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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 Copyright: an c 2017 open-access Florent article Jacquemard distributed under Creative Commons Attribution 3.0 Unported License, et al. the which terms This is of the permits unre- stricted use, distribution, and reproduction in any medium, provided the original author and source are credited. 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 3 1 "! " " 4 " 3 3 4 5 " " " !" !" " " " " "" 3 3 3 " " """ " " "!"!" 7 10 " " 3 3 3 3 3 7 3 3 " " """"" " 5 3 3 " " " """ " " "! " """! 7 3 3 "! " " " " " " " """ " " 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. From the new non-terminals with singleton IOIs, we have 1 0.1 1 0.1 ], q2 i −− →LilyPond h[ 2engraving n h[ 3 ], q3 i −−→ n Music by 2.18.2—www.lilypond.org From the new non-terminal h[ 16 , 31 ], q2 i, we have: h[ 61 , 13 ], q2 i h[ 16 , 13 ], q2 i 0.5 1 −−→ h[ 61 , 12 ], q4 i, h−[ 14 ], q4 i 0.6 −−→ h[ 61 ], q5 i, h[ 16 ], q5 i, h−[ 16 ], q5 i and then: 1 h[ 61 , 12 ], q4 i 0.75 −−−→ 1 1 1 h[ 12 ], q5 i, h−[ 12 ], q5 i, h[ 12 ], q5 i From the other non-singleton non-terminal h−[ 61 , 16 ], q3 i we have: h−[ 61 , 16 ], q3 i 0.5 −−→ h−[ 16 ], q5 i, h[ 16 ], q5 i Finally, from the remaining singleton non-terminals: 1 h[ 12 ], q5 i h−[ 14 ], q4 i h−[ 16 ], q5 i 0.1 −−→ n 0.2 −−→ 0.2 −−→ 1 h−[ 12 ], q5 i h[ 16 ], q5 i 0.2 −−→ 0.1 −−→ n 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: 3 q qqq given givena aweighted weighted rhythm grammar GitsGand and two two non-em non-em Generally Genera PP It rhythm isIt the isIt grammar the isof empty the empty empty sequence sequence sequence when when wh k duration values of leaves. duration values its leaves. the the values. the values. values. hal-01105418 hal-01105418 erate RT erate RT by rep free) free) lan freb 0 0<<i < i <j + j the +1,the 1,onset( onset( ) ) = = rhythm tree t represents the notation of rhythm tree t represents the notation values. values. sequences sequences ofrhythm ofpositive positive rational rational numbers numbers and and free) langu free) lan rhythm rhythm treelet tree t represents tree tlet represents the the notatio the nota n More precisely, . of , our ⌫the be the More precisely, . .t.⌫represents , ⌫. .be enum Example The RT three of Figure 3 avalue have arepresents rhythmic Example 2 The2three of RT Figure 3 have rhythmic value⌫value rules of th rules the first our case ou c micmic the way this rhyth value represents the way this rhy case, t our case mic mic value mic value value represents represents represents the the way the way this way thi rh [6] [6] F. F.Jacquemard, Jacquemard, P. P. Donat-B first ordering, of the of tDonatlabel first of the leaves of tour labelled wth return return aRests... a ,Rests... weighted rhythm rhythm grammar grammar G leaves Gnode such such that that Example Example8 of 8If If, =of= , ,Pitches, , ,Pitches, then ,and then onset( onset( )equivalent. = )notes, =0,notes, 0, ,weighted , 1,Rests... 1ordering, ,2.3 are therefore equivalent. ,2.3, ,2.3 and are therefore Pitches, Grace Grace Grace notes, node ofcal Rd of RT, ex call call that Two RT t , t are called equivalent, Two RT t , t are called equivalen 2.32.3Pitches, Grace notes, Rests... Pitches, Grace notes, Rests... Two Two RT tsequence t(t) ,the tare t= ,merge( are tTheory called are called called equivale eq call that RT call Structural Theory of ofequiv Rhyt Rhy value t,RT is sequence of tRT is|of the L(G L(Gmic )= ) =value {tmic {t2Two 2L(G) L(G) val |tStructural val (t) = merge( ,that )}. ) this node. this node. The of, of these of the val (t ) = val (t ). val (t ) = val (t ). Note Note that Note that in that the in the in above the above above RT RT representation, RT representation, representation, we we consider we consider consider only only only val val (t val (t ) = (t ) val = ) val = (t val (t ). (t ). ). Representations Representations and and Term Term Given Givena list a list of of onsets onsets on, on, we we call call the the inverse inverse transformatransformaof these ev of these Deciding the equivalence ofgiven two given tconsider t only can rules)rules) Deciding thethe equivalence of two RTwe t RT and t and canonly can Note that in above RT representation, consider Note that in the above RT representation, we can be ap two two info tw in This This problem problem is is very very similar similar to to the the one one we and and Computation Computation in in.Mus one one generic one generic generic symbol symbol symbol n to n represent to n represent to represent any any kind any kind of kind event. of event. of event. tion tionIOI(on), IOI(on), defined defined as as follows follows : IOI(on) : IOI(on) = = on on val (t) =(t, dur (t, .Mu .(t , val (t) = dur ⌫ we ), . addressed .⌫two .addresse ,), dur two inform info be done simply by evaluating their respective rhythmic valbe done simply by evaluating their respective rhythmic valis label alabel is allow of nodes ofaser. no of oneone generic symbol n to represent any kind event. generic symbol n part towe represent any kind of event. part 4.could 4.All All we we have have toof to do do now now is to define define the the merge merge op ference, ference, MCM MCM 2015, 2015, se onon. . Example 2and The three RT of Figure 3Figu Example 2new The RT of Figure 3Fao Example Example Example 2is The 2tothree The 2three The three RT three RT of Figure RT ofnodes ofha Straightforwardly, Straightforwardly, Straightforwardly, we we could could introduce introduce introduce new symbols new symbols symbols lalalaof nodes, of ues (it is done in linear time for each tree) comparing ues (it is done in linear time for each tree) and comparing Generall Generally in set set of set RT of Straightforwardly, we could introduce new symbols laStraightforwardly, we could introduce new symbols laator. Intelligence, Intelligence, D.equivalent. D. M. Tom C InInorder ordertotobebeable able to tobe be able able to toreconstitute reconstitute the merged merged Itof isempty empty sequence when k0.Co of , ofencode ,and ,and ,therefore and are and are therefore are therefore therefore equival equi e= isoforder the sequence when kM. =Tom beling beling beling the the leaves the leaves leaves ofator. of RTsof RTs inthe RTs order inIt order in to encode to any any kind any kind of kind of of of ,to and are ,encode ,the are therefore equivale set of RTs set of RT the values. the values. free) lang free) language guage), guage gu Example 9 If = (C, ), (E, ), (G, ) and = (A, ), (C, ), (E function : given two series of extended onsets [1] I. Shmulevich and D.-J. Povel, “Complexity measures function two of extended onsets Example 9of = (C, ),Bandtlow (E, )6an Example 96series If (C, (E, [1] I. Shmulevich and D.-J. Povel, “Complexity measures i 1:any i3of 1of 1 function :given given two series of extended beling the leaves of RTs in0“Complexity order to encode kind beling the leaves of RTs in order to encode function :any given two of extended function :represents two series extended onsets [1] I.I.Shmulevich Shmulevich and D.-J. Povel, “Complexity measures [1] Shmulevich and D.-J. Povel, “Complexity measures [1] I. and D.-J. Povel, measures 16If 1 2 36 6),), 1 given 1= 1 We define aoperator new operator on two rhythmic values and Weated define atwo new onvalues two values and We define aoperator new operator on two rhythmic and 9110, 9110, Oscar Oscar an 2onsets 6(G, 3 define a new operator on rhythmic values and fine a new on two rhythmic values 23 22Bandtlow 6onsets 3 36353–6 2)series 53–64. 2 3 3(G, We define anew new operator on two rhythmic and by the grammars described, as avalues post-processing step, We define aFigure operator on two rhythmic values 2rests 6kind rhythmic rhythmic value, we we also also have have to to attribute attribute to to each each rational rational 1 2),2 1 1value, 1rhythmic 1 and 1 1and tree trests represents notation ifor <pitches, jfor +pitches, 1, onset( 0Transcription )< = itree < j1j=0 +rests = ator. ator. rhythm t,1,note, the a31)th raa finite finite information finite information information such such as such as as pitches, chords, chords, chords, etc. iparticular jonset( i 1etc. 1 1hal-01105418 1Do 11etc. workflow can be found in Figure [FIGURE TO DO] 1 1j1, notation 1)the 1 111),of 1then workflow cancan be found in [FIGURE TO DO] 1 1 P. workflow can be found Figure [FIGURE TO DO] 1(E, workflow canbe befound found Figure [FIGURE TO DO] workflow inininFigure [FIGURE TO DO] “A Approach for Transcription &Supervised is< aTranscription a#rhythm or athen rest, or a,j=0 “A &Supervised Approach Rhythm &Supervised “A Approach Transcription &Supervised Approach Transcription &Rhythm % ""perception %#produc" merge( &"&for &Supervised “A "#"denoting Approach for [5] &&of %9110, &rhyth%""musical "Rhythm % Rhythm Bandtlow and London, ,= = (AC (C, ), ( % #"producthen merge( ),follows = (AC ), ), &"&1rhythms,” %Rhythm "% %"Approach "Rhythm rhythmic also have to ,)is )then (AC ),), (C, (E, (G, ,"Transcription then (AC ),merge( (C, ), (E, ), (G, # "sym # "!symbol "rational #Rhythm # and ,%producthen )# ,= (AC ), (C, (E, (G, is defined as follows : 13(C, given two #&Supervised #"Approach "% "each of musical guage), guage), merge then defined as follows given two rhythmic &% "for "and ,then merge( ):= (C, ),(E, (E, ,= merge( )2), (AC (C, ),rhythm musical Rhythm # and "!"perception ",Transcription #," # merge 16), 2= merge then defined as follows given rhythm "merge( &"&!for then defined as :3given musical rhythms,” producis then : :(AC of musical rhythms,” and "produc&%"&"of 2,1 &new 12is 2 "& rhyth&&new 1Chew. !#"merge "!#"""and 1"merge 2"is ""Rhythm "%Rhythm &$ !""perception !5.2 "#"then 1follows 2 ,which as merge( ,new ),we which returns a1“A rhyth&attribute "Oscar "&“A !""rhythms,” "#perception , ,denoted merge( ),value, returns a“A 1"#Rhythm as merge( returns a21which rhyth3two 6rhy denoted as merge( ,),denoted ),as which returns a12,1new rhyth&$Supervised ! #"Elaine !"""!"!""perception !"merge( ""Merging noted as merge( returns a1 ,new 63two 6 ), 1of 3 6 6 1rhyth3 6 6 3),two ,denoted asmerge( merge( ,rhyth), returns aswhich ), which returns a&&a$to new 3defined 61as 3given rhythmic values 21 2which 3), 6 2by 6rhythm 2 , denoted 2 ), 2,denoted 1 our case, our case, that 2 found found fou in 2 ,212using &" &41"&41""&""Rhythm $"!rhythms,” $ $new " $ " " " simple rewriting rules. 4 " " " " 4 4 4 4 5.2 5.2 Merging Merging rhythmic rhythmic values values Based on Tree Series International Based on Tree Series Enumeration,” International chord... Based on Tree Series Enumeration,” International Based on Tree Series Enumeration,” International Kingdom: Kingdom: Springer Spring Based on Tree Series Enumeration,” in International finite information such as pitches, chords, rests etc. finite information such as chords, rests etc. Based on Tree Series Enumeration,” International on Tree Series Enumeration,” in International number number symbol it it corresponds corresponds to to (either apitches, note, aEnumeration,” note, amic chord, a=#Deciding chord, Kingdom: Springer, Jun. 2015, 12. [Online]. [5] [5] P.rhy Ja the symbol corresponds to (either app. aBased chord, values and merge( )IOI(interleave(on = IOI(interleave tion, 239–244, 2000. values and ,United IOI(interleave(onset tion, pp. 239–244, 2000. 1"in 1 1,United 1merge( 1 2 equivalence 1Jacquemard, 2of 1 1 Deciding Deciding the the the equivalence two ofF.“Towa tw oDPw values and merge( IOI(interleave(on values , 2= )= IOI(interleave(o tion, pp. 239–244, 2000. values and ,equivalence = tion, pp. 239–244, 2000. [6] tion, 239–244, 2000. notations of this value to obtain RTs cor"Example mic value way this 12represents 2,,, then 1the 2)= "in "inOnsets "pp. 1 If represents way this s "in "note, mic value. merge( ,symbol )rhythmic correponds to "(either 1approach 2),2)Donat-Bouillud, 1Deciding 2,merge( 1,)P. 1p. 112 mic value. , the correponds to "invalue "United value. Intuitively, merge( ,)number correponds to Grace Grace Grace notes notes are are events events of of of duration 0. The 0. The The approach preprecmic value. Intuitively, merge( ,)Intuitively, correponds the alue. Intuitively, merge( , 12Intuitively, correponds to the mic value. merge( ,itthe )correponds to the mic value. merge( ,1)the to 1) 2correponds 1to 2merged the equivalence two give Deciding the of two gi 12Intuitively, 2 )merge( 1Intuitively, 8# 5.2 Example ,1rhythmic then onset( =and = ,1merge( ,12,equivalence , 11, preonset( = 0,F. ,rhythm Merging rhythmic values Merging values 2 1 2 # "are # ""events 132approach 1 23 # " 5.2 1)1 2212 "ser. 2 0. #duration 1, 21 "notes "duration 1"2 1 ,21 "" the "ser. "the " the " Music 10, ""Music "If#"(ICMC), 5.1 Problem statement Let onset( )8 be the list onsets corresponding toof afound rhyth5.1 Problem statement in Mu 5.1 Problem statement 5.1 Problem statement 5.1 Problem statement 61 31 2of 61 2 found 3 , 1 in 0 1 Computer Music Conference ser. Proceedings 0 Computer Conference (ICMC), Proceedings 0 1 a in Computer Music Conference (ICMC), ser. Proceedings 0 Computer Music Conference (ICMC), ser. Proceedings 0 Computer Music Conference (ICMC), Proceedings 2.3 Pitches, Grace notes, Rests... 