A Structural Theory of Rhythm Notation based on Tree...
Transcript of A Structural Theory of Rhythm Notation based on Tree...
![Page 1: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/1.jpg)
A Structural Theory of Rhythm Notation based on Tree Representations and Term Rewriting
Jean Bresson Pierre Donat-Bouillud Florent Jacquemard
Ircam Paris, France
ENS Rennes Ircam
Paris, FranceINRIA
![Page 2: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/2.jpg)
Objectives• tree structured encoding of rhythm
• used for reasoning about rhythms with standard theoretical tools for tree processing
• for assisted algorithmic composition with OpenMusic
1. motivation and definition of a tree encoding for rhythm
2. tree languages (tree automata)
3. tree transformations by rewriting (equational theory), application to exploring equivalent rhythmic notations
4. properties, perspectives
![Page 3: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/3.jpg)
Trees Encodings of Rhythm
natural representation of common western notations for rhythms durations are defined hierarchically, by recursive subdivisions
RizoSymbolic music comparison with tree data structures
PhD thesis U. Alicante, 2010
see survey in
![Page 4: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/4.jpg)
Lee The rhythmic interpretation of simple musical sequences
Musical Structure and Cognition, 1985
CHAPTER 3. MUSIC SIMILARITY WITH TREES
!" ##$#% #&' (#(a) First two bars of“Auld Lang Syne”
S! ¯ | <|4
2 +4
2
4
2 ! ˘ “ | <|4
1 +4
1
4
1 ! ˇ “ | > | 8
1 +8
1
8
1 ! ˇ “( | ? | · · ·(b) Grammar for meter 4/4
! "" " " · · · · ""
43 · · · · !!! ! !!
!43 · · · · !" ! !! !## "
! "#" " " · · · · | " ""# $
(c) Some possible parses of (a) usingthe grammar in (b) and that detailed inFig. 3.3a.
44
42
81
81
42
41
41
44
42
41
41
42
41
41
(d) Some possible parses
Figure 3.5: Longuet-Higgins grammar for meter 4/4 (from (Lee, 1985) pages 54-55) andsome possible parses.
62
CHAPTER 3. MUSIC SIMILARITY WITH TREES
!" ##$#% #&' (#(a) First two bars of“Auld Lang Syne”
S! ¯ | <|4
2 +4
2
4
2 ! ˘ “ | <|4
1 +4
1
4
1 ! ˇ “ | > | 8
1 +8
1
8
1 ! ˇ “( | ? | · · ·(b) Grammar for meter 4/4
! "" " " · · · · ""
43 · · · · !!! ! !!
!43 · · · · !" ! !! !## "
! "#" " " · · · · | " ""# $
(c) Some possible parses of (a) usingthe grammar in (b) and that detailed inFig. 3.3a.
44
42
81
81
42
41
41
44
42
41
41
42
41
41
(d) Some possible parses
Figure 3.5: Longuet-Higgins grammar for meter 4/4 (from (Lee, 1985) pages 54-55) andsome possible parses.
62
CHAPTER 3. MUSIC SIMILARITY WITH TREES
!" ##$#% #&' (#(a) First two bars of“Auld Lang Syne”
S! ¯ | <|4
2 +4
2
4
2 ! ˘ “ | <|4
1 +4
1
4
1 ! ˇ “ | > | 8
1 +8
1
8
1 ! ˇ “( | ? | · · ·(b) Grammar for meter 4/4
! "" " " · · · · ""
43 · · · · !!! ! !!
!43 · · · · !" ! !! !## "
! "#" " " · · · · | " ""# $
(c) Some possible parses of (a) usingthe grammar in (b) and that detailed inFig. 3.3a.
44
42
81
81
42
41
41
44
42
41
41
42
41
41
(d) Some possible parses
Figure 3.5: Longuet-Higgins grammar for meter 4/4 (from (Lee, 1985) pages 54-55) andsome possible parses.
62
Syntax Trees
Longuet-Higgins The perception of music
I.S.R., 1978
![Page 5: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/5.jpg)
OpenMusic Rhythm TreesOpenMusic: graphical programming environment for algorithmic composition developed at Ircam
Laurson Patchwork: A Visual Programming Language
Helsinki: Sibelius Academy, 1996
OM RT (nested lists) are a first class data structure for the representation of rhythms in OM
![Page 6: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/6.jpg)
Arbres de rythme dans OpenMusic
Des rapports hiérarchisésLes nombres représentent une durée relative par rapport auxautres nombres de la mesure : c’est un rapport de durée.
Le chiffrage est intégré à la notation.
