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GENERATING EQUIVALENT RHYTHMIC NOTATIONS BASED ON
RHYTHM TREE LANGUAGES
Florent Jacquemard
INRIA Sorbonne Universités
Adrien Ycart
School of Electronic Engineering
and Computer Science
Queen Mary Univ. of London, UK
[email protected]
STMS (IRCAM-CNRS-UPMC)
IRCAM, Paris, France
[email protected]
ABSTRACT
We propose a compact presentation of languages of preferred rhythms notation as formal grammars. It is based on
a standard structure of rhythm trees capturing a wide range
of rhythms in Western notation.
As an application, we then describe a dynamic programming algorithm for the lazy enumeration of equivalent rhythm
notations (i.e. notations defining the same durations), from
the simplest to the most complex. This procedure, based
on the notion of rhythm grammars has been implemented
and may be useful in the context of automated music transcription and computer-assistance to composition.
1. INTRODUCTION
Music notation is for music very much like writing for
language. It serves as a support to convey ideas, to keep
track of them in time, and as a working support for musical expression. In natural languages, there can be several
synonyms to designate the same entity (such as ”Rome”,
”Roma”, ”capital of Italy”, ”city of the seven hills”, ”eternal city”...) and one might prefer using one word over the
others depending on the context, a special connotation, the
linguistic register etc. Similarly, in common Western music notation, there are often many different ways to write
a given sequence of durations, and the choice of one writing over another is up to the composer, and can be driven
by many reasons, among which are the musical context in
which it is written, or a particular interpretation or phrasing
that the composer wants to imply.
This is an important problem in applications related to
score generation, in particular automated music transcription [1], score editors, or composer assistance environments [2]. In this context, it is interesting to assist users
as much as possible in choosing appropriate notations for
what they want to express.
A first question that arises is the definition of the domain
of rhythms notations that one want to consider: which divisions (tuplets) are allowed? at which level? how many levels of division can we nest? In other words, we need some
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Masahiko Sakai
Graduate School of Information Science
Nagoya University, Japan
[email protected]
formalism to describe languages of rhythms. This corresponds for instance to the codebooks in the transcription
procedure of [3], which are defined by subdivision schemas
(the sequence of authorized successive regular subdivisions
of a time interval), or similar notions already found in former work on transcription [4, 5, 6]. Another example is
the choices of quantization parameters in user preferences
of music editors such as Finale.
A second question is the design of efficient algorithms for
exploring the set of rhythm notations in a given language
that satisfies some property such as, for instance, the set of
rhythms defining the same effective sequence of durations.
In this work, we follow a language theoretic approach to
address these problems of rhythm notation. We propose
a notion of rhythm languages defined by formal contextfree (CF) grammars, following [7] (Section 3). It generalizes the formalisms cited above, as well as a related formalism that we have used so far in a new tool for rhythm
transcription [8]. The rhythms defined by such a grammar
are, roughly, the derivation trees of the grammar, seen as
rhythm trees (RT). The latter are a tree structure for the representation of timed sequences of events [9, 10], where the
events are stored in the leaves of the tree and the duration
are encoded in the tree structure, every branching defining
a uniform division of a time interval (Section 2). Similar representations have been supported since many years
as a native data structure for the representation of rhythms
in visual environment for composition assistance based on
functional programming languages such as Open Music or
PWGL [2, 11].
Using standard formalisms such as CF grammars for the
definition of rhythm languages allows to exploit efficient
construction and decision procedures from the literature.
Here, we use an algorithm for the enumeration of RTs defined by a given acyclic grammar [12] (Section 3.4). The
enumeration follows a rank assigned to RTs by weights
added to the grammar’s rules. These weights can be seen
as a measure for rhythm complexity. The main advantages
of this algorithm is that it does not need to compute all the
RTs of the language in order to rank them, but instead,
thanks to a monotony property of the weight functions,
build the best trees from the best subtrees in a lazy way.
CF grammars are a concise and readable formalism to
define RT languages, but they are also expressive. The languages they define are regular tree languages [13], and as
such can be composed by Boolean operations. This en-
2
3
n
2
n 2 n
n
3(n, 2( , n), n)
3
q qqq
n
3
2
3
n n
n
2(n, 3(n, n, ))
2(n, 2(3(n, , n), ))
3
3
! !!
n
!
! !!
Figure 1. Rhythm Trees and corresponding notations.
ables incremental constructions of complex languages by
composition of elementary languages, in a user-friendly
fashion. In particular, it is possible to use Cartesian product in order to construct the intersection of two tree languages. We use this principle (Section 4) to construct a
grammar defining the set of RTs from a given language L
which correspond to a given sequence σ of duration values.
Using the above-cited algorithm, we can then enumerate
by ascending weight the RTs of L with duration value sequence σ. We apply the same principle to provide a way to
find all rhythms that denote the merging of two voices into
one (Section 5).
2. RHYTHM TREES
A rhythm tree (RT) is a hierarchical representation of a sequence of timed events, where the events are in the leaves,
and the inner nodes define the durations by successive subdivisions of an initial time interval.
2.1 Syntax
In our settings, a RT is either:
a symbol n representing the beginning of an event, or
a symbol representing the continuation of an event, or
a tree of the form t = p(t1 , . . . , tp ) where p is a natural
number (smaller than a given bound) and t1 , . . . , tp
is an ordered sequence of RTs.
The node labeled with p is called the root of t, and the
sub-RTs t1 , . . . , tp are called the children of t.
Note that every inner node of a RT is labelled with the
arity (outer degree) of the node. We adopt this redundant
notation only for readability purpose.
Every occurrence of the symbol n represents the starting date (onset) of an event. Some alternative symbols
could be used for distinguishing different kinds of events:
pitched notes, chords, rests etc (see Section 2.3). In this
paper however, we always use n for the sake of simplicity.
The symbol represents a a note that is tied to the previous one. In some cases, those tied notes can be denoted by
dots, see Section 2.3.