0 1 0 1 Computer Music Conference (ICMC), ser. Proceedings To interleave two series of extended onsets on and on , Computer Conference (ICMC), ser. Proceedings 2.3 Pitches, Grace notes, Rests... Available: https://hal.inria.fr/hal-01138642 “Towards an Equational “Tow Th a rest, etc). We thus consider extended rhythmic values 1 23332 3 2 call that R Structural Theory 3 2 333 call that RTs en Formal Form F responding toboth the merged polyrhythm. A schema of this 323 23 23 rhythm by playing both at the same rhythm obtained by playing and athythm the same rhythm obtained by playing both and at the same obtained playing both atrest, the same mthm obtained by by playing both and atplaying the same rhythm obtained and at the same rhythm obtained by playing both at the same 1 We 2 1etc). 2and 1by 2both 1 and 2rest, 1obtained 2a 1 1and 2 2Grace hythm notations of value [2/15 1/5 2/15 1/15 1/5 1/15 2/15 ]Staworko, (schema-05.tx mic value . onset( )Available: is2simply a list defined as https://hal.inria. follows :11for Available: https://hal.inri notations of Ycart, value [F. 1/5 1/15 1/5 1/15 2/15 1/5 ]and (schema-05.tx May hythm notations of value [1/5 1/5 2/15 1/15 1/5 1/15 2/15 1/5 ]1/5 (schema-05.tx hythm notations of value [2/15 1/5 2/15 1/15 1/5 1/15 2/15 1/5 ]Staworko, (schema-05.tx hythm notations of value [Jacquemard, 1/15 1/5 1/15 2/15 1/5 ]Staworko, (schema-05.tx 1by 11 evaluating 1 a etc). We thus thus consider consider extended extended rhythmic rhythmic values values 1simply 1 1 1r 1 1 1 1 1 1 1 [2] A. J. Bresson, S. Grace notes are events of duration 0. The approach prenotes are events of duration 0. The approach pre[2] A. Ycart, F. Jacquemard, J. Bresson, and S. Staworko, [2] A. Ycart, F. Jacquemard, J. Bresson, and S. Staworko, be done be done be simply done by evaluating by evaluating their the [2] A. Ycart, F. Jacquemard, J. Bresson, and S. [2] A. Ycart, F. Jacquemard, J. Bresson, and S. ...Figure 4 of the 42nd International Computer Music Conference ...Figure 4 of the 42nd International Computer Music Conference ...Figure 4 of the 42nd International Computer Music Conference ...Figure 4 of the 42nd International Computer Music Conference ...Figure 4 Example 9 If = (C, ), (E, ), (G, ) and = ( of the 42nd International Computer Music Conference P Two ,1,(C, tby are called Example 9of (E, ),a (G, )their and =2to ), ...Figure 4:2corresponding ...Figure 42nd International Computer Conference the 42nd International Computer Music Conference we recursive function interleave(on )4),Indeed, Two RT t"list ,notes. tIf9toonsets called equivalent, den Example 99notes. Ifare = (C, ),a(G, ))and == Let onset( ) ""be the list Let onset( onsets corresponding )done be the list ofof onsets atonsets rhythcorresponding to &the Example If = (C, ), (E, (G, and (A, If = (C, ),2),in (E, ), (A, sented in this inpaper this paper paper can can deal with deal with grace with grace grace notes. Indeed, Indeed, arhythin aequivalen & "this in Music in M ofworkflow the form (sym, r), r& % &"is&sented number, &and 12 2 "") "be 1 2 ""be 22 2(A, 1on %"" ""sented "can "define &1&"&&and &in 1=1= We define aoperator new operator on two rhythmic values Representations &of % ") &in %of "% Music onset( onset( be list of corresponding to a=2Confer rhy a1(A, rh "be WeWe define atime. new on on two rhythmic values and 1simply 2 can be found Figure 4a%"" rational done evaluating respe evaluating res RT Rewrite Rules onsets on, Given we a+ the onsets inverse on, transformawe inverse transforma2(E, 6(G, 3their " Given We define atime. new operator on two rhythmic values Wedefine define anew new operator on two rhythmic values and a4.3 operator two rhythmic values 6the 3 3 "RT time. "< 2call 6 3) """" Let 6 ), 3and "Example 6in 3Encoding 3a time. e. 1 where "&of &&and "the ! !"deal !list "simply &&&"&% "and "& "%""%""%"""% "" %""" " "%"!Let "call "% "" " of time. !"""a ""list !$"&$ !$ """&&&Supervised ! "< ! ""ithe 0!""the 1, onset( )by & &""!&$ & " “A“A & &"(ICMC), 11 these 1 1non-term 11 hal-01 $ 1of 1events 11 e 1 non-t 1(C, 1(E, 1Forma $ 11& 1$ Formally "merge( "!# "!"Netherlands, ""for ""(ICMC), "Netherlands, Approach Rhythm Transcription &Supervised Rhythm ",",jthen &&"&Supervised Approach for “A for &Supervised of these % "&Approach %#[Online]. " ""Transcription no Utrecht, &“A %""Netherlands, %Netherlands, Sep. [Online]. Sep. 2016. &1“A "(ICMC), "%"# ""number, % "Approach (ICMC), Utrecht, Sep. [Online]. "%"&r "%"IOI(on), (ICMC), Utrecht, , then merge( )(AC =13 ), (AC ),), (C, ),), (E, ),May (G, ""!Utrecht, # ." ,follows then , :112,i1as = (AC (G, [Online]. [Online]. "Transcription #"Sep. then merge( ,)defined ),), value ."#"%Netherlands, onset( mic a"2016. defined onset( as follows )IOI(on) is a 1list for as :16),16), for ,[Online]. then merge( ,= )=2=(AC (AC ), (E, (G, merge( ),2)on = ),13),), (C, (E, (G, "Utrecht, "#Transcription #! )"!"Sep. ""Sep. # tion #"Rhythm "!Approach # 2016. 2015. "2016. 12 "2016. &&&&Utrecht, "!"for "list "Netherlands, "[Online]. "1 %(ICMC), "#and &1&&1&"1"&(ICMC), ""2016. "#Rhythm "for "Rhythm 2 !value "2016. &"$where !"#"Transcription "is , the denoted as merge( ,2),:),which ), which returns asented new Computation &&&Supervised "!# Sep. && &Netherlands, "is "a !"mic "!Utrecht, , denoted as merge( ,of returns a interest new rhythtion defined as IOI(on), :Indeed, defined follows :follows IOI(on) = on of the the form (sym, (sym, r), r), where r is a rational rational number, and 3(C, 6(E, 6(G, as merge( ,), which returns anew new ,denoted denoted as merge( , 21which which returns ainterest new rhythas merge( ,1interest aform rhyth6(C, 6[Online]. 3 Let reformulate the problem of interest :Note 3a 3Ava 6 3 3May 6linear 66), 3ea Let us reformulate the problem of :rhyth2 2 the problem of interest :returns & et us reformulate the problem :of 2us 1 interest s Let reformulate problem ofus " " " i+1 iand i+1 2,2,denoted 2), 2reformulate 112 " " Let us reformulate the problem of interest :" that Let us reformulate the problem of :rhyth" " " " " $ " $ " " " " " $ $ P P " " " ues ues (it ues is (it done is (it done is in done linear in linear in time time for time for " " " " " in this paper can deal with grace notes. in a sented in this paper can deal with grace notes. Indeed, in that in the above RT representation, we consider only 4 Note in the above RT representation, we consider only 4 4 4 4on val = val (tjonset( ).defined ionset( i notes 1list val (t =(t val (tis ).CONCLUSIONS RT, RT, one RT, one note one note by by one by one or or several or several grace notes can notes can be can be be " https://hal.inria.fr/hal-01315689 "ues "Available: "ininInternational " Based "Enumeration,” "Available: " preceded " note Based on Tree Enumeration,” in International "preceded "International on Tree Series Enumeration,” International " Available: "International mic value value .one .ues onset( )=)done is agrace list defined follows follows :Jac Based on Tree if mic on+ is"inonset( empty, return on .1) Based on Tree Enumeration,” Based Tree Enumeration,” in 1))grace 2 https://hal.inria.fr/hal-01315689 [6] F. 2in Available: https://hal.inria.fr/hal-01315689 Available: https://hal.inria.fr/hal-01315689 https://hal.inria.fr/hal-01315689 1 1, 21 Available: https://hal.inria.fr/hal-01315689 (it is linear each tr2 is in linear time for each hal-01105418 hal-0 0 https://hal.inria.fr/hal-01315689 < ipreceded < i several < jExample + "Series 8 done If , 6. , a , then = 0,CONCLUSIONS ,as ,as 1for 5.1 In [17, 18] we have proposed systems rules for rewrit"Series mic value. Intuitively, merge( ) statement correponds to theof ference, MCM "Series 6. CONCLUSIONS "Series micmic value. Intuitively, merge( to the 6.)time CONCLUSIONS jonset( i = on . (it mic value. Intuitively, merge( correponds totoAvailable: the mic value. Intuitively, merge( the 6. CONCLUSIONS value. Intuitively, merge( ,1,), 2correponds to the CONCLUSIONS 6. CONCLUSIONS 6. 1 2correponds 1 , 1Problem j=0 j=0 i1, 12 2),2))correponds # " " " " 0 )<i = #j on "" Conference non-termin non-term 1 " Music "(ICMC), 121116. two inform "# i# . "ser. " " "Music " "# (ICMC), two information P Struct 02312231123112123 1 P Computer Music Conference ser. Proceedings 0 130120the Computer (ICMC), Proceedings Computer Music Conference (ICMC), ser. Proceedings 033to Computer Music Conference (ICMC), ser. Proceedings Computer Conference ser. Proceedings i i 1 1 given a weighted rhythm grammar G and two non-empty & given a weighted rhythm grammar G and two non-empty & given arhythm weighted rhythm grammar G and two non-empty & en arhythm weighted rhythm grammar G and two non-empty & a weighted rhythm grammar G and two non-empty & % & % " " 2 given a weighted rhythm grammar G and two non-empty 3 given a weighted rhythm grammar G and two non-empty & 3 2 % & " % " " % & " % " " % & " % " rhythm obtained by playing both and at the same Intelligence, D. % & " % " " obtained by playing both and at the same " " the values. the values. values. % & " % " % & % " In order to be able to In be order able to reconstitute be able to be the able merged to reconstitute the merged obtained by playing both and at the same " " rhythm obtained by playing both and at the same rhythm obtained by playing both and at the same " " one generic symbol n to represent any kind of event. " " " " one generic symbol n to represent any kind of event. " " & " " 1 2 " " " " " " & " " 1 2 " & " " &Agon, "&in "a RT, note by several can be note by or several notes can be "one &RT, "Agon, 111 222 encoded encoded encoded single in aAssayag, single symbol, symbol, symbol, labelling labelling leaf. a1= For leaf. For instance, For instance, ! "!and !a "grace "aofnotes "J.&Agon, &a ""< " labelling ""1Rueda, "&as ""!or ""1J. &such ! " and ! onset( "and &"C. !in ing RTs into equivalent RTs. This includes rules &[3] !preceded "the "one &[3] ! "42nd "on &International &preceded !"Assayag, "!one !grace ".!C. [3] C. Agon, G. Assayag, Fineberg, and C. [3] C. Agon, G.single J. Fineberg, andthe C.Given Agon, G. Assayag, J. Fineberg, Rueda, 0"Computer 0< iC. i"J.""C. jFineberg, + jreturn 1, 1, )0,leaf. &""42nd C. Agon, G. Assayag, Fineberg, Rueda, [3] C. G. Assayag, Fineberg, C. Rueda, &one & two & on a"Rueda, on, we call the[6] transforma[3] C. G. J.< C. G. Assayag, Fineberg, and $ We define a and new values and &International 1Conference 1+ 1 2 1Jacquemard, 2 L (G) LRepre (G LtM $ $ P. [6] F.q Ja $ $numbers if on "list $ $ ""!and ""Rueda, of International Computer Conference ...Figure "J. jinstance, ""International the values. values. of the Computer Conference 4i1) "[3] "42nd of the 42nd Computer ...Figure the 42nd Music Conference 4= of the Computer 4i4then ""Rueda, "...Figure 1"onset( qDonat-B &of j=0 j=0 %"""" C. %empty, "2""value, 8""""< Example 8onset( =...Figure )...Figure = ,attribute ,onsets 14 onset( ) =inverse 0,F.jof 1to each sequences of &&and sequences ofnumbers positive rational ,"&rhythmic %""""%is "Music "= %2"Music ""If "Music sequences of positive rational numbers ,numbers sequences of positive rational , operator sequences of positive rational numbers and " " Music sequences positive and sequences ofof positive rational numbers """!1"""""! "! "!"" Example "","Conference 12 ""International 1 and time. " Ifhave 9110, Oscar Ban 1rational 2 1 and 2numbers time. &&&2&&&,2&"&&"&$ " """%"""rhythmic of nodes, 1 positive 2 , rational rhythmic to also each have rational to attribute rational time. time. &2&&"&%,""&"&critique ""in "& % ""%""% " “Kant: time. and " "critique " "also 1 1and ! ""A !""then 6! "!,""!3" ,"we 2value, 6defined 32,,we 3 ,asto 2,= 3 ,nodes, ! [ &&,&critique " " and " " tion IOI(on), follows : IOI(on) on & “Kant: of pure quantification,” in Proc. of $ A critique of pure quantification,” Proc. of “Kant: A of pure quantification,” in Proc. of $ “Kant: A of pure quantification,” in Proc. of $ $ “Kant: A critique of pure quantification,” in Proc. of Acknowledgments " Acknowledgments “Kant: critique pure Proc. ofofSep. pure quantification,” ininProc. ofofThe Acknowledgments " [Online]. ""Netherlands, "A Acknowledgments " " [Online]. ""Netherlands, " " "weSep. as ),"2which returns a Utrecht, new "[Online]. "A"""critique Acknowledgments , )the→ orproblem 2(n, )interest → n merge( or 3 2(x ),"2(x xsymbol ), 2( Structural Theory of Rhyth Struc Acknowledgments "introduce "“Kant: "Sep. (ICMC), Utrecht, "" Netherlands, (ICMC), Utrecht, 2016. [Online]. 