4
3 1 2 1 1
2 1 1 1 1
FIGURE : L’arbre de rythme (? (((4 4) ( 3 (1 ( 2 1)) 2 1 (1 ( 1 11)) ))))
Pierre Donat-Bouillud (ENS Cachan Bretagne) Quantification rythmique dans OpenMusic 4 Juillet 2013 4 / 18
Arbres de rythme dans OpenMusic
Des rapports hiérarchisésLes nombres représentent une durée relative par rapport auxautres nombres de la mesure : c’est un rapport de durée.
Le chiffrage est intégré à la notation.
4
3 1 2 1 1
2 1 1 1 1
FIGURE : L’arbre de rythme (? (((4 4) ( 3 (1 ( 2 1)) 2 1 (1 ( 1 11)) ))))
Pierre Donat-Bouillud (ENS Cachan Bretagne) Quantification rythmique dans OpenMusic 4 Juillet 2013 4 / 18
OpenMusic Rhythm Trees
• infinite alphabet (integers) • processing require arithmetics
Objective to use Term Rewriting tools: ➡ purely syntactic processing ➡ labeling with finite alphabet
![Page 7: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/7.jpg)
Durations in Semi-Structured Music Encodings
in MEI, MusicXML, etc a score is a tree (XML doc) durations are attributes of notes
score transformations can be defined using (tree) patterns but not for rhythms…
➡ encode durations in the tree structure
<mei:note pname="c" oct="5" dur="4"/>
![Page 8: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/8.jpg)
Rhythm Trees (sum-up)
• hierarchical encoding of durations in tree structures • labeling with finite (and small) alphabet
➡ expression of symbolic constraints (e.g. sum = 1) ➡ definition of schemas (types) for rhythms ➡ transformations defined syntactically
![Page 9: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/9.jpg)
Rhythm Trees (RT)
ordered ranked trees over a signature: • inner nodes labeled by prime numbers (= arity) • leaves labeled by n, r, s, d, o
2
n 2
n n
denoted: 2(n(2(n,n))
![Page 10: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/10.jpg)
Distinguished Nodes and Node Relations
3
2
r n
2
r n
2
r n
![Page 11: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/11.jpg)
Distinguished Nodes and Node Relations
root
3
2
r n
2
r n
2
r n
![Page 12: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/12.jpg)
Distinguished Nodes and Node Relations
root
3
2
r n
2
r n
2
r n
leaves
![Page 13: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/13.jpg)
Distinguished Nodes and Node Relations
root
3
2
r n
2
r n
2
r n
![Page 14: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/14.jpg)
Distinguished Nodes and Node Relations
root
3
2
r n
2
r n
2
r n
siblings
![Page 15: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/15.jpg)
Distinguished Nodes and Node Relations
root
3
2
r n
2
r n
2
r n
siblings
siblings
![Page 16: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/16.jpg)
Distinguished Nodes and Node Relations
root
3
2
r n
2
r n
2
r n
siblings
![Page 17: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/17.jpg)
Distinguished Nodes and Node Relations
root
3
2
r n
2
r n
2
r n
siblings
cousins
![Page 18: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/18.jpg)
Semantics: Rhythmic Value
we associate durations to nodes:
dur(root) = 1 beat or 1 measure
dur(node) = dur(parent)arity(parent)
when previous cousin is not o
2
n 2
n n
![Page 19: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/19.jpg)
Semantics: Rhythmic Value
we associate durations to nodes:
dur(root) = 1 beat or 1 measure
dur(node) = dur(parent)arity(parent)
when previous cousin is not o
2
n 2
n n
rhythmic value = sequence of ratios = duration of leaves (in dfs traversal) in the case of n and r
1
2
1
4
1
4
![Page 20: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/20.jpg)
Rhythmic Value
3
2
r n
2
r n
2
r n
1
6
�1
6
1
6
�1
6
1
6
�1
6rhythmic value
![Page 21: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/21.jpg)
Rhythmic Value
3
2
r n
2
r n
2
r n
1
6
�1
6
1
6
�1
6
1
6
�1
6
notes
rhythmic value
![Page 22: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/22.jpg)
Rhythmic Value
3
2
r n
2
r n
2
r n
1
6
�1
6
1
6
�1
6
1
6
�1
6
rests notes
rhythmic value
![Page 23: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/23.jpg)
