1. II CONCEPT SPACES
II.6 (Thu Feb 22) Denotators—definition of a universal concept space, notations, and examples

2. Sylvain Auroux: La sémiotique des encyclopédistes (1979)
Jean le Rond D‘Alembert
Denis Diderot
1751
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) encyclopedic ordering representation

3. ramification type ~ completeness
reference ~ unity
linear ordering ~ discourse

4. concepts are points in concept spaces
(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

5. D1 F1 Ds-1 Ds Fn <denotator_name><form_name>(coordinates)
<form_name><type>(coordinator) F1 Fn

6. Simple Forms = Elementary Spaces

7. Simple 1
<form_name><type>(coordinator)
example:
‘Loudness’
A = STRG = set of strings (words) from a given alphabet
Simple
<denotator_name><form_name>(coordinates)
example:
‘mezzoforte’
a string of letters
example: mf

8. Simple 2
<form_name><type>(coordinator)
example:
‘HiHat-State’
A = Boole = {NO, YES} (boolean)
Simple
<denotator_name><form_name>(coordinates)
example:
‘openHiHat’
boolean value
example:
YES

9. Simple 3
<form_name><type>(coordinator)
example:
‘Pitch’
A = integers Ÿ = {...-2,-1,0,1,2,3,...}
Simple
<denotator_name><form_name>(coordinates)
example:
‘thisPitch’
integer number
from Ÿ
example:
b-flat ~ 58

10. Simple 4
<form_name><type>(coordinator)
example:
‘Onset’
A = real (= decimal) numbers —
Simple
<denotator_name><form_name>(coordinates)
example:
‘myOnset’
real number
from —
example:
11.25

11. <form_name><type>(coordinator)
Simple +
<form_name><type>(coordinator)
<denotator_name><form_name>(coordinates)
Simple
example:
‘Eulerspace’
Extend to more general mathematical spaces M!
example:
‘myEulerpoint’
point in M
e.g. Euler pitch spaces....
octave
fifth
third

12. <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 = Ÿ12, Ÿ3, Ÿ4 for pitch classes
M = Ÿ Ÿ365  Ÿ24  Ÿ60  Ÿ60  Ÿ28 (y:d:h:m:s:fr) for time
M = ¬, Polynomials R[X] etc. for sound, analysis, etc.

13. Compound Forms = Recursive Spaces

14. spaces/forms exist three compound space types: product/limit
union/colimit
collections/powersets

15. Limit <form_name><type>(coordinator)
example:
‘Note’
sequence F1, F1,... Fn of n forms
Limit
<denotator_name><form_name>(coordinates)
example:
‘myNote’
n denotators from F1, F1,... Fn
example (n=2):
(‘myOnset’,’thisPitch’)

16. Limit K-nets (networks!) <form_name><type>(coordinator) F1
<denotator_name><form_name>(coordinates)
Limit
extend to diagram of
n forms + functions F1 Fn Fi
example:
‘Interval’
K-nets (networks!)
example:
‘myInterval’
n denotators, plus arrow conditions
example:
(‘note1’,’on’,’note2’)
Note Onset Note

17. Db} J1 J2 J3 J4
Klumpenhouwer (hyper)networks

18. Ÿ12 T4 T2
T5.-1
T11.-1 D 3 7 2 4 
(3, 7, 2, 4)  lim(D)

19. Colimit <form_name><type>(coordinator)
example:
‘Orchestra’
sequence F1, F1,... Fn of n forms
Colimit
<denotator_name><form_name>(coordinates)
example:
‘mySelection’
one denotator for i-th form Fi
example:
Select a note from celesta

20. Gluing together spaces of musical objects!
<form_name><type>(coordinator)
diagram of
n forms F1 Fn Fi
Colimit
Idea: take union of all Fi and identify corresponding points under the given maps.
Gluing together spaces
of musical objects!

21. Powerset <form_name><type>(coordinator) one form F
Example:
‘Motif’
one form F
Powerset
<denotator_name><form_name>(coordinates)
A set of
denotators
of form F
example:
{n1,n2,n3,n4,n5}
F = Note
example:
‘thisMotif’

22. <form_name><type>(coordinator)
Colimit(Dn)
Chord
Dn = Tn
Chord = Power(PitchClass)
PitchClass = Simple(Ÿ12)
Tn{c1,c2,...,ck} = {n+c1, n+c2,..., n+ck} mod 12 (transposition by n semitones)
Result = set of n-transposition chord classes!
BTW: What would Limit(Dn) be?

23. Note
Onset
Loudness
Duration
Pitch — Ÿ
STRG —

24. GeneralNote Pause Note Onset Duration Onset Loudness Duration Pitch —Ÿ
STRG —

25. FM-Synthesis

26. FM-Object Node Support Modulator Amplitude Phase Frequency FM-Object —
FM-Synthesis
FM-Object
Node
Support
Modulator
Amplitude
Phase
Frequency
FM-Object — — —

27. Examples:
A FM form and a denotator for this function:

29. Examples:
The form of Schumann‘s Träumerei score (see also Gérard Milmeister‘s documentation with Mariana Montiel‘s realization of form and denotator)

30. hounted