2 , denoted 1 ,1",2x (ICMC), Utrecht, Netherlands, Sep. 2016. [Online]. " 2016. "rhyth"quantification,” (ICMC), Utrecht, Sep. 2016. 2016. 3 ,in 4single Example 2Acknowledgments The three RT Figure 3C Straightforwardly, could new symbols Example 2preceded RT of Figure 3chord, have Straightforwardly, we could introduce new symbols la-by the the symbol symbol n1994, ncould nsymbol, could describe describe one note one note note preceded by kainverse grace by kla-grace k grace in athe symbol, labelling aa52–9. leaf. For instance, encoded aICMC, labelling leaf. For instance, Let us reformulate the of interest United Kingdom LetLet us reformulate problem of interest : : : : : encoded aproblem (ICMC), Let us reformulate the problem ofofinterest number symbol itand corresponds number the symbol tothe (either itthree acorresponds note, chord, (either a of note, aL Letus usreformulate reformulate the problem interest the of "k "1994, "one kNetherlands, kpp. "1994, " " "describe " "pp. " ICMC, "1994, " "single "the ferenc on .a preceded L (G) := (G) ICMC, Aarhus, Denmark, Aarhus, Denmark, ICMC, Aarhus, Denmark, 1994, ICMC, Aarhus, 1994, pp. 52–9. ICMC, Aarhus, Denmark, pp. 52–9. " "https://hal.inria.fr/hal-01315689 otherwise, let (S , o1pp. )52–9. (S oinverse first element of toRepresentations " Available: "Denmark, Aarhus, Denmark, pp. ICMC, Aarhus, 1994, pp. 52–9. return weighted rhythm grammar Gsuch such that "Available: "list " Denmark, " " of "correponds "52–9. q q return a weighted rhythm grammar G such that return a weighted rhythm grammar G such that mic value. Intuitively, merge( , ) to the a weighted rhythm grammar G such that and Term Repr nurn a weighted rhythm grammar G such that Given a onsets on, Given we call list the of transformawe call the transformareturn a weighted rhythm grammar G that return a weighted rhythm grammar G such that 152–9. 2 ,onsets 2 ) be on, https://hal.inria.fr/hal-01315689 Available: 1 2 https://hal.inria.fr/hal-01315689 Available: https://hal.inria.fr/hal-01315689 Available: https://hal.inria.fr/hal-01315689 set of RTs set of RTs gene 2(x5 , x6 ) → 2 3(x1 , x2 , x3 ), 3(x4 , x5 , x6 ) . 6. CONCLUSIONS 11able 1people, 6.may CONCLUSIONS You may acknowledge people, projects, 1extended 1You 1to 1projects, 1 2[ 2Intelli 1acknowledge 1toacknowledge 1may 1rest, 1etc). 6.6. CONCLUSIONS may acknowledge people, projects, funding CONCLUSIONS 6. CONCLUSIONS You may people, projects, funding agencies, You may people, projects, funding agencies, Available: acknowledge projects, funding agencies, In order be be able toacknowledge reconstitute the merged You may acknowledge people, funf a rest, etc). We thus aconsider We thus rhythmic consider values extended rhythmic values You people, projects, Rhythmic values on and respectively. )2 = 2 |two val (t) = merge( ,4. )}. Example Example 8"tion 8inYou If = ,any ,of ,kind ,afollows then ,and then onset( onset( )equivalent. = )i+1 = 0, 0, ,the ,second,https:/ 1fund ,[ 1 of ,be are therefore equivale of RTs order to encode any kind of L(G )| L(G = L(G) |GL(G) val (t) = merge( ,4.1beling )}. ,rhythmic ,2.3 and are therefore L(G {t 2 val (t) merge( ,and )}. beling the of RTs in to of ){t = {t 2 |L(G val (t) = merge( ,|)}. )}. Computation in and Musi rhythm obtained by playing both and at%The the same L(GL(G ) given = |L(G) val (t) merge( 2.3 Pitches, 2.3 Pitches, Pitches, Grace Grace Grace notes, notes, notes, Rests... Rest R notes. The definition The definition of of rhythmic of value value of value aincan RT of RT should aafter RT should then then tion defined as follows :preceded IOI(on) defined as = on :should IOI(on) = on L(G ){t = {t 2L(G) |G val (t) = merge( ,notes. )grammar = {t 2{t val (t) = merge( 2of 1& )}. 2&Chopin Figure 4. Nocturne No op. 9which and 6= alternative 1Figure 2,Chopin Figure 4. Chopin Nocturne 3No op. 93rhythmic and 6If alternative 12 2 Nocturne No 3definition op. and 6"one 1 Figure 4. Chopin Nocturne No 3Huang op. 9"IOI(on), and 6%order alternative 1notes. 1 ,and Figure Nocturne No 3L. 9Chopin and 69%"k-best alternative i, i+1 iand 2 4. Chopin Nocturne 3on op. 9IOI(on), and 6"which alternative 2 4. Chopin Nocturne 3“Better op. 9encode and 6of alternative given aL(G) weighted rhythm grammar two non-empty the symbol could describe note preceded by k grace the symbol n describe by k grace &"the given a )2 weighted rhythm GL(G) two non-empty &&)}. given aL(G) weighted rhythm grammar G and two non-empty given aweighted weighted rhythm grammar G2and and two non-empty a= rhythm grammar non-empty &leaves &"&exploration "leaves %1"in "No %n "parsing,” value, we also have to attribute to each rational &&"&&&&&and "%[4] "Figure %No "rhythmic % ""%2Figure &the "Fineberg, ""!op. 2 2in 6,is6 6(sym, 3 3 2 23and 39110, etc. which can be included after the seco "k ""alternative 2 3 etc. which can bethen included after the second-lev k etc. which be included after the second-level heading "could "one 2 6 3 "“Better "C. etc. can be included after the second-level heading "G. ""“Better etc. can included the second-level heading [4] and D. Chiang, “Better parsing,” in &Agon, "Chiang, "%""parsing,” etc. which can be included after second [4] L. Huang and D. “Better k-best parsing,” of the form (sym, r), where the form r a rational r), where number, r is and a rational number, "note "“Better etc. which can be included after the [4] L. Huang and D. Chiang, parsing,” [4] L. 1[3] Huang D. Chiang, “Better k-best in "k-best "Agon, " " " & " " " " " [4] L. Huang and D. Chiang, k-best in ! " ! " [4] L. Huang and D. Chiang, k-best parsing,” in L. Huang and D. Chiang, k-best parsing,” &$ ! " ! " q!q q!q ,... q! ! ! " " ! " & ! " ! " We have considered using RT rewriting for [3] C. Agon, G. Assayag, J. and C. Rueda, C. Agon, G. Assayag, J. Fineberg, and Rueda, 1 1 & [3] C. G. Assayag, J. Fineberg, and C. Rueda, & [3] C. Agon, Assayag, J. Fineberg, and C. Rueda, [3] C. G. Assayag, J. Fineberg, and C. Rueda, 2.3 Pitches, Grace notes, Rests... 2.3 Pitches, Grace notes, Rests... & & & $ guage), th $ guage), than in $$merging ". both "1st "of "bar "merging "1st "on "half " " " " ference, MCM 2015, feren ser. " " " " " " time. on . notations merging both hands for 1st half of bar 9. notations merging both hands for 1st half of bar 9. notations both hands for half of bar 9. notations merging both hands for 1st half of bar 9. notations merging both hands for half of bar 9. i i notations both hands for 1st half bar 9. notations merging hands for 1st of 9. Unite sequences of positive rational numbers and , number the symbol it corresponds to (either a note, a chord, sequences of of positive rational numbers and , sequences ofofpositive positive rational numbers and , sequences positive rational numbers and , sequences rational numbers and , 2 of 1 111 12 2Proc. 22 Proc. “Acknowledgments” (with no numbering “Acknowledgments” (with no numbering). “Acknowledgments” (with numbering). (with no numbering). “Acknowledgments” (with Proc. of the 9th Int. Workshop on Parsing Technology. Proc. of 9th Int. Workshop Parsing Technology. “Acknowledgments” (with nonumbering). numbering). “Acknowledgments” no Proc. of the 9th on Parsing Technology. ofin the 9th Int. on Parsing Technology. the 9th Int. on Parsing Technology. "Workshop "in Proc. of the 9th Int. Workshop on Parsing Technology. "Workshop " pitches, of the 9th Int. on Technology. "Workshop "Proc. "Int. " " "Ifon finite information such chords, rests etc. finite information such as pitches, rests etc. This issimilar very similar to one we addressed in problem is to the one we addressed be extended be be extended accordingly, accordingly, accordingly, adding by adding adding to the to the to sequence the sequence sequence kno times k times kthe(with times “Kant: A critique of quantification,” in Proc. of This problem isThis very similar to very the one we addressed in his problem is very similar toproblem the one we addressed in critique ofthe pure quantification,” Proc. of problem is very similar to the one we addressed into “Kant: A critique of pure quantification,” in Proc. of “Kant: A critique of pure inParsing of “Kant: A critique of pure quantification,” Proc. of This problem is very similar thethe one we addressed in This problem is very similar to the one we addressed Acknowledgments oby = ovalue ,“Acknowledgments” return (o ,Acknowledgments add(S , Sable ))no ::numbering). interleave(rest(on Acknowledgments Acknowledgments Acknowledgments "Workshop of equivalent rhythm notations. However, we gave up due "in "pure 1 2Proc. 1 1 2merged 1 ), rest(on 2 )) " rhythmic " A:"definition " as " "extended " quantification,” Intelligence, D. M. Tom Intell Co notes. The definition of rhythmic value of a RT should then notes. The of of a RT should then Let us reformulate the problem of“Kant: interest In order to be•by ablechords, toin In be order able to reconstitute be able to be the to reconstitute merged Availa ato rest, etc). We thus consider extended rhythmic values Association for Computational Linguistics, 2005, pp. Association for Computational Linguistics, 2005, pp. Association for Computational Linguistics, 2005, pp. Association for Computational Linguistics, 2005, pp. found in aR Association for Computational Linguistics, 2005, pp. found in arepres trave Association for Computational Linguistics, 2005, pp. Association for Computational Linguistics, 2005, pp. q!q q!q ,.. Note Note that Note that in the in above the above above RT RT represent RT rep part 4. All we have to do now isthat tothat define the merge oper4. All we have to do now tois define thethe merge operICMC, Aarhus, Denmark, 1994, pp. 4.return All we to do now is to define the merge opertpart 4. return All we have do now isrhythm to define the merge operICMC, Aarhus, Denmark, 1994, pp. 52–9. All we have do is to define the merge oper1 ,...,q p 1 Given Given a52–9. list arhythmic list of of onsets onsets on, on, we we call the inverse inverse transform transfo ICMC, Aarhus, Denmark, 1994, pp. 52–9. ICMC, Aarhus, Denmark, 1994, pp. 52–9. ICMC, Aarhus, Denmark, 1994, pp. 52–9. part 4.All All we have todo is to define the merge operpart 4. we have to to define merge operreturn anow weighted rhythm grammar G such a to weighted rhythm grammar Gdo such that return apart weighted rhythm grammar Gnow such return ato weighted grammar Gis such that ahave weighted rhythm grammar Gnow such that the form (sym, r), where rcall is the ain rational number, and 9110, Oscar Bandtlow 9110 an rhythmic value, also to value, attribute we also to each have rational to that attribute tothe each rational to the complexity of this syntactic approach. Rewrite rules Deciding the equivalence of two g the equivalence of two given • we Else, if ohave < oof return (oYou ,may Sacknowledge )7. ::acknowledge interleave(rest(on ),projects, on ).funding may acknowledge people, funding age You may people, projects, funding agencies, You acknowledge people, projects, agenci Grace notes are events of duration 0. The approach preYou may people, projects, funding agenc You acknowledge people, projects, agencie Grace notes are events of duration 0. The approach prethe the value the value value 0 at 0 the at 0 the at appropriate the appropriate appropriate place. place. place. The The lang Th la 1Deciding 2obtain 1may 1the 1REFERENCES 2funding 53–64. 53–64. 53–64. 53–64. 7. REFERENCES 53–64. 7. REFERENCES Note that in above RT representation Note that in the above RT representat 7. REFERENCES 53–64. 53–64. REFERENCES 7. REFERENCES 7. REFERENCES 7. notations of this merged rhythmic value to RTs corgiven a weighted rhythm grammar G and two non-empty ator. notations of this merged rhythmic value to obtain RTs corator. notations of this merged rhythmic value to obtain RTs cornotations of this merged rhythmic value to obtain RTs corr.ator. notations of this merged rhythmic value to obtain RTs corbe extended accordingly, by adding to the sequence k times be extended accordingly, by adding to the sequence k times notations of this merged rhythmic value to obtain RTs corator. notations of this merged rhythmic value to obtain RTs corator. L(G ) = {t 2 L(G) | val (t) = merge( , )}. L(G ) =))= {t 2{t L(G) | val =(t)= merge( L(G L(G) val (t) ==merge( merge( L(G )=={t {t L(G) |(t) val merge( , 212)}. )}. L(G 222L(G) | |val (t) ,1,)}. 