Rhythmic Value
1
5
1
5
1
15
1
15
1
15
1
5
1
5
5
n n 3
n n n
n n
rhythmic value
![Page 24: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/24.jpg)
Rhythmic Value
3
2
2
n n
2
n n
n 2
2
n n
2
n n
1
12
1
12
1
12
1
12
1
3
1
12
1
12
1
12
1
12rhythmic value
![Page 25: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/25.jpg)
Ties and Dots
2
2
n n
d
2
2
n n
s
we sum durations for subsequences of leafs of the form n s … s or n d or n d d
1
4
3
4
1
4
3
4rhythmic value
![Page 26: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/26.jpg)
Simplifiable RT with Ties
1
2
1
2
3
n 2
s n
s
![Page 27: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/27.jpg)
Summation with Symbol o (a)
1
4
1
2
1
4
2
2
n o
2
n n
dur(node) = dur(parent)arity(parent)
where d’(node) = dur(previous cousin) when the previous cousin labelled with o and 0 otherwise
+ dur’(node)
we ignore the leaves labeled with o in the computation of rhythmic value
![Page 28: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/28.jpg)
Ratios with Symbol o
2
2
n o
2
3
n n n
r
1
4
1
6
1
6
1
6
1
4
�
3 in the time of 2
3
10
3
10
3
10
3
10
3
10
1
2
2
2
o o
2
5
n n n n n
n
5 in the time of 3
dur(root) = 2
3
2dur =1
2dur =
![Page 29: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/29.jpg)
Tuplet Beaming: one beat
3
2
n n
2
n n
2
n n
2̄
3
n n n
3
n n n
3̄
2
n n
2
n n
2
n n
![Page 30: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/30.jpg)
Tuplet Beaming: one bar
2̄
2
2
r n
2
n n
2
2
n n
2
n r
2̄
2̄
2
r n
2
n n
2̄
2
n n
2
n r
2̄
2̄
2̄
r n
2̄
n n
2̄
2̄
n n
2̄
n r
![Page 31: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/31.jpg)
Regular Tree Languagesdefined by tree automata
(embed all current XML schema languages)Murata et al
Taxonomy of XML schema languages using formal language theory ACM Trans. Internet Technol., 5:660–704, 2005
definition of well-formed trees
definition of user preferences
e.g. every d must occur in one of the following patterns2
n 2
d 2
d x
2
2
2
x n
d
d
2
n 2
d x
2
2
x n
d
![Page 32: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/32.jpg)
Rewrite Ruleslocal transformations of RT
d ! s
p̄(x1, . . . , xp) ! p(x1, . . . , xp) p 2 P
symbols with same semantics
(1)
(2)
replacement of a subtree (matching the left pattern) by a subtree
![Page 33: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/33.jpg)
Rewrite Rules
addition of rests
p(r, . . . , r| {z }p
) ! r p 2 P
r; s ! r; r
o; r ! r; r
(3)
(4)
(5)
; denotes the cousin relation replacement of a sequence of cousins
by a sequence of cousins of same length
![Page 34: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/34.jpg)
Rewrite Rules
normalization of ties
(6)
(7)
p(s, . . . , s) ! s p 2 P
p(n, s, . . . , s) ! n p 2 P
![Page 35: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/35.jpg)
Rewrite Rules
elimination of o
(8)
(9)
o; s ! s; s
sum and division by 1 o; n ! n; s
sum and division by 2 o; 2(x1, x2) ! x1;x2
o; o; o; 2(x1, x2) ! o;x1; o;x2
o; o; 3(x1, x2, x3) ! x1;x2;x3sum and division by 3
o; . . . ; o| {z }kp�1
; p(x1, . . . , xp) ! o; . . . ; o| {z }k�1
;x1; o; . . . ; o| {z }k�1
;x2; . . . ; o; . . . ; o| {z }k�1
;xp
(10)simulated with intermediate rules and auxiliary symbols
![Page 36: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/36.jpg)
6.3.4 General
o; . . . ; o| {z }kp�1
; p(x1, . . . , xp)! o; . . . ; o| {z }k�1
;x1; o; . . . ; o| {z }k�1
;x2; . . . ; o; . . . ; o| {z }k�1
;xp (10)
Simulated in several steps, using auxiliary symbols of ⇥rn.
6.4 Subdivision Equivalence
6.4.1 Examples
2(x1, x2) ! 3(2(o, o), 2(x1, o), 2(o, x2))
2(x1, x2) ! 5(2(o, o), 2(o, o), 2(x1, o), 2(o, o), 2(o, x2))
3(x1, x2, x3) ! 2(3(o, x1, o), 3(x2, o, x3)) . . .
6.4.2 General
p(x1, . . . , xp)! p
0�p(u1,1, . . . , u1,p), . . . , p(up0,1, . . . , up0,p)
�(11)
where p, p
0 2 P, p 6= p
0, for all 1 i p
0, 1 j p, ui,j 2 {o, x1, . . . , xp} and the
sequence u1,1, . . . , u1,p, . . . , up0,1, . . . , up0,p has the form o, . . . , o, x1| {z }p0
, . . . , o, . . . , o, xp| {z }p0
.