Example 1 We present in Figure 1 three examples of RTs
and the corresponding rhythm notations according to the
duration semantics defined in Section 2.2.
The first RT on the left is a triplet. Its second child is further divided in 2 parts, and the first part is a continuation.
This means that the corresponding note (second note of the
RT) is tied to the previous note (first note).
In the second RT, we have a division by 2 and then by 3,
and a tie between the two last notes.
The last RT presents three levels of division, and two ties.
Note that the depth in the RTs reflects the beaming level
in the notation. For readability purposes, some tied notes
have been merged, for instance in the second example, the
last two tied sixteenth notes have been merged into one
single eighth note.
2.2 Semantics
To every RT t, we associate a sequence of positive rational
numbers called rhythmic value of t. This numbers correspond to the inter-onset intervals (IOIs) between the onset
of events described in the RT. In the following, when considering the leaves of a tree, we will interchangeably use
the term IOI instead of positive rational number.
In order to define the rhythmic value of the RT t, we associate to every node ν of t a positive rational number called
duration value, and denoted by dur (t, ν).
Intuitively, the duration of a node is divided uniformly in
the duration of its children, and is used to sum duration
of leaves. Formally,
If ν is the root of t, then dur (t, ν) = 1.
Otherwise, dur (t, ν) =
dur (t,ν0 )
t(ν0 )
+ cdur (t, ν), where
ν0 is the parent of ν in t,
t(ν0 ) is the label of ν0 in t (i.e. its arity),
cdur (t, ν) = dur (t, ν 0 ) if ν is a leaf and it has a
next leaf ν 0 labelled with ,
cdur (t, ν) = 0 otherwise.
The choice of duration 1 for the root node is arbitrary. It
means that the fractions refer to divisions of one beat but
with different root durations, they could as well refer to
other orders of magnitude (bar etc). Note that the durations are expressed in beats (time units relative to a tempo),
which makes the definition of a tempo useless here.
The events represented by a RT are stored in its leaves.
Hence, the event’s durations, i.e. the actual rhythm corresponding to a RT, denoted val (t), is defined from the duration values of its leaves. More precisely, let ν1 . . . , νk be
the enumeration, in depth-first ordering, of the leaves of t
labelled with n. The rhythmic value of t is the sequence
val (t) = dur (t, ν1 ), . . . , dur (t, νk ).
It is the empty sequence when k = 0. Intuitively, if a
rhythm tree t represents the notation of a rhythm, its rhythmic value represents the way this rhythm sounds.
Two RT t1 , t2 are called equivalent, denoted t1 ≡ t2 iff
val (t1 ) = val (t2 ).
3
n 2 n
3(n, 2(d, n), n)
d n
!" ! !
3
Figure 2. Alternative representation for dots.
Example 2 The three RT of Figure 1 have a rhythmic value
of [ 12 , 16 , 13 ] and are therefore equivalent.
Deciding the equivalence of two given RT t1 and t2 can
be done simply by evaluating their respective rhythmic values (in linear time for each tree) and comparing the values.
2.3 Pitches, Rests, Grace Notes, Dots...
In the above RT representation, we consider only one generic
symbol n to represent any kind of event. Straightforwardly,
we could introduce new symbols labeling the leaves of RTs
in order to encode any kind of finite information on events
(pitches for notes or chords, trills, rests etc.
Alternatively, we could maintain along with a RT t a list
of event’s information of same length as the rhythm value
of t. This list can be used later to recover this information
(e.g. for rendering). These details are left out of this paper.
Grace notes are events of duration 0. The approach presented in this paper can deal with grace notes. Indeed, in a
RT, one note preceded by one or several grace notes can be
encoded in a single symbol, labelling a leaf. For instance,
the symbol nk could describe one note preceded by k grace
notes. The definition of the rhythmic value of a RT should
then be extended accordingly, by adding to the sequence k
times the value 0 at the appropriate place.
Moreover, in some case we could use in RTs a dot symbol
d in place of the tie symbol, with the same semantics. For
instance, in a RT containing a node n whose next sibling
has the form 2( , n), the could be replaced by the symbol d representing explicitly a dot notation, see Figure 2.
The CF grammar formalism presented in Section 3 can do
matching of patterns such as the previous one, hence it it is
expressive enough to express the correct placement of dot
symbols. In the rest of the paper, we voluntarily omit the
dot symbols in grammars for simplicity purpose.
2.4 Related Rhythm Tree Formalisms
The definition of RTs used in this paper differs slightly
from the earlier definitions of rhythm trees in Patchwork [9]
and Open Music [10] (OMRT). In OMRTs, inner nodes
are labelled by integer values denoting the relative length
of durations of sibling nodes. For example, if an OMRT
has 2 sons labelled with 1 and 2, the second son is twice
as long as the first. Hence this tree corresponds to a triplet
where the two last notes are tied. This is exactly equivalent
as if the sons were labelled 2 and 4: only the ratios matter.
Note that these numbers have nothing to do with the numbers used in the above RT encodings, which are just the
arity of the inner nodes and carry no information. There-
fore, processing OMRTs requires some integer arithmetics
computations.
The standard formalism that we are using below to represent languages of rhythms (and the associated algorithms
e.g. for language enumeration) can only handle finite sets
of symbols. Therefore they are not appropriate to deal with
OMRTs and we preferred here an all-symbolic encoding of
RTs without integral values.
3. RHYTHM LANGUAGES
In this section, we propose a general finite representations
for sets of RTs called rhythm grammars. Their purpose is
to fix the kind of rhythm notations that we want to consider,
using declarative rules.
For instance, one rule may express that at some level, division by three is allowed but division by five is not. One
other rule can express that at some level we can have a leaf
labeled with a note (n) or labeled with a continuation ( ).