2 Figure 4. Chopin Nocturne No 3 op. 9 and 6 alternative 1 , 112 United Kingdom: Springer Unite Figure 4. Chopin Nocturne No 3 op. 9 and 6 alternative 2)}. Figure 4. Chopin Nocturne No 3 op. 9 and 6 alternative Figure 4. Chopin Nocturne No 3 op. 9 and 6 alternative Figure 4. Chopin Nocturne No 3 op. 9 and 6 alternative number the symbol it corresponds number the symbol to (either it a corresponds note, a chord, to (either a note, a chord, Formally Formally, the etc. which can be included the second-level he one one generic one generic generic symbol symbol nafter to nsecond-level represent to n= represe to repre a etc. which can be included the second-level etc. which can be included after the second-level headi etc. which can be included after the second-level head etc. which can be included after the headin tion tion IOI(on), IOI(on), defined defined as follows follows :symbol IOI(on) :rest(on IOI(on) =heading on on [4] L. and D. Chiang, “Better k-best parsing,” in [4] L. Huang and D. Chiang, “Better k-best parsing,” in [4] L. Huang and D. Chiang, “Better parsing,” [4] L.of Huang and D. Chiang, “Better k-best parsing,” [4] L. Huang and D. “Better k-best parsing,” ininin as above can indeed be applied at responding any node aHuang RT, as ihttps://hal.inria. iPovel, i+1 i+ to the merged polyrhythm. A schema of this sequences of positive rational numbers and ,Chiang, responding to the merged polyrhythm. A schema of to the merged polyrhythm. A schema of this responding tothis the merged polyrhythm. A schema of this •k-best Otherwise, (othis ,thus Sthis )as ::consider interleave(on ,after )). responding to the merged polyrhythm. A schema of this responding to the merged polyrhythm. A schema of responding to merged polyrhythm. Agrace schema of 1 responding 2both 2 2this 1in 2D.-J. be done simply by evaluating their re be done simply by their respecti notations merging hands for 1st half of bar 9. Available: Avail notations merging both hands for 1st half of bar 9. notations merging both hands for 1st half of bar 9. notations merging both hands for 1st half of bar 9. notations merging both hands for 1st half of bar 9. athe rest, etc). We thus aTechnology. consider rest, etc). extended We rhythmic values rhythmic values [1] I. Shmulevich and D.-J. “Com in this paper can deal with notes. Indeed, ano [1] I.symbol Shmulevich and D.-J. Povel, “Complexity sented in paper can deal with grace notes. Indeed, in a(with [1] I. Shmulevich and D.-J. Povel, “Complexity measures [1] I. Shmulevich and D.-J. Povel, “Complexity measures [1] I.return Shmulevich and D.-J. Povel, “Complexity measures [1] I.evaluating Shmulevich and D.-J. Povel, “Comple [5] P. Donat-Bouillud, F. Jacquemard, and M. Sakai, [1] I.extended Shmulevich and Povel, “Complex [5] P. Donat-Bouillud, F. Jacquemard, and M. Sakai, [5] P. Donat-Bouillud, F. Jacquemard, and M. Sakai, [5] P.sented F. Jacquemard, and M. Sakai, [5] P.the 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, “Acknowledgments” (with no numbering). “Acknowledgments” (with no numbering). “Acknowledgments” (with no numbering). one generic symbol n to represent any k one generic n to represent an Rhythm tree “Acknowledgments” (with numbering). “Acknowledgments” no numbering). Proc. of the 9th Int. Workshop on Parsing Technology. Proc. of the 9th Int. Workshop on Parsing Technology. Proc. of the 9th Int. Workshop on Parsing Proc. of the 9th Int. Workshop on Parsing Technology. Proc. of the 9th Int. Workshop on Parsing Technology. the value 0 at the appropriate place. The langua value 0 at the appropriate place. The lang This problem is very similar to the one we addressed inDonat-Bouillud, 5.2 Merging values This problem isMerging very similar torhythmic the one we addressed in 5.2 rhythmic This problem isis very similar totovalues the one we addressed in 5.2 Merging rhythmic values This problem very similar the one we addressed in can This problem isMerging very similar to the one we addressed in Merging rhythmic values Merging rhythmic values workflow can be found in Figure [FIGURE TO DO] 5.2 Merging rhythmic values workflow can be found in Figure [FIGURE TO DO] 5.2 rhythmic values workflow can be found in Figure [FIGURE TO DO] long as the subtree at this node matches the left-hand-side workflow can be found in Figure [FIGURE TO DO] workflow be found in Figure [FIGURE TO DO] workflow can be found in Figure [FIGURE TO DO] workflow can be found in Figure [FIGURE TO DO] non-termi non-terminal q on on .Linguistics, .Linguistics, ofan the form (sym, r), ofRhythm where the form rpp. ispp. (sym, amusical rational r),Straightforwardly, where number, r musical is and aof rational number, and musical rhythms,” Rhythm percep Straightforwardly, Straightforwardly, we we could we could could intro in rhythms,” Rhythm perception an of musical rhythms,” Rhythm perception and producof rhythms,” Rhythm perception and producof musical rhythms,” Rhythm perception and producofmusical musical rhythms,” Rhythm perceptio “Towards an Equational Theory of Rhythm Notation,” of rhythms,” Rhythm perception “Towards an Equational Theory of Notation,” “Towards an Equational Theory of Rhythm Notation,” “Towards an Equational Theory of Rhythm iLinguistics, i::Notation,” “Towards an Equational of Rhythm Notation,” “Towards an Equational Theory of Rhythm Notation,” “Towards Equational Theory of Rhythm Notation,” return a define weighted rhythm grammar GAssociation 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 isdefine to the merge operues (it is done in time for each part 4. part All we have tohave do now isThis to the merge operues (it is done inoftion, for each tree Here, denotes the concatenation operator, andlinear add(S ,linear Stime ) part 4.4.All All we have totodo do now define the merge operpart Allwe wehave have donow now define themerge merge operpart 4. to isisisdefine tototo the opernote preceded by one or several grace notes can be 1 pp. 2239–244, RT, one note preceded by one or several grace notes can be of rule. gives generally, ain(t) given non tion, 2000. pp. 239–244, 2000. tion, pp. 239–244, 2000. tion, pp. 239–244, 2000. Let )the be the list of onsets corresponding aone rhythtion, pp. 239–244, 2000. onset( ) be list of onsets corresponding to aRT, rhythin Music Encoding Conference 2015, Florence, Italy, tion, pp. 239–244, 2000. Let onset( beLet the list ofonset( onsets corresponding rhythtion, pp. 239–244, 2000. onset( be)the list ofthe onsets corresponding torhytha)to rhythintrivial Music Encoding Conference 2015, Florence, Italy, set( ) be) the onsets corresponding to aonsets Music Encoding Conference 2015, Florence, Italy, in Music Encoding Conference 2015, Florence, Italy, Straightforwardly, we could introduc Straightforwardly, we could introd Let onset( thelist list of onsets corresponding to rhythonset( ) )the bebe of to aato rhythinfor Encoding Conference 2015, Florence, Italy, in Music Encoding Conference 2015, Florence, Italy, inmerged Music Encoding Conference 2015, Florence, Italy, L(G =acorresponding {t 2 L(G) |Music val = merge( , )}. 53–64. 53–64. 53–64. 53–64. 1 2 53–64. 7. 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( )rewrite a: list defined as follows : afor mic .a list onset( )asis)of 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 aofthis 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. A 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 (G) := [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, F. Jacquemard, J.order Bresson, an [2] A. Ycart, F. Jacquemard, J.Staworko, Bresson, and 1 i 1 This problem iP 1 iP i is 1 1ivery q [1] 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, “Complexity 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 a new operator on two rhythmic values and We define a new operator on two rhythmic values and hal-01105418 We define a new operator on two rhythmic values and hal-01105418 hal-01105418 0 < i < j + 1, onset( ) = We define a new operator on two rhythmic values and hal-01105418 We define a new operator on two rhythmic values and 0 < i j + 1, onset( ) = rhythmic rhythmic value, value, we we also also have to to attribute attribute to to each each ratio rat 1 0i< j + 1, onset( ) = hal-01105418 hal-01105418 << + 1, onset( ) = 1 j<5.2 +ij< 1, onset( ) = 1 1 1 0 < i < j + 1, onset( ) = 0 < i < j + 1, onset( ) = i j 1 1 i j [ i j i rhythmic j values i rhythmic jvalues [ finite finite information finite information information such such as pitches, as pitche as pitc i i j=0j=0 j j rewrite j=0 5.2 Merging rhythmic j=0j=0 Merging rhythmic values divergence in the application of rules. To be applij=0j=0 5.2 Merging rhythmic values 5.2 Merging values 5.2 Merging workflow can be found in Figure [FIGURE TO DO] workflow can be found in Figure [FIGURE TO DO] workflow can be found Figure [FIGURE TO DO] workflow can befound found Figure [FIGURE TODO] DO] workflow can be inininFigure [FIGURE TO Figure 5. Workflow to obtain notations corresponding to asuch gives aRhythm note, etc). “A Supervised Approach for Rhyth “A Supervised Approach for Rhythm Tra “A Supervised Approach for Rhythm Transcription “A Supervised Approach for Rhythm Transcription “A Supervised Approach for Rhythm Transcription “A Supervised Approach for Rhythm “A 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,” Rhythm perception and prod of musical rhythms,” Rhythm perception and produc “Towards an Equational Theory of Rhythm Notation,” “Towards an Equational Theory of Notation,” “Towards an Equational Theory of Rhythm Notation,” , denoted as merge( , ), which returns a new rhyth“Towards an Equational Theory of Rhythm Notation,” the symbol n could describe one note preceded by k grace “Towards an Equational Theory of Rhythm Notation,” , denoted as merge( , ), which returns a new rhyththe symbol n could describe one note preceded by k grace , denoted as merge( , ), which returns a new rhyth, denoted as merge( , ), which returns a new rhyth, denoted as merge( , ), which returns a new rhyth, denoted as merge( , ), which returns a new rhyth, denoted as merge( , ), which returns a new rhyth12 2 2 1 2 212 2k 1 121 2 12 2 2 2 2 k[6] Based on Tree Series Based on Tree Series Enumeration,” in Int merge is then defined follows :on given two rhythmic Based Tree Series Enumeration,” ina International on Tree Series Enumeration,” in International 11 2 to ato 2 beP.considered 1 12corresponding Based on Tree Series Enumeration,” in International finite such as pitches, chord finite information such as pitches, 11 1corresponding 1[6] 1 ator. [6] F.Intuitively, Jacquemard, P.merge( and J.information Bresson, “A Based on Tree Series Enumeration,” restrictive strategy must 1)the 1some 1be Based on Tree Series Enumeration,” in 1be 2 F.P. Jacquemard, P. Donat-Bouillud, and J.)Italy, Bresson, “A number number the the symbol itthe it corresponds corresponds to to (either (either note, aEnumeration,” note, adura cho ach ch 1)1 be 1)1) 1 onset( F. Donat-Bouillud, and J.Donat-Bouillud, Bresson, “A [6] F.21,0, P. Donat-Bouillud, and J. Bresson, “A 1Jacquemard, 2Jacquemard, F.