6.5 Examples
2
2
n o
2
n s
��!(9) 2
2
n n
2
s s
��!(6) 2
2
n n
s
��(1) 2
2
n n
d
3
n
2
s n
s
���!(11) 2
3
o n o
3
2
s n
o s
���!(10) 2
3
o n s
3
n o s
���!(9,8) 2
3
n s s
3
n s s
��!(7) 2
n n
8
Rewriting Equivalent Rhythms
![Page 37: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/37.jpg)
Rewrite Rules
equivalent subdivisions
2(x1, x2) ! 3(2(o, o), 2(x1, o), 2(o, x2))2(x1, x2) ! 5(2(o, o), 2(o, o), 2(x1, o), 2(o, o), 2(o, x2))
3(x1, x2, x3) ! 2(3(o, x1, o), 3(x2, o, x3)) . . .
p(x1, . . . , xp) ! p
0�p(u1,1, . . . , u1,p), . . . , p(up0,1, . . . , up0,p)
�
where p, p
0 2 P, p 6= p
0,
for all 1 i p
0, 1 j p, ui,j 2 {o, x1, . . . , xp}
and the sequence u1,1, . . . , u1,p, . . . , up0,1, . . . , up0,p
has the form o, . . . , o, x1| {z }p0
, . . . , o, . . . , o, xp| {z }p0
.
(11)
![Page 38: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/38.jpg)
Reduction Sequence
6.3.4 General
o; . . . ; o| {z }kp�1
; p(x1, . . . , xp)! o; . . . ; o| {z }k�1
;x1; o; . . . ; o| {z }k�1
;x2; . . . ; o; . . . ; o| {z }k�1
;xp (10)
Simulated in several steps, using auxiliary symbols of ⇥rn.
6.4 Subdivision Equivalence
6.4.1 Examples
2(x1, x2) ! 3(2(o, o), 2(x1, o), 2(o, x2))
2(x1, x2) ! 5(2(o, o), 2(o, o), 2(x1, o), 2(o, o), 2(o, x2))
3(x1, x2, x3) ! 2(3(o, x1, o), 3(x2, o, x3)) . . .
6.4.2 General
p(x1, . . . , xp)! p
0�p(u1,1, . . . , u1,p), . . . , p(up0,1, . . . , up0,p)
�(11)
where p, p
0 2 P, p 6= p
0, for all 1 i p
0, 1 j p, ui,j 2 {o, x1, . . . , xp} and the
sequence u1,1, . . . , u1,p, . . . , up0,1, . . . , up0,p has the form o, . . . , o, x1| {z }p0
, . . . , o, . . . , o, xp| {z }p0
.
6.5 Examples
2
2
n o
2
n s
��!(9) 2
2
n n
2
s s
��!(6) 2
2
n n
s
��(1) 2
2
n n
d
3
n
2
s n
s
���!(11) 2
3
o n o
3
2
s n
o s
���!(10) 2
3
o n s
3
n o s
���!(9,8) 2
3
n s s
3
n s s
��!(7) 2
n n
8
(simplification)2(x1, x2) ! 3(2(o, o), 2(x1, o), 2(o, x2))2(x1, x2) ! 5(2(o, o), 2(o, o), 2(x1, o), 2(o, o), 2(o, x2))
3(x1, x2, x3) ! 2(3(o, x1, o), 3(x2, o, x3)) . . . (11)
![Page 39: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/39.jpg)
Properties
➡ explore the space of rhythms with same value as a given rhythm
➡ suggest alternative notations
for well-formed trees
Every two trees in relation by rewriting have the same rhythmic value (equivalent)
![Page 40: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/40.jpg)
Properties (perspectives)
under restriction for termination (bounded depth)
• confluence
➡ canonical representation of equivalence classes of rhythms
• rewrite strategies e.g. top-down ➡ for efficiency ➡ prove completeness?
T
T’
T1 T2
![Page 41: A Structural Theory of Rhythm Notation based on Tree ...repmus.ircam.fr/_media/jacquemard/strn-mcm.pdf · A Structural Theory of Rhythm Notation based on Tree Representations and](https://reader034.fdocuments.us/reader034/viewer/2022051508/5ab2eee87f8b9aea528dc5d1/html5/thumbnails/41.jpg)
Applications and Perspectives➡ framework for rhythm transcription (by quantization)
in OpenMusic, based on RT ➡ conversions ‣ RT → OMRT for rendering ‣ RT ↔ standard encodings
➡ alternative: rewriting and tree automata with build-in arithmetic
Conclusion• tree structured encoding of rhythm • defining well formed tree languages (schemas) • tree rewriting rules defining rhythm equivalence