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

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.

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

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!

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 2 - 10 notes & span 2 bars -> 237,736 motives for S take all motives of whole composition plus submotives of 2 - 10 notes & span 1 bar -> 711,198 motives Span & # notes More examples from MeloRubette later!

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)

COM shape type Motif1 Motif2 COM(Motif1) COM(Motif2)

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)

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???

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

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

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!

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!

900 1800 9 1200 900 180 =inversion 90 refl. =fourth circle 11 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

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

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)

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*

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!

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

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

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

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

12. MeloRubette, Examples, and Discussion OsX

Schumann‘s Träumerei

Webern‘s Variationen op. 27/II

Beethoven‘s 5th Symphony

Beethoven‘s 5th Symphony pitch weight onset