=rhythJacquemard, Donat-Bouillud, and J. Bresson, “A 11)corresponding 21 [6] F. Jacquemard, P. Donat-Bouillud, and J.Based Bresson, “A [6] F. Jacquemard, P. Donat-Bouillud, and J.as Bresson, “A polyrhythm. tion, pp. 239–244, 2000. pp. 239–244, 2000. tion, pp. 239–244, 2000. Let be the list avalue. tion, pp. 239–244, 2000. tion, pp. 239–244, 2000. onset( list of onsets corresponding in Music Encoding 2015, Italy, onset( list ofof onsets ato Let onset( the onsets a)rhythonsets torhytha[6] rhythMusic Encoding 2015, Florence, Example 8= If ,3312613,,0, , then )mic = ,1in ,5.1 1in Music Encoding Conference 2015, Florence, Italy, inMusic Music Encoding Conference 2015, Florence, Italy, mic value. merge( ,symbol correponds to the in Encoding Conference 2015, 8,list If8list ,=onsets ,= ,6,,= then onset( )to = 0, ,0, 1value. mic value. ,Florence, correponds to Example If =,cable, ,be ,the then onset( ,onset( 1 onset( Intuitively, merge( ,)Conference correponds to the ample 8Let If8onset( =Example onset( )20, =1corresponding 0, ,then Grace Grace Grace notes notes notes are are events are events events of duratio of of dPsein mic value. merge( ,)Intuitively, correponds the ple 8LetIfLet = ,,)the then )of = 1rewrite mic Intuitively, merge( correponds to the 8of If then )rhyth13Intuitively, If = mic value. )Italy, to the mic value. Intuitively, merge( correponds totion, the 1) 2correponds 1to 2,1),Florence, 12Intuitively, 2 )merge( 12 1 ,Conference 2 1 2 2 2= 6 20, 32 5.1 2 3onset( 2,,13onset( 36, then 3 2 , 62, Example 36Example 2, ,3622 2, ,35.1 3,5.1 Problem statement Problem statement Problem statement 5.1 Problem statement Problem statement Computer Music Conference (ICMC) Computer Music Conference (ICMC), ser.ser Computer Music Conference (ICMC), ser. Proceedings values and ,Notation merge( ,RT )on = IOI(interleave(onset( onset( Computer Music Conference (ICMC), ser. Proceedings Computer Music Conference (ICMC), ser. Proceedings notes. The definition of rhythmic value of a RT should then Pitches, Grace notes, Rests... Structural Theory of Rhythm Notation based on Tree Computer Music Conference (ICMC), Computer Music Conference (ICMC), notes. The definition of rhythmic value of a should then 2.3 Pitches, Grace notes, Rests... Structural Theory of Rhythm Notation based on Tree Structural Theory of Rhythm Notation based on Tree Structural Theory of Rhythm Notation based on Tree Structural Theory of Rhythm Notation based on Tree 1 2 12.3 2on 1 ), 2 ))). Structural Theory of Rhythm based Tree Structural Theory of Rhythm Notation based Tree mic value .this onset( )aais a defined list defined as follows : obtained forobtained micmic value . and onset( ) strategies is)))aisisis list defined as follows : for May 2015. [Online]. Available: https://hal.inria.fr/ mic value onset( list defined as follows : :for for mic value onset( alist list defined as follows for value . . .onset( as follows : May 2015. [Online]. Available: https://hal.inria.fr/ has to be proven complete. May 2015. [Online]. Available: https://hal.inria.fr/ May 2015. [Online]. Available: https://hal.inria.fr/ rhythm obtained by playing both and at the same May 2015. [Online]. Available: https://hal.inria.fr/ rhythm obtained by playing both and at the same rhythm obtained by playing both and at the same rhythm by playing both and at the same rhythm by playing both and at the same rhythm obtained by playing both and at the same rhythm obtained by playing both and at the same 1 2 1 2 1 2 1 2 1 2 1 2 1 2 q!q ,...,q q!q ,...,q 2R atwo aTerm rest, rest, etc). etc). We We thus thus consider consider extended extended rhythmic rhythmic valu va P Pthe 1S. P P P 1Conference p [2] A. F. Jacquemard, J. Bresson, and Staw Grace notes events of duration 0. Grace notes are events of duration [2] A. Ycart, F.are J. Bresson, and S. Staworko, [2] A. Ycart, F. Jacquemard, J.paper Bresson, and S. Stawork [2] A. Ycart, F. Jacquemard, J. Bresson, and S. Stawor [2] A. Ycart, F. Jacquemard, J.Music Bresson, and S. Staworko iwe 1on, iof of the 42nd International Computer iinverse 1on, ofJacquemard, the 42nd International Computer Music itransforma1 1Merging of the 42nd International Computer Music of the 42nd International Computer Conference of the 42nd International Computer Music Conference Representations and Term Rewriting,” in Mathematics of the 42nd International Computer Mus of the 42nd International Computer Musi Representations and Rewriting,” in Mathematics sented sented sented inYcart, this in this in paper this paper can can deal can deal with deal wit gMC Representations and Term Rewriting,” in Mathematics Given a)call list onsets we call the inverse transformaRepresentations Term Rewriting,” in Mathematics Representations Rewriting,” in Mathematics Given a1, list of on, call the inverse transformalist of on, transformaof on, we the inverse transformaRepresentations and Term Rewriting,” in Mathematics Representations and Term Rewriting,” inand niven aGiven list of onsets on, we call the 5.2 rhythmic values Given acall list of)= onsets on, we call the inverse transformaGiven awe of onsets we call the inverse transformaWe define aoperator new operator on two rhythmic values and We define aand new on two rhythmic values hal-01105418 We define new operator on two rhythmic values and hal-01105418 We define aTerm new operator ontwo rhythmic values We define aaand new operator on rhythmic values hal-01105418 0ii< < jonset( + onset( =1ij=0 hal-01105418 hal-01105418 time. 0a< i< < jionsets +ijjonsets 1, )list time. 1 Mathematics time. 00a< < 1, onset( )ionsets < j++ 1,onset( onset( )iinverse time. 0list + 1, = 1 and time. 111and time. time. ij=0 i = j j=0 i= j j jjfound In comparison, we the above semantic approach, j=0 j=0 be extended accordingly, by adding to the sequence k times extended accordingly, by adding to the k times 1sequence 1Utrecht, 1Supervised “A Supervised Approach for Rhythm Transcr “A Supervised Approach for Rhythm Transcription “A Approach for Rhythm Transcripti “A Supervised Approach for Rhythm Transcript “A Supervised Approach for Rhythm Transcriptio (ICMC), Utrecht, Netherlands, Sep. (ICMC), Utrecht, Netherlands, Sep. 2016. (ICMC), Utrecht, Netherlands, Sep. 2016. [Online]. (ICMC), Utrecht, Netherlands, Sep. 2016. [Online]. (ICMC), Netherlands, Sep. 2016. [Online]. and Computation in Music: 5th International Con(ICMC), Utrecht, Netherlands, Sep. 2 (ICMC), Utrecht, Netherlands, Sep. 20 and Computation in Music: 5th International Conand Computation in Music: 5th International ConIOI(on), defined :be IOI(on) = on and Computation in Music: 5th International Conand Computation in Music: 5th International Contion IOI(on), defined :on IOI(on) = on Example 9 If = (C, ), (E, ), (G, ) and IOI(on), defined as follows :defined IOI(on) =follows ntion IOI(on), defined astion follows : IOI(on) =follows on and Computation in Music: 5th International Conand Computation in Music: 5th International ConOI(on), defined astion follows : IOI(on) on tion IOI(on), follows IOI(on) = on IOI(on), defined as : :IOI(on) = on i i+1 i i+1 1 , denoted as merge( , ), which returns a new rhythias i+1 i as i+1 , denoted as merge( , ), which returns a new rhythias=follows i+1 of the the form (sym, (sym, r), r), where where rnote rSeries is ishttps://hal.inria.fr/hal-01315689 apreceded acan rational rational number, number, merge( ,of which returns anew new ,denoted denoted as merge( , 212), which returns ainterest new rhythireformulate ,2,denoted asas merge( ,1interest which aform rhythi Let i+1 return a 1of weighted rhythm G such that Let reformulate the problem of interest :Note us reformulate the problem interest :rhyth2 2 the problem interest :returns us reformulate the problem of :of 2us 1 interest 6grammar 3 LetLet usonsets theLet problem ofus 2i+1 2),:), 2reformulate 112 Let usrhythreformulate the problem of interest Let us reformulate the problem of :2:1 that σ sented in this paper can deal with grace sented in this paper deal with gra that in the above RT representa Note in the above RT representation, ) be the12list corresponding to aP. RT, RT, one RT, one note one preceded note preceded by one by one or one sev oroaw 1Notes Based on Tree Series Enumeration,” in Interna based values, less complex. Based on Tree Series Enumeration,” inby International Based on Tree Series Enumeration,” ininInternationa Internation 1Let onset( 2ference, Based on Tree Series Enumeration,” Internatio Based on Tree Enumeration,” in 1dramatically 1 i .on [6] F. Jacquemard, P. Donat-Bouillud, and J. “A Available: https://hal.inria.fr/hal-0131 212, of 111rhythmic [6] F. Jacquemard, Donat-Bouillud, and J. Bresson, “A Available: Available: https://hal.inria.fr/hal-01315689 [6] F.value. Jacquemard, P. Donat-Bouillud, and J.J. Bresson, “A Available: https://hal.inria.fr/hal-01315689 [6] F.Jacquemard, Jacquemard, P. Donat-Bouillud, and Bresson, “A [6] F.value. P.ser. Donat-Bouillud, and J. Bresson, “A Available: https://hal.inria.fr/hal-01315689 ference, MCM ser. Lecture Notes in Artificial Available: https://hal.inria.fr/hal-013156 Available: https://hal.inria.fr/hal-0131568 ference, MCM 2015, ser. Lecture in ference, MCM 2015, Lecture Notes in Artificial ference, MCM 2015, ser. Lecture Notes in Artificial MCM 2015, ser. Lecture Notes in Artificial .Ifon =1,)2015, (A, ), (C, ),Bresson, (E, )Artificial ,in then . i .Example ference, MCM 2015, ser. Lecture Notes Artificial ference, MCM 2015, ser. Lecture Notes in Artificial Example ,13,13,1613,then ,,then , then onset( )0, =122312,0, , 1 mic value. Intuitively, merge( , ) correponds to the 21 ion 8 on If ,=12612,= onset( ) =))0, ,0, 1 mic value. Intuitively, merge( , correponds to the Example If8= ,263,6,12,16,then onset( = , , 1 Example Ifi .on onset( ) = , , 1 Example 88i8on = onset( = 0, , 1 mic Intuitively, merge( ) correponds to the i .If mic Intuitively, merge( , ) correponds to the mic value. Intuitively, merge( , ) correponds to the 1 2 the value 0 at the appropriate place. The langu 1 2 3 6 2 the value 0 at the appropriate place. The language 1 2 2 2 3then 2 3 2= 2 2333l. onset( ) is a list defined as folL(G )Collins =and {ton ∈and L(G) |vol. val (t) =Music σ kConference σConference }.Conference mic value of length σEds., 1on 1non-empty 1vol. 1Music Computer Music (ICMC), ser. Procee Computer (ICMC), ser.ser. Proceedings Computer Conference (ICMC), ser. Proceedin Computer (ICMC), ser.Proceeding Proceedi Computer Music (ICMC), Structural Theory of Rhythm Notation based Tree Structural Theory of Rhythm Notation based on Tree Structural Theory of Rhythm Notation on Tree Structural Theory of Rhythm Notation based on Tree Structural Theory of Rhythm Notation based Tree Intelligence, D. M. Tom and A. Volk, Eds., Intelligence, D. M. Tom Collins and A. Volk, Eds., Intelligence, D. M. Tom Collins and A. Volk, vol. In order to be to be able to reconstitute the merged D. M. Tom Collins and A. Volk, Eds., vol. given aplaying weighted rhythm grammar G two Intelligence, D. M. Tom and A. Volk, Eds., vol. In to be able toable be able to reconstitute the merged aCollins weighted rhythm grammar Gbased two non-empty In to order to able betoable toorder beto able to reconstitute the nder order to able be toorder be able reconstitute the merged Intelligence, D. M. Tom and A. Volk, vol. given arhythm weighted rhythm G and two non-empty merge( ,and )the =at ), (C, ), (E, ), preceded (G, )1Music Intelligence, D. M. Tom A. Volk, Eds., vol. be be able reconstitute merged given arhythm weighted rhythm grammar G and two given a Intelligence, weighted rhythm grammar G and two non-empty In order beable able tobe be able toreconstitute reconstitute the merged P In totobe tothe able tomerged the merged given agrammar weighted rhythm grammar G(AC and two non-empty aby weighted rhythm grammar G and two non-empty rhythm obtained by playing both at the same 1 obtained by playing both and at same obtained by playing both and at the same rhythm obtained both and the same rhythm obtained by playing both the same one generic symbol nand to represent an one generic nAgon, to represent any kin 1Collins 2and igiven 1given 1 2Collins RT, one note by one or several one note by one or seve 3RT, 6Eds., 6symbol 3preceded 1and 11non-empty 2222at encoded encoded encoded in aInternational in single a2Conference in aAssayag, single symbol, symbol, symbol, labelli lab [3] C. Agon, G. Assayag, J. Fineberg [3] C. Agon, G.single J. Fineberg, and [3] C. Agon, G. Assayag, J. Fineberg, Rueda, [3] C. Agon, G. Assayag, J.International Fineberg, and C. Rueda, [3] C. Agon, G. Assayag, J. Fineberg, C.and Rueda, [3] C. G.Computer Assayag, J. Fineberg, Fineberg, [3] C. Agon, G. Assayag, J.C. anal lows :the for 0inverse < iattribute < lattribute + onset( )iOscar = of the 42nd Computer Music Confe of the 42nd International Computer Music Conference of the 42nd International Computer Music Conferen of the 42nd Computer Music Confere of the 42nd International Music Conferenc Representations and Term Rewriting,” in Mathematics jsequences Representations and Term Rewriting,” in Mathematics ahave list of onsets on, we call the inverse transformaRepresentations and Term Rewriting,” in Mathematics Representations Term Rewriting,” in Mathematics Representations and Term Rewriting,” in Mathematics Given list of onsets on, we call the inverse transforma9110, Oscar Bandtlow and Elaine Chew. London, Given list ofof onsets on, we call the inverse transformaGiven alist list onsets on, we call the transformaGiven aarhythmic of onsets on, we call inverse transforma9110, Oscar Bandtlow and Elaine Chew. London, j=0 9110, Oscar Bandtlow and Elaine Chew. London, rhythmic