3.1 Rhythm Grammars
A rhythm grammar (RG) is a context-free grammar G =
(Q, q0 , R) where Q is a finite set of non-terminals, q0 is
the initial non-terminal and R is a finite set of production
rules of one of the forms
q → q1 , . . . , qp with q, q1 , . . . , qp ∈ Q,
q → a with q ∈ Q and a ∈ {n, }.
Intuitively, these rules are applied from top to bottom to
generate RTs by replacement of non-terminals by subtrees.
A rule of the first kind (called inner rule) generate an inner
node of a RT, expressing a division by p of the duration of
this node. A rule of the second kind (called terminal rule)
generate a leaf of a RT, and expresses that the label a is
allowed at this leaf.
Generally in the literature, one considers the (contextfree) language of words generated by CF grammars – in
our case, they would be words over {n, }. However, recall that RTs encode events in the leaves but also durations
of these events in the inner nodes. Since we need these
two informations for rhythm encoding, we need both kind
of nodes, and therefore we will be more interested in the
set of RTs generated by a RG (which is a regular tree language), than in the (context-free) language of the words
found in a traversal of the leaves of these RT.
Formally, the language Lq (G) of a RG G = (Q, q0 , R) in
non-terminal q ∈ Q is defined recursively by
Lq (G)
[:= {aif q → a ∈ R} ∪
p(t1 , . . . tp ) | t1 ∈ Lq1 (G), . . . , tp ∈ Lqp (G) .
q→q1 ,...,qp ∈R
The language of G is L(G) = Lq0 (G).
Example 3 Let us consider the following RT G, with initial non-terminal q0 .
q0 −
→n
q0 −
→ q2 , q2
q0 −
→ q3 , q3 , q3
q4 −
→
q4 −
→n
q4 −
→ q5 , q5 , q5
q2
q2
q2
q2
q5
q5
−
→
−
→n
−
→ q4 , q4
−
→ q5 , q5 , q5
−
→
−
→n
q3 −
→
q3 −
→n
q3 −
→ q5 , q5
3.2 Weighted Rhythm Grammars
We extend RG into weighted rhythm grammars (WRG)
by adding a weight (real value) to each production rule,
w
w0
with the notations q −−→ q1 , . . . , qp and q −−→ a respectively for weighted inner and terminal rules with weights
w and w0 .
Example 4 We describe below a weighted extension of the
RG of Example 3.
0.1
0.2
q4 −−→
0.1
q4 −−→ n
0.75
q4 −−−→ q5 , q5 , q5
q2
q2
q2
q2
0.2
−−→
0.1
−−→ n
0.5
−−→ q4 , q4
0.6
−−→ q5 , q5 , q5
q3
n
q5
q0
q3
q3
n
q5
n
q2
n
q5
n
n
q2
q5
n
q2
q4
q5
q5
n
q5
q4
q5
n
Figure 3. Runs on the RTs of Fig. 1 of the WRG of Ex. 4.
The three rules with left-hand side q0 express that the root
of RT in G’s language can be either a single event n, or a
division by 2, with children in q2 , or a division by 3, with
children in q3 .
At the next level, starting with q2 , we can have leaves
labeled with or n, or division by 2 (with children in q4 ),
or division by 3 (with children in q5 ). Starting with q3 ,
we can have only leaves or division by 2 (with children
in q5 ). At the next level, q5 will generate only leaves ( or
n) whereas at q4 we can have a last division by 3.
Note that rules of G are acyclic, and hence define a finite
RT language.
All the three RTs of Figure 1 are in the language of this
RG. The precise computations to obtain these RT are described in Example 5 below.
q0 −−→ n
0.35
q0 −−−→ q2 , q2
0.45
q0 −−−→ q3 , q3 , q3
q0
q0
q2
0.2
q3 −−→
0.1
q3 −−→ n
0.5
q3 −−→ q5 , q5
0.2
q5 −−→
0.1
q5 −−→ n
The weights can be thought as penalties for symbols or
divisions. For instance, every note n induces a minimal
penalty of 0.1 whereas a continuation induces a penalty
of 0.2, in order to penalize ties. At level 0 (root), a duplet induces a penalty of 0.35 whereas for triplet it is a bit
bigger (0.45). The situation is similar at lower levels. See
Section 3.3 for lengthier discussion on choosing weights.
The weights in WRG are used to rank the RTs (by ascending weight). In order to associate to every RT a unique
weight by a WRG, we use the following notion of run. Intuitively, a run represents the sequence of application of
the grammar’s production rules in order to obtain a RT.
Formally, a run of a WRG G = (Q, q0 , R) on a RT t is a relabelling of the nodes of t with production rules of R, such
w
that for every inner node ν labeled with q −−→ q1 , . . . , qp ,
for every 1 ≤ i ≤ p, the ith children node of ν is labeled
by a rule of R of left-hand side qi .
Example 5 In Figure 3, we display runs of the WRG of
Example 4 on the three RTs of Figure 1. For the sake of
readability, we only display the left-hand side of rules on
inner nodes and the non-terminal and symbol on leaves
(the whole run can be easily recovered from this).
Note that there can be several runs for one RT of the language (although it is not the case in the above example).
The weight of a run is the sum of weights of the rules at
all nodes of this run. The weight associated to a RT t by a
WRG G is either undefined if there exists no run of G on t,
or the minimum of the weights of the runs of G on t.
Example 6 The weights associated by the WRG of Example 4 to the RTs of Figure 1 are respectively:
3(n, 2( , n), n) : 1.45
2(n, 3(n, n, )) : 1.45
2(n, 2(3(n, , n), )) : 2.3
3.3 Choice of the Weight Values
The choice of the weights in production rules is crucial,
as it is what will determine how the various RTs will be
ranked. Generally speaking, we want RTs to be ranked
according to their complexity. However, rhythmic complexity is difficult to define because it is highly subjective,
and depends strongly of the context in which the rhythm
is embedded. Some studies have tackled the perception
of rhythmic complexity (see [14] for various definitions),
but to the best of our knowledge, none have treated of the
complexity of rhythm notation.
