino Mazzola (Spring 2018 ©): Music Informatics Seminar FM-Synthesis FM-Object Node Support Amplitude — Modulator Frequency Phase — — FM-Object

Guerino Mazzola (Spring 2018 ©): Music Informatics Seminar FM-Synthesis FM-Object Node Support Amplitude — FM-Object Frequency Phase — —

Guerino Mazzola (Spring 2018 ©): Music Informatics Seminar Embellishments Schenker Analysis GTTM Composition ! s e i h c r a r e i h ?

Nodify Guerino Mazzola (Spring 2018 ©): Music Informatics Seminar score macroscore Flatten Note node macroscore Note onset — pitch loudness Ÿ STRG duration — voice Ÿ

Guerino Mazzola (Spring 2018 ©): Music Informatics Seminar a score form by Mariana Montiel

Guerino Mazzola (Spring 2018 ©): Music Informatics Seminar The denote. X notation forms and denotators

Guerino Mazzola (Spring 2018 ©): Music Informatics Seminar 1. Forms Name: . TYPE(Coordinator); • Name = word (string) • TYPE = 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, . . . Fn of form names - TYPE = Powerset: One form name F

Guerino Mazzola (Spring 2018 ©): Music Informatics Seminar 2. Denotators Name: @FORM(Coordinates); • Name = word (string) • FORM = name of a defined form • 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, . . . xk} xi = F-denotators, only names xi: - FORM: . Limit(F 1, . . . Fn), then x = (x 1, x 2, x 3, . . . xn) xi = Fi-denotators, i = 1, . . . n - FORM: . Colimit(F 1, . . . Fn), then x = denotator of one Fi (i>x, only names x: )

Guerino Mazzola (Spring 2018 ©): Music Informatics Seminar Exercise: • A FM form and a denotator for this function: f(t) = -12. 5 sin(2 5 t+3)+cos(t -sin(6 t+sin( t+89))) FM-Object Node Support Amplitude — Frequency — FM-Object Phase —