value, we also have to to each rational 9110, Bandtlow and Elaine Chew. London, sequences of positive rational numbers and , 9110, Oscar Bandtlow and Elaine Chew. London, value, we also have to attribute to1,each rational rhythmic value, we also have to attribute to each rational of positive rational numbers and , thmic value, we also have to attribute to each rational 9110, Oscar Bandtlow and Elaine Chew. London, sequences of positive rational numbers , 9110, Oscar Bandtlow and Elaine Chew. London, mic value, weaGiven also to attribute to each rational sequences of positive rational numbers and , of positive rational numbers and , rhythmic value, we also have to attribute tosequences each rational rhythmic value, we also have to to each rational sequences of positive rational numbers and , sequences of positive rational numbers and , 1 2 1 2 time. 2 2 problem time. 1 1 12 this time. time. time. 1 1 is very 2 2 similar to “Kant: Therefore, the one we adA critique of pure quantifica “Kant: A critique of pure quantification,” “Kant: A critique of pure quantification,” in Proc. of “Kant: A critique of pure quantification,” in Proc. of “Kant: A (ICMC), critique ofUtrecht, pure quantification,” in Proc. ofquantificatio “Kant: A critique of pure “Kant: A critique of pure quantification (ICMC), Utrecht, Netherlands, Sep. 2016. [On Netherlands, Sep. 2016. [Online]. (ICMC), Utrecht, Netherlands, Sep. 2016. [Onlin (ICMC), Utrecht, Netherlands, Sep. 2016. [Onlin (ICMC), Utrecht, Netherlands, Sep. 2016. [Online 5. POLYRHYTHMS and Computation in Music: 5th International Conand Computation in Music: 5th International Contion IOI(on), defined as follows : IOI(on) = on and Computation in Music: 5th International Conand Computation in Music: 5th International Conand Computation 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), defined as follows : IOI(on) = on tion IOI(on), defined as follows : IOI(on) = on United Kingdom: Springer, Jun. 2015, p. 12. [Online]. Straightforwardly, we could introd United Kingdom: Springer, Jun. 2015, p. 12. [Online]. number the symbol it corresponds to (either a note, a chord, United Kingdom: Springer, Jun. 2015, p. 12. [Online]. Straightforwardly, we could introduce United Kingdom: Springer, Jun. 2015, p. 12. [Online]. number the symbol it corresponds to (either a note, a chord, i1 i+1 number the symbol it corresponds to (either aa note, mber symbol it corresponds to (either acorresponds note, a chord, Kingdom: Springer, Jun. 2015,encoded p.12. 12. [Online]. i aichord, i+11i+1 Springer, p. [Online]. the the symbol symbol symbol n1994, nthe could nsymbol, could describe describe describe one on na5ni er the the symbol it corresponds to aitnote, chord, i ito(either i+1 i+1 number the(either symbol itcorresponds (either note,Let chord, number the symbol to note, aachord, that we could merge two rhythmic valencoded in athe single symbol, labelling in aICMC, single labelling Let us reformulate the problem of interest : 2015, us reformulate the problem of Note interest : in: :Jun. 1aa 1 United 2United Let us reformulate the problem ofof interest Letus us reformulate theKingdom: problem interest :principle, Let reformulate the problem of interest 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 such that return a weighted rhythm grammar G such that dressed in Section 4 and can be solved with same proreturn a weighted rhythm grammar G such that return a weighted rhythm grammar G such that return a weighted rhythm grammar G such that return a weighted rhythm grammar G such that return a weighted rhythm grammar G such that Available: https://hal.inria.fr/hal-01315689 Available: https://hal.inria.fr/hal-01315689 Available: https://hal.inria.fr/hal-01315689 Available: https://hal.inria.fr/hal-01315689 Available: https://hal.inria.fr/hal-01315689 2 6 3 2 3 ference, MCM 2015, ser. Lecture Notes in Artificial ference, MCM 2015, ser.ser. Lecture Notes in Artificial on .thus ference, MCM 2015, ser. Lecture Notes inin Artificial ference, MCM 2015, ser. Lecture Notes Artificial ference, MCM 2015, Lecture Notes Artificial on .etc). Available: https://hal.inria.fr/hal-01138642 on .ithus on . iWe on Available: https://hal.inria.fr/hal-01138642 Available: 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 aconsider rest, We thus consider extended rhythmic values aetc). rest, thus consider extended rhythmic values est, etc). extended rhythmic values Available: https://hal.inria.fr/hal-01138642 iWe Available: https://hal.inria.fr/hal-01138642 extended rhythmic values i .iWe rest, etc). We thus consider extended rhythmic values aaconsider rest, etc). We thus consider extended rhythmic values ues that are notin the same length. Nevertheless, here, a L(G )2 = {t 2 L(G) |two val (t) = merge( , )}. beling the of RTs in order tp L(G )|Tom = {t L(G) |Gand (t) = merge( , 1vol. L(G )Intelligence, {t 2 val (t) merge( ,and )}. beling the leaves of RTs in order to enc L(G ){t = {t 2 |D. val (t) = merge( ,|)}. )}. L(G ) given = 2 |L(G) val (t) merge( notes. notes. The The definition The definition definition of rhythmic of rhythmic of rhyth va L(G )Tom = {t 2Collins L(G) |G val (t) = merge( ,notes. L(G )grammar = {t 2 L(G) val (t) = merge( 1 )}. 2 )}. We present an of our approach to obtain good 1symbol 2,1)}. 1 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 to the merged D. M. Collins and A. Volk, Eds., vol. Intelligence, D. M. Tom Collins and A. Volk, Eds., vol. M. Tom Collins A. Volk, Eds., vol. given aL(G) weighted rhythm grammar two non-empty the nAgon, describe one note the symbol nG.could could describe one not In(sym, order to be able to be able reconstitute the merged given aIntelligence, weighted rhythm Gand two non-empty In(sym, order to be able to be able totoreconstitute reconstitute the merged cedure. Inorder order tobe be tobe able reconstitute the merged In to able able to given aL(G) weighted rhythm grammar Gval and two non-empty given aweighted weighted rhythm grammar G2A. and two non-empty a= rhythm grammar non-empty of the (sym, r), where rrational amerged rational number, and ofr), the form (sym, r), where rnumber, is ais number, and of the form (sym, r), where rrational is atoapplication rational and the form where rto is abe rational number, and form r), where rable isform aform number, and of the (sym, r), where is ais rational number, and of the form (sym, r), where rrof athe rational number, and khave kleaves rhythmic value is always normalized to aG. total du[4] L. Huang and D. Chiang, “Better kGiven areconstitute list onsets on, we call the inverse transforma[4] L. Huang and D. Chiang, “Better k-best p [4] L. Huang and D. Chiang, “Better k-best parsing,” inRued [4] L. Huang and D. Chiang, “Better k-best parsing,” [4] L. Huang and D. Chiang, “Better k-best parsing,” in [4] L. Huang D. Chiang, “Better k-be [4] L. Huang and D. Chiang, “Better k-bes [3] C. Agon, G. Assayag, J. Fineberg, and C. R [3] C. Agon, Assayag, J.and Fineberg, andand C.in Rueda, [3] C. Agon, Assayag, J. Fineberg, and C. Rued [3] C. Agon, G. Assayag, J. Fineberg, and C. Rue [3] C. G. Assayag, J. Fineberg, C. 9110, Oscar Bandtlow and Elaine Chew. London, 9110, Oscar Bandtlow and Elaine Chew. London, rhythmic value, we also have to attribute to each rational 9110, Oscar Bandtlow and Elaine Chew. London, 9110, Oscar Bandtlow and Elaine Chew. London, 9110, Oscar Bandtlow and Elaine Chew. London, sequences of positive rational numbers and , rhythmic value, we also have to attribute to each rational sequences of positive rational numbers and , rhythmic value, we also have to attribute to each rational rhythmic value, we also have to attribute to each rational rhythmic value, we also have to attribute to each rational sequences of positive rational numbers and , sequences of positive rational numbers and , sequences of positive rational numbers and , 2 of notations for polyrhythms, following studies invery [19]. Po1 111 12 2Proc. 22 Proc. ration ofsimilar Proc. of the 9th Int. Workshop on Par tion IOI(on), defined asThis follows : is IOI(on) = on Proc. of the 9th Int. Workshop on Parsing Teo Proc. of the Int. Workshop on Parsing Technology. of“Kant: the 9th Int. Workshop on Parsing Technology. the 9th Int. Workshop on Parsing Technology. Proc. of the 9th Int. Workshop on Parsin Proc. of the 9th Int. Workshop on Parsing i This i+1 finite information such as pitches, ch finite information such as pitches, chords, This problem is very to the we addressed inA9th This problem is very similar to1.the one weone addressed inextended be be extended be extended accordingly, accordingly, accordingly, by by adding by addi a “Kant: A critique of pure quantification,” in Pr This problem is very similar to the one we addressed in problem is similar to the one we addressed in critique of pure quantification,” in Proc. of This problem very similar to the one we addressed in “Kant: A critique of pure quantification,” in Proc. “Kant: A critique of pure quantification,” in Proc “Kant: A critique of pure quantification,” in Proc. This problem is very similar to the one we addressed in problem is very similar to the one we 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 symbol corresponds (either anote, note, achord, chord, number symbol ititit corresponds to (either aanote, aamore chord, Example 10 Figure 6the presents afor list of rhythms obtained Association for Computational Lingu Association for Computational Linguistics, ontoi .(either An example is12. given Figure 5. Association for Computational Linguistics, 2005, pp. Association Computational Linguistics, 2005, pp.Linguisti Association for Computational Linguistics, 2005, pp. Association for Computational Linguis Association for Computational lyrhythms occurs when orchord, that part 4. All we have to do now is to define merge oper4. All we have to do now is tois define thethe merge operICMC, Aarhus, Denmark, 1994, pp. 52–9. 4.return All we have to do now is to define the merge operpart 4.rhythmic All we have do now isrhythm to define the merge operICMC, Aarhus, Denmark, 1994, pp. 52–9. part 4. part All we have to do is to define the merge operICMC, Aarhus, Denmark, 1994, pp. 52–9. ICMC, Aarhus, Denmark, 1994, pp. 52–9. ICMC, Aarhus, Denmark, 1994, pp. 52–9. part 4. All we have to is to define merge operpart 4. All we have to do now to define operreturn anow weighted rhythm grammar G such that return aAvailable: weighted rhythm grammar Gdo such that return apart weighted rhythm grammar Gnow such return ato weighted grammar G such that afigures weighted rhythm grammar that Available: https://hal.inria.fr/hal-01138642 1G 1 such 1that 1 1 themerge Available: https://hal.inria.fr/hal-01138642 aetc). rest, etc). thus consider extended rhythmic values 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). WeWe thus consider extended rhythmic values aaarest, etc). We thus consider extended rhythmic values Grace notes are events of duration Grace notes are events of duration 0. Th the the value the value value 0 at 0 the at 0 the at appropriate the appropriate appropriate place pla 53–64. 53–64. 53–64. 53–64. 53–64. for [ , ] k [ , , ], using the complex WRG mentionned 53–64. 53–64. ator. ator. ator. ator. ator. be extended accordingly, by adding to th be extended accordingly, by adding to ator. ator. ){t = {t 2 |(t) val (t) = merge( ,23)}. L(G ) L(G =)other, {t 2{t L(G) |L(G) val =(t)2= merge( are not perceived as directly one the or L(G )= L(G) val (t) ==merge( merge( L(G )== {t 2L(G) L(G) |(t) val merge( , 212)}. )}. L(G 22 | |val ,1,3)}. 2 2 )}. 1 ,3112 of the form (sym, r), where rrational a rational number, and from of of the form (sym, r),r), where r isrrraisis number, andand ofofthe the form (sym, r), where rational number, and theform form (sym, r),where where is aderiving rational number, and (sym, aais rational number, [4] L. Huang and D. Chiang, “Better k-best parsin [4] L. Huang and D. Chiang, “Better k-best parsing,” in i [4] L. Huang and D. Chiang, “Better k-best parsing,” [4] L. Huang and D. Chiang, “Better k-best parsing,” [4] L. Huang and D. Chiang, “Better k-best parsing,” in Example 9. sented in this paper can deal with gra sented in this paper can deal with grace no as simple manifestations of the same meter are played si[5] P.Int. Donat-Bouillud, F.Jacquemard, Jacquemard, [5] P.the Donat-Bouillud, F.on Jacquemard, andana [5] P. Donat-Bouillud, F. Jacquemard, and M. Sakai, [5] P. Donat-Bouillud, F. Jacquemard, and M. Sakai, [5] P.the Donat-Bouillud, F. Jacquemard, andon M. Sakai, [5] P. Donat-Bouillud, F. [5] P. Donat-Bouillud, F. Jacquemard, Proc. of the 9th Int. Workshop on Parsing Techno Proc. of 9th Int. Workshop Parsing Technology. Proc. 9th Int. Workshop on Parsing Technolog Proc. of the 9th Int. Workshop on Parsing Technolo Proc. the 9th Workshop Parsing Technolog the value 0Equational at the appropriate place. 