One naive definition would be to consider basic measures
such as the size and the depth of the RT: the more nodes
there are and the deeper the tree is, the more ”complex”
the rhythm is likely to be. The resulting weight function
d+s∗p
would be of the form : q −−−−−→ q1 , . . . , qp where d and
s are two positive coefficients penalizing the depth and the
size of the tree, respectively.
This definition shows its limits when comparing for example a quintolet and a sextolet. The quintolet is quite unusual, and will often be seen as complex, while a sextolet
is much more common. Size alone cannot account for the
complexity of a notation. Same with depth: Õ Õ Õ is generally considered less complex than a quintolet, even though
its tree is deeper.
To go around this problem, a measure for notation complexity that tries to take into account these musical considerations was proposed in [8]. It is based on a ranking of
divisions complexity proposed in [15], in which arities are
ranked as follows, from less complex to more complex : 1,
2, 4, 3, 6, 8, 5, 7,... We can thus define a function β(p)
representing the penalty associated to an arity p, and the
β(p)
weight function becomes q −−−→ q1 , . . . , qp
Still, in this measure, the function β(p) is arbitrarily determined, even though its contour is determined based on
musical considerations. To determine it more relevantly,
we could perform large-scale analysis of music score corpora, and choose β(p) according to the frequency of each
arity (the higher the frequency, the lower the weight). We
could go even further in this corpus-based estimation of
weights by assigning to each production rule of our RG a
weight inversely proportional to its frequency in a corpus
of scores. This would allow us to not only take into account the arity of the nodes in the RT, but also the depth at
which they are found, and even the series of subdivisions
they are in, relatively to the grammar chosen.
Another advantage of this corpus-based approach is that
it can capture specific styles of notation. For example, if
the weights were computed with a corpus of contemporary
classical music scores, quintolet, septolet, and other artificial subdivisions would be less penalised than if a corpus
of baroque music was used. It would even be possible for a
composer to get personalised recommendations matching
their own preferences of notation by using a corpus of their
previous pieces.
3.4 k-best Parsing Algorithm
The k-best Parsing Algorithm [16] is a dynamic programming algorithm that, given a weighted context-free grammar G = (Q, q0 , R), enumerates its k best runs (ranked by
their weight), where k is a parameter given by the user. In
[8], we proposed an implementation of this algorithm from
which the following stems. Here, we apply this algorithm
to the WRG G, and enumerate its least-weight RTs.
The algorithm uses a table which associates to every nonterminal q ∈ Q two ordered lists of runs of G:
bests[q], containing the runs of G on RTs with q at the
root and of minimal weight, as well as their weights.
cands[q] (candidate bests), containing runs of G among
which the next best will be chosen.
In each of those lists, the runs are not stored in-extenso.
The elements of those
lists are lists of pairs of the form
hq1 , i1 i, . . . , hqp , ip i , where every qj ∈ Q and every ij is
w
an index in bests[qj ], and R contains a rule q −−→ q1 , . . . , qp .
Those lists will also be called runs in what follows.
3.4.1 Initialization of the table
For each q ∈ Q, bests[q] is initialy empty, and
cands[q] is
initialized with one run hq1 , 1i, . . . , hqp , 1i for each rule
w
q −−→ q1 , . . . , qp in R, with an unknown weight, and one
w0
run () of weight w0 for each rule q −−→ a in R.
3.4.2 Algorithm
The algorithm evaluates the weights of candidates in the
table in a lazy fashion, and transfers candidates of minimal
weights in the best list. It works recursively, by computing the weight of each run from the weights of its sons.
The main function, best(k, q), returns the k-th best run of
root q:
1. If best[q] already contains k elements or more, then
return the the k-th run of this list and its weight.
2. Otherwise, evaluate the weight of allruns in cand [q] as
follows: for a run hq1 , i1 i, . . . , hqp , ip i ∈ cand [q] whose
weight is unknown, call recursively best(ij , qj ) for each
1 ≤ j ≤ p, and then evaluate the weight by summing
the weights of the sub-runs and the weight w of the rule
w
q −−→ q1 , . . . , qp .
3. Once all the weights of the runs in cand [q] have been
evaluated, remove the run of smallest weight from this list,
add it to best[q] (together with its weight). Then add to
cand [q] the following next
runs, with unknown weight: hq1 , i1 +1i, . . . , hqp , ip i , hq1, i1 i, hq2 , i2 +1i, . . . , hqp , ip i ,
. . . , hq1 , i1 i, . . . , hqp , ip + 1i . This step ensures us that
the next best is in the candidate list. Repeat 2. and 3. until
best[q] or cand [q] is empty (in the later case, best(k, q) is
undefined).
4. ENUMERATION OF EQUIVALENT RHYTHMS
Now we have all the elements to represent and enumerate
the set of rhythms equivalent to a given rhythm. Let us first
reformulate precisely, in the above settings, the problem
we are interested in:
given a weighted rhythm grammar G and a non-empty sequence σ of positive rational numbers (IOIs),
return a weighted rhythm grammar Gσ such that
L(Gσ ) = {t ∈ L(G) | val (t) = σ}.
Hence, given a RT t, the WRG Gval(t) will represent the
set of WRT of G equivalent to t. Moreover, using the algorithm of Section 3.4, we can enumerate this set.
4.1 Grammar Product Construction
Let G = (Q, q0 , R). The construction of Gσ works as a
Cartesian product, following the similar construction for
tree automata [13]. The non-terminals of Gσ are pairs of
the form hτ, qi where q ∈ Q and τ is a part of σ, in a sense
explained below. Its initial non-terminal is hσ, q0 i, and evw
ery production rule of Gσ is either of the form: hτ, qi −−→ a
w
such that q −−→ a ∈ R and τ is a singleton sequence, or
w
w
hτ, qi −−→ hτ1 , q1 i, . . . , hτp , qp i such that q −−→ q1 , . . . , qp ∈
R and τ1 , . . . , τp is a partition of τ in p parts of equal
length, where the length of a sequence of positive rational
numbers is the sum of its elements.
