1. II CONCEPT SPACES
II.4 (Thu Feb 08) Concepts and software for a theory of motifs: The MeloRubette

2. Motif Space MOT
Give concept of a note „point“: note = sequence of numerical coordinates n = (on, pn, ln,...) (onset o, pitch p mandatory, the others may be present but we do not use them here)
Definition: A motif M in the given note space (fixed for given context) is a set M = {n1, n2,... nk} of notes with oni ≠ onj for i ≠ j.

3. MOT = space of all motives (of notes of given type)
Yes!
No!

4. 2. Motif Space MOT(S) of a Composition S
S = set of notes in our note space. Attention: S has only one type of notes, although one could generalize.
Want the Submotif Extension Axiom SEA for MOT(S):
Similar motives have similar submotives
Will make that more precise later!

5. More examples from MeloRubette later!
Soprano voice in „Träumerei“, selection by Bruno Repp,
for S take all 28 Repp motives plus submotives of 2 or more notes -> 1,483 motives
for S take all motives plus submotives of notes & span 2 bars -> 237,736 motives
for S take all motives of whole composition plus submotives of notes & span 1 bar -> 711,198 motives
Span & # notes
More examples from MeloRubette later!

6. 3. Contour Types (abstraction/shapes for Reti)
t: MOT  t (= space of contours)
Examples:
COM(M) = matrix of +1, -1, or 0
Dia(M) = sequence of successive pitch differences
India(M) = codiagonal of COM(M)
Elast(M) = sequence of angles + rel. lengths
Rigid(M) = sequence of op-notes in onset order
Toroid(M) = pitch classes of Dia(M)

7. COM shape type
Motif1
Motif2
COM(Motif1)
COM(Motif2)

8. Example: space of notes with onset, pitch, duration
COM(M) = matrix of +1, -1, or 0
Dia(M) = sequence of succ. pitch differences
India(M) = codiagonal of COM(M)
Elast(M) = sequence of angles + rel. lengths
Rigid(M) = sequence of op-notes in onset order
Toroid(M) = pitch classes of Dia(M)

9. 4. Distances Between Contours („Metrics“)
We mostly take Euclidean distance between vectors x, y (sequences x = (x1, x2,... xk) of numbers) x y
Toroid type???

10. Ÿ12  Ÿ3 x Ÿ4 z ~> (z mod 3, -z mod4) 4.u+3.v <~ (u,v) 8 11 4 31 2 3 4 5 6 7 8 9 10 11 11 10 8 1 2 3 4 5 6 7 9

11. 2 5
minor third 10
major third
d(x,y) = min. # major/minor thirds from x to y

12. 5. Paradigmatic Groups (von Ehrenfels‘ „Gestalt“, Reti‘s „imitation“, Ruwet‘s and Nattiez‘ „paradigmatic theme“)
P = collection of transformations on notes, such as transpositions T, „da capo“, inversions I, retrograde R, and combinations thereof TI, IR, TIR, etc.
„Group“ = P must be closed under combinations of its members.
P acts on motives M = {n1, n2,... nk}: if g is transformation in P, then we define
gM = {gn1, gn2,... gnk}
Want also corresponding action on contours!

13. 6. Isometries
Want that the paradigmatic group actions preserve distances among the selected motif contours, i.e., the transformations are isometries.
This is evident for all contours except the toroid type!
What if we make a seventh or fourth transformation?
x ~> 7x x ~> 5x
MIRACLE ON THE THIRD TORUS!

14. 900 1800 9 1200 900 180 =inversion 90 refl. =fourth circle11 10 8 1 2 3 4 5 6 7 9
1800
900
1200 9
180 =inversion
90 refl. =fourth circle
90 rot.=minor third chain
120 tilt.=major third chain

15. 7. Gestalts
Given a contour type t: MOT  t and a paradigmatic group P, the gestalt of a motif M is the set of all motives N such that there is a transformation g in P, which transforms M to a motif gM having the contour of N: t(gM) = t(N) N gM
t(gM) = t(N) t M

16. 8. Gestalt Distances
for two motives M, N of same # of contour entries, take nearest distances of motif contours from the two gestalts.
gestalt(M)
gestalt(N)
gd(M,N)

17. 9. Motif Topology
Like for metrical analysis, we have „ball neighbourhoods“ for motives M. Give contour type t and number  > 0. Then two cases:
B(M) contains all N with same contour # as M such that gd(M,N) < .
2. B(M) contains also all N* with more contour # than M, but such that N* contains one submotif N of same # as M and gd(M,N) < . N M N*

18. 10. Inheritance
The Ball neighbourhoods not always intersect properly!
Need inheritance property! M M‘
Inheritance property: If two motives are near to each other, then their submotives must also be near to each other.
India type violates this!

19. 11. Melodic weights of motives and notes for given shape & group and  > 0
presence
content
weight(Motif) = presence(Motif) x content(Motif)
presence
content
weight(note x) =  weights of motives containing note x

20. “... is everywhere in the composition...“

21. Chantal Buteau‘s work (PhD 2003 thesis A Topological Model of Motivic Structure and Analysis of Music: Theory and Operationalization. Dissertation, Zürich 2003)
Motivic evolution tree

22. From Contour Similarity to Motivic Topologies
Musicae Scientiae Sept. 2000, Ch. Buteau & G. Mazzola

23. 12. MeloRubette, Examples, and Discussion
OsX

24. Schumann‘s Träumerei

25. Webern‘s Variationen op. 27/II

26. Beethoven‘s 5th Symphony

27. Beethoven‘s 5th Symphony
pitch
weight
onset