0oftheofEquational at the appropriate place. This problem is very similar to the one we addressed 5.2 Merging values This problem isMerging very similar torhythmic the one we addressed in 5.2 rhythmic This problem isis very similar totovalues the one we addressed 5.2 Merging rhythmic values This problem very similar the one weaddressed addressed in invalue This problem isMerging very similar to the one we inin Merging rhythmic values 5.2 5.2 Merging rhythmic values 5.2 Merging rhythmic values 5.2 rhythmic values “Towards an Theory of R “Towards an Equational Theory of Rhythm “Towards an Theory of Equational Rhythm Notation,” “Towards an Equational Theory of Rhythm Notation,” “Towards an of Rhythm Notation,” “Towards anEquational Equational Theory of Rhy “Towards an Theory of Rhyth Association for Computational Linguistics, 200 Association forTheory Computational Linguistics, 2005, pp. multaneously [20]. In our context, it can be as having Association for Computational Linguistics, 2005, p Association for Computational Linguistics, 2005, Association for Computational Linguistics, 2005, pp part 4. All we to do is to define the merge operpart 4. seen All we have tohave do now is now to define the merge operpart 4.4.All All we have do now to define the merge operpart All we have donow now is to define the merge operpart 4. we have tototodo isis to define the merge operRT, one note preceded by one or seve RT, one note preceded by one or several gr All we to doinnow to define kin operator. ToConference aConference )the be the list of onsets corresponding a 53–64. rhythonset( ) be list of onsets corresponding to rhythMusic Encoding 2015 Let onset( beLet the list ofonset( onsets corresponding to acorresponding rhythonset( be)the list ofLet onsets corresponding torhythaneed rhythinthe Music Encoding Conference 2015, Flore LetLet onset( ) be) the list ofLet onsets corresponding to aonsets inis Music Encoding Conference 2015, Florence, Italy, inatoMusic Encoding Conference 2015, Florence, Italy, Let onset( the list of onsets corresponding to rhythonset( ) )the bebe list of aato rhythMusic Encoding Conference 2015, Florence, Italy, in Music Encoding Conference 2015, F in Music Encoding 2015, Fl 53–64. 53–64. 53–64. 53–64. ator. ator. ator. ator. ator. simultaneously two RTs that do not share the same rootmic value . as onset( )a:aislist a: list defined as :pa]for mic .a list onset( )asis)follows defined as= follows for rhythmic value σencoded ,follows .: .2015. .:in ,: for d[Online]. we associate a[Online]. sequence May 2015. [Online]. Available: h mic value . onset( )value defined as follows : for value . onset( )value aislist defined for May 2015. Available: https://h micmic value . onset( ) mic ismic ais list defined follows for May 2015. [Online]. Available: https://hal.inria.fr/ May [Online]. Available: https://hal.inria.fr/ onset( )aisP islist list defined as[d follows for value . . onset( defined as follows May 2015. Available: https://hal.inria.fr/ May 2015. [Online]. Available: http 2015. [Online]. Available: 1encoded in aMay single symbol, labellin single symbol, ahttps le PiP1 iP1 i 1 PP P Plabelling i 1 1i 1 level arities. i−1andand P. Donat-Bouillud, F. Jacquemard, and S [5] P. F. F.Jacquemard, M. Sakai, [5] Donat-Bouillud, F.F.Jacquemard, Jacquemard, and M. Sak [5][5] Donat-Bouillud, Jacquemard, and M.M. Sak [5] Donat-Bouillud, M. Saka hal-01105418 hal-01105418 hal-01105418 hal-01105418 ij< + 1, )iij=0 =1ij=0 hal-01105418 0onset( < i< += onset( )onsets 0i< jonset( + 1,Merging hal-01105418 hal-01105418 +5.2 1,5.2 onset( )rhythmic 0 <0i<< j<5.2 +ij< 1,5.2 )rhythmic < i)j< < j1, 1, 0i 0= i=< ++ 1, onset( )i)i== i = j j=0 ij=0 i0rhythmic j onset( jj jj j = j=0 j=0 rhythmic values Merging values j=0 Merging rhythmic values ofjonset( onset(σ) [o . ,P.Donat-Bouillud, oP.P.p+1 ] defined by o = d Merging values 5.2 Merging values 1 , . .“Towards i j j=1 “Towards an Equational Theory of Rhythm Nota an an Equational Theory ofone Rhythm Notation,” “Towards an Equational Theory ofofRhythm Rhythm Notation “Towards anEquational Equational Theory Rhythm Notatio the symbol n could describe one no “Towards Theory of Notation the symbol n could describe note prec Merging two RTs that have different arities is not a trivial k k 1 1 that 2 1 12corresponding 11 1corresponding 1[6] 1 for [6] F. Conference Jacquemard, P.2015, an 1)the 1 list 1, then 2 1be 1be [6] F.P. Jacquemard, P.and and J.Ital B all < i ,onset( ≤ prhythNote oP.Music = 0. 1)1 be 1)1) 1corresponding 1 onset( F. Donat-Bouillud, and J.Donat-Bouillud, Bresson, “A [6] F.21,0, P. Donat-Bouillud, J. Bresson, “A 1Jacquemard, F.=rhythDonat-Bouillud, and J.Donat-Bouillud, Bresson, “A 1 [6] F.Encoding Jacquemard, Donat-Bouillud, and [6] F. Jacquemard, P.P. Donat-Bouillud, and 1Music Let be list to a1. onset( list of onsets corresponding to in Music Conference 2015, Florence, onset( list ofof onsets a+ Let onset( the onsets a)rhythonsets toto rhythMusic Encoding 2015, Florence, Italy, Example 8= If ,33126130 )Jacquemard, = ,1in 1ininMusic Encoding Conference 2015, Florence, Ita Encoding Conference 2015, Florence, ItJ Encoding Conference Florence, 8, the If8list ,=onsets ,= ,6,,= then onset( )toa[6] = 0, ,0, Example If =, Example ,be ,the then onset( )0, 1 aonset( Example 8Let If8onset( =Example onset( )20, =1corresponding ,then Example 8Let IfLet = ,,)the then )of = 123then 8of If )rhyth,23223,Jacquemard, 13 ,in If ,,0, = 2= 6 20, 32 2, 1 2 3onset( 2,,13onset( 36, then 2 , 62, merging 36Example 2, ,3622 task when considering only the trees. However, notes. The definition of rhythmic val Structural Theory of Rhythm Notatio notes. The definition of rhythmic value of Structural Theory of Rhythm Notation base Structural Theory of Rhythm Notation based onNotation Tree Theory of2015. Rhythm Notation based on Tree Theory of Rhythm Notation based on Tree Structural Theory of Rhythm Notation Structural Theory of Rhythm b mic value . onset( )aais a defined list defined as follows :1Structural for micmic value . onset( ) is)))aisisis list defined as as follows : for May [Online]. Available: https://hal.in mic value onset( list defined asasfollows follows :Structural for mic value onset( alist list defined follows : for value . . .onset( : for May 2015. [Online]. Available: https://hal.inria.fr/ May 2015. [Online]. Available: https://hal.inria. May 2015. [Online]. Available: https://hal.inria May 2015. [Online]. Available: https://hal.inria.f 1 1 1 2 P Pthe P P P iwe 1on, iof ,transforma,transforma], hal-01105418 then onset(σ) =Rewriting,” [0,and 1]. Example 11 Ifinverse σthe =inverse [ 2Representations iinverse itransforma1 11on, two rhythms is trivial when considering only their onsets: Representations and Term Rewriting, Representations Term Rewriting,” in M and Term in Mathematics Given a)call list onsets we and Term Rewriting,” in Representations and Term Rewriting,” in Mathematics Given a1, list of on, call thecall transformalist of on, transformaGiven of on, we the inverse transformaRepresentations and Term Rewriting,” Representations Term Rewriting,” ini Given aGiven list of onsets on, we call the Given acall list of)= onsets on, we call the inverse Given awe of onsets we call the inverse transformahal-01105418 6Representations 3hal-01105418 2 ,and 3 ,Mathematics hal-01105418 0ii< < jonset( + onset( =1ij=0 hal-01105418 0a< i< < jionsets +ijjonsets 1, )list 00a< < 1, onset( )ionsets < j++ 1,onset( onset( )iinverse 0list + 1, = ij=0 i = j j=0 i= jjj j j=0 j=0 be extended accordingly, by adding t be extended accordingly, by adding to the and in Music: 5th Ins Computation inInternational Music: 5th Internati and Computation in Music: 5th ConIOI(on), defined : IOI(on) = on Computation in and Music: 5th International ConMusic: 5thComputation International Contion IOI(on), defined :on IOI(on) =i and tion IOI(on), defined as follows :defined IOI(on) =follows tion IOI(on), defined astion follows : IOI(on) =follows on and Computation inMusic: Music: 5th Inter Computation in 5th Intern IOI(on), defined astion follows : IOI(on) on tion IOI(on), follows IOI(on) =ii+1 on IOI(on), defined as : :IOI(on) =Computation on i+1 in and iand ias i+1 i as i+1 we only have to merge the twotion onset lists into one list, and ias=follows i+1 ion i+1 i+1 The inverse transformation, called IOI , associates to a 1 1 1 2 1 1 2 1 [6] F. Jacquemard, P. Donat-Bouillud, and J. Bresso 1 1 1 2 1 1 1 1 2 1 [6] F. Jacquemard, P. Donat-Bouillud, and J. Bresson, “A 1 1 1 2 1 [6] F. Jacquemard, P. Donat-Bouillud, and J. Bresson, [6] F. Jacquemard, P. Donat-Bouillud, and J. Bresson, [6] F. Jacquemard, P. Donat-Bouillud, and J. Bresson, ““ ference, MCM 2015, ser. Lecture N ference, MCM 2015, ser. Lecture Notes in ference, MCM 2015, ser. Lecture Notes in Artificial on . 2015, ser. Lecture Notes inser. Artificial ser. Lecture Notes in place. Artificial .Ifon .greater. ference, MCM 2015, ser. Lecture Note ference, MCM 2015, Lecture Notes onthe . ion Example ,3,3,63,then ,,then , then onset( ) ,0, =232,0, , value 1 ference, i .Example 8 on If ,i=262,= onset( ) =))0, ion Example If8= ,3,6,2,6,then onset( ,1,3,23,1,,131ference, Example Ifi .i .If onset( )==0, Example 88i8on = onset( = theMCM value 0Intelligence, at the appropriate place. 0MCM at2015, the appropriate 3then order onsets from the smaller to 2= 20, 26 23the Structural Theory of Rhythm Notation based on Structural Theory of of Rhythm Notation based on Tree Structural Theory ofofRhythm Rhythm Notation based on Tr Structural Theory Rhythm Notation based onTre T Structural Theory based Intelligence, D.Notation M. Tom Collins and A D. M. Tom Collins and A.on Volk 1 1 +5 1,3 "" #onset( = i5 now < 53j5Example +5 1, )6. 8 If6.1= , 6. , CONCLUSIONS , then jonset(6.) = 0, , CONCLUSIONS , 1ference, 5 " " " "3 " 05 < 5 5 " " " 0to3)<ido 2015, ference, ser.Linguistics, Lecture MCM Notes 2015, inpp. ser. Artificial Lecture Notes in Artificial 6.CONCLUSIONS "3have CONCLUSIONS 5i55 <5 5j on i 2= on . jonset( CONCLUSIONS CONCLUSIONS Association for MCM Computational 2005, 6. 6. CONCLUSIONS j=0 j=0 3 3 is ito 3 3 3" " " 3 " 33 4. 3 3 " 5 221 " 3#3 3All 3 3 3 part 3 3 3 3 define "3 "# "#3i# .we “A Supervised Approach “AYcS %" &2&&& 3as % of " Rhythm " new Theory Notation based on Tree [2] A. "21, ,denoted "2 ),rhyth16 3 oper0 1330120130231312311202312the 13121merge " "as itheable i1merged 1to denoted merge( ), which merge( returns , and returns aGand new !Structural "D. !which "onTom 5 5 5 5 "5 5% " 3 2 , reconstitute 2rhythm 1operator define a newaG rhythmic values "3" 3""! ""to In 3be & " Intelligence, %""" !""!""!"" ! ""In &5&"&& &&%"&"& %"" "% " !"" 5 5!""&53&5&55&! 5"3&"&35&"3&5&&"&3"&%&"""& "3%"5%"%""3 53"%53%"""5 ator. Tom Intelligence, and D. A.M. Volk, Eds., Collins vol. A.rhythVolk, Eds., vol. given a weighted given grammar a"Collins weighted rhythm two non-empty and two non-empty 1 and ""be % " 3 3"%able order ableon,towe be the merged $53–64. "" "order 3" 3 %3"be "reconstitute 3!""" 3 " "We " M. " two "grammar 3 3"" "to !able ""a list "to ""1to ! " ! ! " ! Based on Tree Series Base Enu & & Representations and Term Rewriting,” in Mathematics “A S Given of onsets call the inverse transforma$ & & 1 1 1 1 1 2 1 2 1 $& 3 & 3&3&3 &&3 & & & % &" $ [3] C. Agon, G. As [6] F. Jacquemard, P. [6] Donat-Bouillud, F. Jacquemard, and P. J. Donat-Bouillud, Bresson, “A and J. Bresson, “A " " " $ " " %""8"""%"""If " If " "...Figure ...Figure 4 onset( " "" ", Example i i j j 4 "% ""% ""% "$""%"""" """""" " """ " Example ""% " "%""""= ...Figure 4 ...Figure 4 ...Figure 4 " " " " " mic value. Intuitively, mic merge( value. Intuitively, , ) correponds merge( to the , ) correponds to the , denoted as merge( , ), which returns a new rhythj=0 j=0 , , then 8 onset( = ) = , 0, , , , then , 1 ) = 0, , , 1 % & & 2 1 2 1 2 1 2 " 1 1 2 " " " " & " " " 9110, Oscar Bandtlow 9110, and Oscar Elaine Bandtlow Chew. London, and Elaine Chew. London, sequences of positive rational sequences numbers of positive and rational , numbers and , have to2value, attribute also each have rational to attribute to"i+1 each rationaland Computation " rhythmic " ! " " "also 2value, 6defined 32 we 3 asto 2 = 3 on &"& & &"&"&&&&&&"&"" "& ! " ! "! "! " ! " " " "" " "rhythmic ! " 6! "! "!3" "we 1 2 Computer 1 2 Based " 2.18.2—www.lilypond.org inby Music: 5th International Con-on Tree IOI(on), follows : IOI(on) Music Conferenc Com i Acknowledgments " "tion Acknowledgments Acknowledgments engraving byengraving LilyPond Music engraving byNotation LilyPond 2.18.2—www.lilypond.org " " "" "Acknowledgments Music Music engraving LilyPond by LilyPond 2.18.2—www.lilypond.org 2.18.2—www.lilypond.org “Kant: A critiqu Structural Theory ofMusic Rhythm Structural Notation Theory based of Rhythm on Tree based Acknowledgments Acknowledgments " """ "" " " """"""""" " "" " " " " "" Acknowledgments " " $ "$$&$ $ mic value. Intuitively, merge( , ) correponds to the 1 k 1 k [5] P. Donat-Bouillud, F. Jacquemard, and M. Sakai, 1 2 rhythm obtained by playing rhythm both obtained and by playing at the both same and at the same 1United 2 ser. 1 [Online]. 