The only tricky point for partitioning τ in p parts is that
it may require to split some rational number r in two parts
r1 and r2 , such that r = r1 + r2 , where r1 will be the last
element of some τi and r2 will be the first element of τi+1
(necessarily a continuation). For instance, the partition of
[ 21 , 16 , 31 ] in two parts is [ 21 ], [ 16 , 13 ], but the partition of [ 16 , 13 ]
1
in two parts of equal length is [ 16 , 12
], [ 14 ], and in this par1
tition, 4 has to be a continuation since the duration 13 has
1
+ 14 .
been cut into 12
Let us state this precisely. We consider non-empty sequences of positive rational numbers with a sign: −τ means
that the first element of the sequence τ is a continuation
and +τ means that it is not (the sign + may be omitted).
Now we define the concatenation of signed sequences of
positive rational numbers by:
δ[r1 , . . . , rn ] +[s1 , . . . , sm ] = δ[r1 , . . . , rn , s1 , . . . , sm ]
δ[r1 , . . . , rn ] −[s1 , . . . , sm ] =
δ[r1 , . . . , rn + s1 , s2 , . . . , sm ]
where δ is + or − and n, m ≥ 1.
The length of a signed
Pn sequence σ = δ[r1 , . . . , rn ] of
rational numbers is i=1 ri , denoted kσk. A p-partition
of a signed sequence τ (for p > 0) is a sequence τ1 , . . . , τp
of signed sequences such that τ1 . . .τp = τ and kτ1 k =
. . . = kτp k = kτpk . Note that it is unique for a given couple
(τ, p).
Example 7 +[ 12 , 16 , 13 ] = +[ 12 ] +[ 16 , 13 ] and
1
+[ 16 , 13 ] = +[ 61 , 12
] −[ 14 ].
These two concatenation are 2-partitions.
Now we can describe precisely the construction of Gσ
from the RG G and the sequence σ. Every non-terminal of
Gσ is a pair made of a signed sequence τ of positive rational numbers and a non-terminal q of G, denoted by hτ, qi.
1. Gσ contains the non-terminal h+σ, q0 i (initial).
2. For every non-terminal hτ, qi of Gσ , and every producw
tion rule q −−→ q1 , . . . , qp of G, Gσ contains the production
w
rule hτ, qi −−→ hτ1 , q1 i, . . . , hτp , qp i such that τ1 , . . . , τp
is the p-partition of τ , and Gσ contains the non-terminals
hτ1 , q1 i,. . . , hτp , qp i.
3. For every singleton non-terminal h+[r], qi of Gσ (r ∈
w
Q+ ) and every production rule q −−→ n of G, Gσ contains
w
the production rule h+[r], qi −−→ n.
4. For every singleton non-terminal h−[r], qi of Gσ and
w
every production rule q −−→ of G, Gσ contains the prow
duction rule h−[r], qi −−→ .
The size of Gσ is at most the size of G times the size of σ.
The correctness of construction follows from the property
that: Lhτ,qi = {t ∈ Lq | val(t) = |τ |}, where |τ | denotes
the absolute value of τ , i.e. the sequence τ without its sign.
4.2 Examples
Example 8 Let us consider the application of the above
procedure to the WRG G of Example 4 and the rhythmic
value σ = [ 12 , 16 , 13 ].
The initial non-terminal of Gσ is hσ, q0 i (we omit the +
sign).
From the production rules of G starting with q0 , and 2and 3-partitions of σ, we obtain
hσ, q0 i
hσ, q0 i
0.35
−−−→
0.45
−−−→
h[ 12 ], q2 i, h[ 16 , 13 ], q2 i
h[ 13 ], q3 i, h−[ 16 , 61 ], q3 i, h[ 13 ], q3 i
otations of value [ 1/2 1/6 1/3 ] (sche
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Figure 4. Enumeration for σ = [ 21 , 16 , 13 ] and a more elaborate WRG than in Example 4. The RTs are separated by
double bars. Single bar separate two version with or without dots of the same RT.
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we have:
h−[ 61 , 16 ], q3 i
0.5
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Finally, from the remaining singleton non-terminals:
1
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], q5 i
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h[ 16 ], q5 i
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The language of this grammar contains the three RTs displayed in Figure 1, with the weights listed in Example 6.
Example 9 With a more elaborate WRG G (10 non-terminals
and 55 production rules) and the same σ as in Example 8,
we obtain a WRG Gσ with 73 non-terminals and 79 production rules, whose language contains the 7 RTs displayed
Figure 4 (with or without dots).
One can notice in the examples that some of the notations
proposed do not seem to be generated by the grammars
described. For instance, in the middle example in Figure 1,
the notation displayed does not match exactly the structure
of the RT above: the matching notation should indeed be:
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Donat-Bouillud,
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Jacquemard,
and
M.
Sakai,
[5]
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Donat-Bouillud,
F.
Jacquemard,
and
M.
Sakai,
“Acknowledgments”
(with
no
numbering).
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(with
no
numbering).
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This
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workflow
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5.2
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TO
DO]
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TO
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return
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grammar
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such
that
Association
for
Computational
2005,
Association
forTheory
Computational
Linguistics,
2005,
pp.
Association
for
Computational
2005,
Association
for
Computational
Linguistics,
2005,
pp.
for
Computational
2005,
pp.
4.
All
we
to
do
now
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to
the
merge
operues
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is
done
in
time
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preceded
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2239–244,
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ain(t)
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tion,
2000.
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239–244,
2000.
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pp.
239–244,
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239–244,
2000.
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be
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2000.
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Music
Encoding
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2015,
Florence,
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tion,
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2000.
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REFERENCES
7.
REFERENCES
7.
REFERENCES
7.
REFERENCES
7.
REFERENCES
ator.list ofLet
notations
of
this
merged
rhythmic
value
to
obtain
RTs
cor5.1
Problem
statement
In
In
order
order
to
to
be
be
able
able
to
to
be
be
able
able
to
to
reconstitute
reconstitute
the
the
merg
me
ator.
notations
ofstatement
this
merged
rhythmic
value
to
obtain
RTs
cor5.1
Problem
statement
ator.
notations
ofof
this
rhythmic
value
to
obtain
RTs
corator.
5.1
Problem
notations
this
merged
rhythmic
value
to
obtain
RTs
corator.
is
a
fonction
that
adds
the
two
musical
symbols
(a
note
notations
of
this
merged
rhythmic
value
to
obtain
RTs
cor5.1
Problem
statement
5.1
Problem
statement
5.1 Problem
Problemstatement
statement
5.1
beling
beling
beling
thethe
leaves
the
leaves
leaves
of of
RTsof
RTs
inRTs
order
in ord
in
mic
value
. asonset(
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defined
as
follows
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mic
.a list
onset(
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defined
as as
follows
: 2015.
for
May
2015.
[Online].
Available:
https://hal.inria.fr/
value
. onset(
)value
defined
as
follows
: positions
for
cmic
value
. onset(
)value
aislist
defined
follows
for
May
2015.
[Online].
Available:
https://hal.inria.fr/
alue
. onset(
) RT,
ismic
ais
list
defined
follows
for
May
2015.
[Online].
Available:
https://hal.inria.fr/
May
[Online].
Available:
https://hal.inria.fr/
. onset(
onset(
)aisP
islist
list
defined
asfollows
follows
:induces
for
mic
value
.choices
a:aislist
defined
:in
for
May
2015.
[Online].
Available:
https://hal.inria.fr/
May
2015.
[Online].
Available:
https://hal.inria.fr/
May
2015.
[Online].
https://hal.inria.fr/
many
and
high
the
values.
the
values.
encoded
in
athe
single
symbol,
labelling
aYcart,
leaf.
For
instance,
encoded
single
symbol,
labelling
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leaf.
For
instance,
responding
to
the
merged
polyrhythm.
A
schema
of
this
responding
to
the
merged
polyrhythm.
A
schema
of
P
responding
to
the
merged
polyrhythm.
AAschema
schema
of
this
responding
toathe
merged
polyrhythm.
schema
of
this
responding
to
merged
polyrhythm.
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this
added
to Available:
another
note
gives
a[1]
chord,
aI.Shmulevich
note
added
to
aBresson,
rest
P
A.
Ycart,
F.
Jacquemard,
J.
Bresson,
L
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:=
[2]
A. [2]
Ycart,
F.
Jacquemard,
J.
Bresson,
and
S.
L
(G)
:=
{a
[2]
A.
Ycart,
F.
Jacquemard,
J.D.-J.
Bresson,
and
S.
[2]
A.
F.
Jacquemard,
J.
and
S.
Staworko,
[2]
A.
Ycart,
F.
Jacquemard,
J.
Bresson,
and
S.
Staworko,
1
iP
1i P
[2]
A.
Ycart,
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Jacquemard,
J.order
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[2]
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and
1 i 1 This problem
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I.have
Shmulevich
and
D.-J.
Povel,
“Complexity
mea
q
I.
Shmulevich
and
D.-J.
Povel,
“Complexity
measures
similar
to
the
one
we
addressed
in
beling
the
leaves
of
RTs
in
order
to
eci
beling
the
leaves
of
RTs
in
to
[1]
I.
Shmulevich
and
Povel,
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measur
[1]
Shmulevich
and
D.-J.
Povel,
“Complexity
measu
[1]
I.
and
D.-J.
Povel,
“Complexity
measure
[5]
P.
Donat-Bouillud,
F.
Jacquemard,
and
M.
Sakai,
[5]
P.
Donat-Bouillud,
F.
Jacquemard,
and
M.
Sakai,
[5]
P.
Donat-Bouillud,
F.
Jacquemard,
and
M.
Sakai,
We
define
a
new
operator
on
two
rhythmic
values
and
[5]
P.
Donat-Bouillud,
F.
Jacquemard,
M.
Sakai,
[5]
P.
Donat-Bouillud,
F.
Jacquemard,
and
M.
Sakai,
We
define
a
new
operator
on
two
rhythmic
values
and
hal-01105418
We
define
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new
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rhythmic
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We
define
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We
define
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and
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rhythmic
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finite
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pitches,
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i i j=0j=0
j j rewrite
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5.2
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j=0j=0
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values
divergence
in
the
application
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rules.
To
be
applij=0j=0
5.2
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values
5.2
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values
5.2
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workflow
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be
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in
Figure
[FIGURE
TO
DO]
workflow
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[FIGURE
TO
DO]
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[FIGURE
TO
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workflow
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TODO]
DO]
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Transcription
“A
Supervised
Approach
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Supervised
Approach
for
Rhythm
of
musical
rhythms,”
Rhythm
perception
and
pr
of
musical
rhythms,”
Rhythm
perception
and
producpart 4. All we have to
do
now
is
to
define
the
merge
operof
musical
rhythms,”
Rhythm
perception
and
produ
of
musical
rhythms,”
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perception
and
prod
of
musical
rhythms,”
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perception
and
produc
“Towards
an
Equational
Theory
of
Rhythm
Notation,”
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an
Equational
Theory
of
Notation,”
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Equational
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,
denoted
as
merge(
,
),
which
returns
a
new
rhyth“Towards
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Equational
Theory
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Rhythm
Notation,”
the
symbol
n
could
describe
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note
preceded
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k
grace
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Equational
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Representations
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onsets
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Representations
Term
Rewriting,”
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and
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Rewriting,”
in
Mathematics
Given
list
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call
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j=0
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rhythmic
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to
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and
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sequences
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value,
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mic
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adA
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“Kant:
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critique
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pure
quantification,”
“Kant:
A
critique
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pure
quantification,”
in
Proc.
of
“Kant:
A
critique
of
pure
quantification,”
in
Proc.
of
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critique
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pure
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in
Proc.
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critique
of
pure
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critique
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pure
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(ICMC),
Utrecht,
Netherlands,
Sep.
2016.
[On
Netherlands,
Sep.
2016.
[Online].
(ICMC),
Utrecht,
Netherlands,
Sep.
2016.
[Onlin
(ICMC),
Utrecht,
Netherlands,
Sep.
2016.
[Onlin
(ICMC),
Utrecht,
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Sep.
2016.
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5.
POLYRHYTHMS
and
Computation
in
Music:
5th
International
Conand
Computation
in
Music:
5th
International
Contion
IOI(on),
defined
as
follows
:
IOI(on)
=
on
and
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in
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International
Conand
Computation
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Music:
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Conand
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in
Music:
5th
International
Contion
IOI(on),
defined
as
follows
:
IOI(on)
=
on
United
Kingdom:
Springer,
Jun.
2015,
p.
12.
[Online].
tion
IOI(on),
defined
as
follows
:
IOI(on)
=
on
tion
IOI(on),
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as
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:
IOI(on)
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tion
IOI(on),
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:
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=
on
United
Kingdom:
Springer,
Jun.
2015,
p.
12.
[Online].
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we
could
introd
United
Kingdom:
Springer,
Jun.
2015,
p.
12.
[Online].
number
the
symbol
it
corresponds
to
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a
note,
a
chord,
United
Kingdom:
Springer,
Jun.
2015,
p.
12.
[Online].
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we
could
introduce
United
Kingdom:
Springer,
Jun.
2015,
p.
12.
[Online].
number
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i1
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mber
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Kingdom:
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Jun.
2015,encoded
p.12.
12.
[Online].
i aichord,
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Springer,
p.
[Online].
the
the
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symbol
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on
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er
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rhythmic
valencoded
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athe
single
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in
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Let
us
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the
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us
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2United
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us
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k
k Aarhus,
kpp.
ICMC,
Denmark,
1994,
pp.
Example
8
If
=
,
,
,
then
onset(
)
=
0,
,
,
1
Aarhus,
Denmark,
1994,
pp.
52–9.
ICMC,
Aarhus,
Denmark,
1994,
52–9.
ICMC,
Aarhus,
Denmark,
1994,
pp.
52–9.
ICMC,
Aarhus,
Denmark,
pp.
52–9.
ICMC,
Aarhus,
Denmark,
1994,
pp.
52–
ICMC,
Aarhus,
Denmark,
1994,
pp.
52–9
return
a
weighted
rhythm
grammar
G
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weighted
rhythm
grammar
G
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that
dressed
in
Section
4
and
can
be
solved
with
same
proreturn
a
weighted
rhythm
grammar
G
such
that
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rhythm
grammar
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ference,
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ues
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L(G
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definition
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of
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1symbol
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2
1
2
1 ,and
Intelligence,
D.
M.
Tom
and
A.
Volk,
Eds.,
2
Intelligence,
D.
M.
Collins
Volk,
Eds.,
vol.
2
In
order
to
be
able
to
be
able
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the
merged
D.
M.
Collins
and
A.
Volk,
Eds.,
vol.
Intelligence,
D.
M.
Tom
Collins
and
A.
Volk,
Eds.,
vol.
M.
Tom
Collins
A.
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given
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reconstitute
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be
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able
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rrational
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du[4]
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in
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Chiang,
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[3]
C.
Agon,
G.
Assayag,
J.
Fineberg,
and
C.
R
[3]
C.
Agon,
Assayag,
J.and
Fineberg,
andand
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Rueda,
[3]
C.
Agon,
Assayag,
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Fineberg,
and
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Rued
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C.
Agon,
G.
Assayag,
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Fineberg,
and
C.
Rue
[3]
C.
G.
Assayag,
J.
Fineberg,
C.
9110,
Oscar
Bandtlow
and
Elaine
Chew.
London,
9110,
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Bandtlow
and
Elaine
Chew.
London,
rhythmic
value,
we
also
have
to
attribute
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rational
9110,
Oscar
Bandtlow
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Chew.
London,
9110,
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Bandtlow
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9110,
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Bandtlow
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Chew.
London,
sequences
of
positive
rational
numbers
and
,
rhythmic
value,
we
also
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each
rational
sequences
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positive
rational
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and
,
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value,
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also
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value,
we
also
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rational
sequences
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and
,
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rational
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,
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and
,
2 of
notations for polyrhythms,
following
studies
invery
[19].
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22 Proc.
ration
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Proc.
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Workshop
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tion IOI(on), defined
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Proc.
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Proc.
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Parsin
Proc.
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i This
i+1
finite
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ch
finite
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This
problem
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very
to
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This
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inextended
be
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be
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“Kant:
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Pr
This
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problem
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critique
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This
problem
very
similar
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we
addressed
in
“Kant:
A
critique
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pure
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in
Proc.
“Kant:
A
critique
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pure
quantification,”
in
Proc
“Kant:
A
critique
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pure
quantification,”
in
Proc.
This
problem
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similar
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we
addressed
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problem
is
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similar
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addressed
in
notes.
The
definition
of
rhythmic
value
notes.
The
definition
of
rhythmic
valu
United
Kingdom:
Springer,
Jun.
2015,
p.
12.
[Online].
Kingdom:
Springer,
Jun.
2015,
p.
12.
[Online].
number
the
symbol
it corresponds
to
(either
aa note,
a chord, United
United
Kingdom:
Springer,
Jun.
2015,
12.
[Online].
United
Kingdom:
Springer,
Jun.
2015,
12.
[Online].
United
Kingdom:
Springer,
Jun.
2015,
p.p.p.
[Online].
number
thethe
symbol
it corresponds
a two
note,
number
the
symbol
corresponds
toto
(either
number
the
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anote,
note,
achord,
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number
symbol
ititit
corresponds
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Example
10
Figure
6the
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list
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lyrhythms
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4.
All
we
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now
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merge
oper4.
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we
have
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operICMC,
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Denmark,
1994,
pp.
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4.return
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we
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All
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pp.
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part
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have
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1994,
pp.
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Aarhus,
Denmark,
1994,
pp.
52–9.
ICMC,
Aarhus,
Denmark,
1994,
pp.
52–9.
part
4.
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Grace
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,
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,
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],
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ator.
ator.
ator.
ator.
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REFERENCES
7.
REFERENCES
7.
REFERENCES
7.
REFERENCES
7.
REFERENCES
Rhythm
tree
7.
REFERENCES
7.
REFERENCES
53–64.
notations
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to
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to
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1
1
2
1
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[6]
F. aator.
Jacquemard,
P. Donat-Bouillud,
J.
Bresson,
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Figure
4.Chopin
Chopin
Nocturne
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and
6 alternative
United
Kingdom:
Springer,
United
Jun.
Kingdom:
2015,
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Springer,
12.
[Online].
Jun.
2015,
p.
12.
[Online].
Figure
4. 4.
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op.
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given
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series
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[1]
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Povel,
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Shmulevich
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I.Shmulevich
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Povel,
“Complexity
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I.extended
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D.-J.
Povel,
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“Acknowledgments”
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number
symbol
ititait
corresponds
tototo
(either
aanote,
aachord,
Here,
the
operator
∪
denotesAvailable:
the
interleaving
of sequences
Available:
https://hal.inria.fr/hal-01138642
https://hal.inria.fr/hal-01138642
aetc).
rest,
etc).
We
thus
consider
extended
rhythmic
values
Available:
https://hal.inria.fr/hal-01138642
Available:
https://hal.inria.fr/hal-01138642
Available:
https://hal.inria.fr/hal-01138642
a rest,
We
thus
consider
extended
rhythmic
values
rest,
etc).
We
thus
consider
extended
rhythmic
values
rest,
etc).
We
thus
consider
extended
rhythmic
values
aaarest,
etc).
We
thus
consider
extended
rhythmic
values
merged rhythmic value (as an IOI sequence).
We
then
enuof
the
form
(sym,
r),
where
rrational
a rational
number,
of of
the
form
(sym,
r),r),
where
r of
isrrraisonsets,
number,
andand
performed
asand
expected to respect the ordering of
ofofthe
the
form
(sym,
r),
where
isis
rational
number,
and
theform
form
(sym,
r),where
where
arational
rational
number,
and
(sym,
aais
number,
merate the equivalent notations of this
merged
rhythmic
onset
values.
value to obtain RTs corresponding to the merged polyrhythm.
During the above processing, a special care must be given
A schema of this workflow can be found in Figure 5.
to the handling of the events associated to the initial rhythThe rhythmic value obtained by merging two rhythmic
mic values σ1 and σ2 , in order to reassign them properly
values σ1 and σ2 will be denoted as σ1 k σ2 . Intuitively,
in σ1 k σ2 (see Section 2.3). We leave these details out of
σ1 k σ2 corresponds to the sequence of durations obtained
this paper.
by playing both σ1 and σ2 at the same time.
Example 12 Figure 7 shows examples of rhythms obtained
Let us reformulate the problem of interest :
for σ1 k σ2 with σ1 = [ 51 , 15 , 15 , 15 , 51 ] and σ2 = [ 13 , 31 , 13 ]
corresponding to a split of Violin I part in Stravinsky’s Rite
given a weighted rhythm grammar G and two non-empty
of Spring.
sequences of positive rational numbers σ1 and σ2 ,
tations of value [ 1/3 1/6 1/6 1/3 ] (sch
3
1 !
4
! ! !
3
3
!
9
We proposed a formalism based on formal grammars to
define languages of weighted rhythm trees, and a procedure to lazily enumerate these trees by ascending weight.
It is applied, via grammar constructions, to the problem
of enumerating rhythm trees defining the same given sequence of durations (IOIs), and enumerating merges of two
polyrhythmic voices into one.
This has been implemented in C++ 1 , with command line
prototypes outputing the enumerations of RTs which were
then translated into a graphical representation in Lilypond.
The Lilypond code was moreover post-processed for the
improvements described at the end of Section 4.2.
After evaluation with this proof-of-concept prototype, this
implementation will be integrated as a dynamic library in
OpenMusic, for use in particular as a backend procedure
for the transcription framework of [8].
3
!
! ! ! !
3
! !!!!! !
5
3
5
3
3
! !"!"! !
!
3
! !! ! !!
3
11
3
3
3
3
6. CONCLUSION
!" !" !
!
! ! ! !
!
! ! !
3
3
3
3
7
!
! !!! !
!
3
3
3
5
3
3
! ! !! ! ! ! !! !
3
Note that in principle, we could merge two rhythmic values that are not the same length. Nevertheless, here, a
rhythmic value is always normalized to have a total duration of 1.
7
3
3
Acknowledgments
!" !! ! ! ! !!" ! ! ! !! ! !! ! !
The authors would like to thank Karim Haddad for his
valuable knowledge and advices on rhythm and notation.
Music
by the
LilyPond
Figureengraving
6. RTs for
merge2.18.2—www.lilypond.org
of [ 1 , 1 ] and [ 1 , 1 , 1 ].
2 2
3 3 3
7. REFERENCES
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[3] A. T. Cemgil, P. Desain, and B. Kappen, “Rhythm
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!
# 41 !!
!
3
3
5
# !
!
# !!
!
!
!
5
3
!
3
!
!
!
5
!
!
!
5
5
! !
3
!
!
!! ""
!
! !
3
!
!
3
!
!
!"
!
5
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!!
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3
!!
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3
!
3
!
5
!
!
!
5
5
!
5
!
!
3
!
!"
! !
!
!
3
Figure 7. Stravinsky, The Rite of Spring (Violin I) and 6
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[10] C. Agon, K. Haddad, and G. Assayag, “Representation
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