2 2015, p. 12. [Online]. United Kingdom: Springer, Jun. Kingdom: 2015, p. Springer, 12. Jun. number the it corresponds number the symbol to (either it a corresponds note, a chord, to (either a note, a chord, " " symbol " " " " " " " " " " " " " ference, MCM 2015, Lecture Notes in Artificial Comp on . 5.2 Merging rhythmic values i 42nd International of thC " """ " " a" "list " " of onsets5 on, rhythm obtained by playing both at the same ICMC, Aarhus, D Representations and Term Representations Rewriting,” and in Mathematics Rewriting,” in of Mathematics Given Given we1acall list the of inverse call the inverse transforma1 and 2the return aprojects, weighted rhythm return grammar aM.weighted GTerm such rhythm that “Towards an Equational Theory of Rhythm Notation,” time. time. 5 5 5 555 555 6. CONCLUSIONS 11on, 1people, 6.may CONCLUSIONS You may acknowledge people, projects, funding 1extended 1transforma1we 1projects, 1 2funding 2Intelligence, 1acknowledge 1toacknowledge 1rest, 1onsets 6. CONCLUSIONS You may acknowledge people, projects, agencies, 6. CONCLUSIONS 6. CONCLUSIONS You may people, projects, funding agencies, You may people, projects, funding agencies, D.3agencies, Tom Collins and A.grammar Volk, Eds.,Gvol.such that Available: https://hal.inria.fr/hal-01138642 Available: https://hal.inria.fr/hal-01138642 You may acknowledge projects, funding agencies, of the In order be able to be able toacknowledge reconstitute the merged You may acknowledge people, funding agencies, a rest, We aconsider etc). We thus rhythmic consider values extended rhythmic values You people, funding agencies, 3 3 3 3 3thus 3 3 3 333 333 3 3 3etc). 3 3 3 values Figure 5. Chopin Nocturne No op. 9inand 6 L(G) alternative Rhythmic time. (ICMC), Utrecht, Netherl (ICM and Computation in and Music: Computation 5th International Music: Con5th International Contion IOI(on), defined tion as follows IOI(on), : IOI(on) defined as = follows on : IOI(on) = on Figure 4. Chopin Nocturne No 3 op. 9 and 6 alternative Figure 4. Chopin Nocturne No 3 op. 9 and 6 alternative 4. Nocturne No 3 op. 9 and 6 alternative L(G ) = {t 2 L(G) | L(G val (t) ) = merge( {t 2 , | val )}. (t) = merge( , )}. 5Chopin gure 4. 5&5Chopin Nocturne No 3 op. 9 and 6 alternative eFigure 4. 5Chopin Nocturne No 3 op. 9 and 6 alternative i i+1 i i+1 Let onset( ) be the list of onsets corresponding to a rhythFigure 4. Chopin Nocturne No 3 op. 9 and 6 alternative Figure 4. Chopin Nocturne No 3 op. 9 and 6 alternative 5 in Music Encoding Conference 2015, Florence, Italy, & 1 2 1 2 & Let us reformulate the Let problem us reformulate of interest the : problem of interest : & & & % & " % " " 9110, Oscar Bandtlow and Elaine Chew. London, % " % " " (ICM rhythmic value, we also have to attribute to each rational % & " % " " % & " % " " % & " % " " " " 2 2 6 6 3 3 2 2 3 3 etc. which can be included after the second-level heading " " 2 6 3 etc. which can be included after the second-level heading which be etc. included after the second-level heading 6the 3form can be included after the second-level heading "!""" ! "! "! " ! "" " " "" "" " " of" the form "!""" ! "! "(sym, which can included after the second-level heading etc. which included after theand second-level heading r), of"which where rbeiscan (sym, a rational r), where number, rcan is and a notations rational number, && & &"&$ which can bebe included after the second-level heading " " etc. "2 ! "" etc. &$$ &&&&&&"&"&"& " !"" "etc. [4] L.https://hal.inria. Huang and D Let MCM us reformulate the problem of interestAvailable: :in Artificial merging both for 1st2015, half bar Avail $ merging $merging 1 inof 2 1Artificial 29. Here, ". both "of "a "merging "1st "on "on "half MCM 2015, ference, ser.hands Lecture Notes ser. ""1st "of "merging "for "half .")"half notations both hands for 1st of bar 9. merging both hands for 1st half bar 9. notations both hands half of bar ations both hands for 1st half of bar 9. ons merging bothnotations hands for of bar 9. imic iof value . 9. onset( is list as(with follows : for notations both hands for 1st bar notations merging hands for 1st half bar 9.9.defined United Kingdom: Jun.Lecture 12.12 [Online]. May 2015. Available: https://hal.inria.fr/ Availa 12015, 1p.2 Notes 2 number the symbol it (with corresponds (eitherference, a note, chord, 1a(with 1no 1numbering). 1 [Online]. 1 1 1 1 Springer, “Acknowledgments” no “Acknowledgments” (with no numbering). “Acknowledgments” (with notonumbering). “Acknowledgments” no numbering). “Acknowledgments” no numbering). “Acknowledgments” “Acknowledgments” (with " sake of readability, grouped the ties between " For the " " " " we P 1 2ornoI Proc. of1 the2 9th =(with ,values ,noproblem ,numbering). ,numbering). ,Tom and =asimilar ,https://hal.inria.fr/hal-01138642 ,problem . M. Acknowledgments Acknowledgments ireconstitute 1etc). 1 weighted 2weighted This is very This to the one is very we similar addressed to the in one we addressed in Acknowledgments Acknowledgments Acknowledgments "j "be " """ " """ "< "i" "<to Intelligence, D. M. Intelligence, Collins and D. A. Volk, Tom Eds., Collins vol. and A. Volk, Eds., vol. given a rhythm given grammar a G and rhythm two non-empty grammar G and two non-empty 5 5 5 5 5 3 3 3 sym is a symbol denoting a particular In0 order able to In be order able to to be able to be the able merged to reconstitute the merged Available: sym is a symbol denoting a particular note, sym is a symbol denoting particular note, or a rest, or a sym is a symbol denoting a particular note, or a rest, or a a rest, We thus consider extended rhythmic sym is symbol denoting a particular note, or a rest, or a sym is a symbol denoting a particular note sym is a symbol denoting a particular note, given a weighted rhythm grammar G[3] andC. twoAgon, non-empty hal-01105418 + 1, onset( )i = j=0 j G. Assayag, [3] C. J. AC [3] C. forAg 1 1of 2numbers equal note values. For instance, in tothe example, part 4.and All we have to chord... part doOscar now 4. All is to we have the to2chord... do now is operto define the2merge oper1define 1chord... 1merge 2numbers 21 and 2and of theprevious form (sym, r),rational where r is a rational number, 9110, Oscar Bandtlow 9110, and Elaine Bandtlow Chew. London, and Elaine Chew. sequences of positive rational sequences numbers positive and rational ,rational ,Association rhythmic value, we also rhythmic have value, attribute we also to each have to attribute topeople, each rational chord... sequences of chord... chord... chord... 1 positive 2 1London, 2, You may acknowledge projects, funding agencies, You may acknowledge people, projects, funding agencies, Rhythm trees You may acknowledge people, projects, funding You may acknowledge people, projects, funding agencies, You may acknowledge people, projects, funding agencies, 1 2 1 2 Music by2.18.2—www.lilypond.org LilyPond 2.18.2—www.lilypond.org Music engraving byagencies, LilyPond 2.18.2—www.lilypond.org Music engraving byengraving LilyPond “Kant “Kant: A critique of “Kan pure 7. REFERENCES 7. REFERENCES 7. REFERENCES 7. REFERENCES 7. REFERENCES Rhythm tree 7. REFERENCES 7. REFERENCES 53–64. notations of this merged rhythmic value to obtain RTs cornotations of this merged rhythmic value to obtain RTs cornotations of this merged rhythmic value to obtain RTs corations of this merged rhythmic value to obtain RTs corons of this merged rhythmic value to obtain RTs cornotations of this merged rhythmic value to obtain RTs cornotations of this merged rhythmic value to obtain RTs cor1 1 1 2 1 ator. [6] F. aator. Jacquemard, P. Donat-Bouillud, J. Bresson, “A Figure 4.Chopin Chopin Nocturne No and 6 alternative United Kingdom: Springer, United Jun. Kingdom: 2015, p. Springer, 12. [Online]. Jun. 2015, p. 12. [Online]. Figure 4. 4. Chopin Nocturne No 3No op. 93symbol and 69linked alternative To two series offollowing onsets, we de Figure 4.4.Chopin Chopin Nocturne No 3the op. and alternative Figure Nocturne 3op. op. and 6alternative Figure Nocturne No 3last 999If and 66corresponds number it number the symbol toonset( (either itetc. acorresponds note, abe toincluded (either aafter note, chord, To interleave two series of onsets, wewe define the To interleave two series ofinterleave onsets, we define the Toreturn interleave two series ofinterleave onsets, we define the following To interleave twoand series of onsets, we define the following To interleave two series of onsets, wedefine defin we grouped the two sixteenth notes one To series of onsets, Example 8op. = ,alternative )which =which 0,into ,chord, ,included 1be be included after the second-level heading etc. which can after the second-level heading etc. which be included after the second-level heading ICMC etc. can included after the second-level heading etc. which can be second-level heading 2this 6 , 3 , then 2can 3can a weighted rhythm grammar Gtwo such that ICMC, Aarhus, Denmark, ICMC 1 ihttps://hal.inria.fr/hal-01138642 ibe i+1 i+1 return athe weighted rhythm return grammar a weighted such rhythm that grammar G :such that In order to able to be ableGhttps://hal.inria.fr/hal-01138642 to reconstitute the merged responding to the merged polyrhythm. A of this responding to the merged polyrhythm. A of this 1extended responding to the merged polyrhythm. A schema of ponding tonotations the merged polyrhythm. Amerged schema of this 1of ding tonotations the merged polyrhythm. A schema of this responding to the polyrhythm. schema of this responding tohands merged polyrhythm. AA of this Structural Theory of Rhythm based on Tree merging both hands for 1st half of bar 9. Available: Available: notations merging both hands for 1st half of bar 9. function given two series of on notations merging both for 1st half ofof bar 9. notations merging both hands for 1st half bar 9. merging both hands for 1st half of bar athe rest, etc). We thus a9. consider rest, etc). extended We thus rhythmic consider values rhythmic values [1] I. (with Shmulevich and D.-J. Povel, “Complexity measures function :val given two series extended onsets function : two given two series extended onsets [1] I.D.-J. Shmulevich and D.-J. Povel, “Complexity measures function :Notation given two series of extended onsets [1] I.simpler, Shmulevich and D.-J. Povel, “Complexity measures function : given series of extended onsets [1] I.schema Shmulevich and Povel, “Complexity measures [1]schema I.schema Shmulevich and D.-J. Povel, “Complexity measures function :given given two series ofextended extended onse function :of two series of onset [1] I.Shmulevich Shmulevich and D.-J. Povel, “Complexity measures [1] I.extended and D.-J. Povel, “Complexity measures “Acknowledgments” (with no numbering). L(G ) = {t 2 L(G) | (t) = merge( , )}. [5] P. Donat-Bouillu “Acknowledgments” (with no numbering). “Acknowledgments” (with no numbering). “Acknowledgments” (with no numbering). “Acknowledgments” no numbering). eighth note. This allows us to display more id1 2 L(G ) = {t 2 L(G) | L(G val (t) ) = {t merge( 2 L(G) , | val )}. (t) = merge( , )}. rhythmic value, we also have to attribute to each rational can be found in Figure [FIGURE TO DO] workflow be found in Figure [FIGURE TO DO] workflow be found incan Figure [FIGURE TO DO] rkflow be found inworkflow Figure [FIGURE TO DO] ow cancan be can found in Figure [FIGURE TO DO] 1 2 1 2 workflow can be found in Figure [FIGURE TO DO] workflow can be found in Figure [FIGURE TO DO] 5.2 Merging rhythmic 5.2 values Merging rhythmic values [4] L. Hu Representations and Term Rewriting,” in Mathematics Given a list of onsets on, we call the inverse transformasym is a symbol denoting a particular note, or a res sym is a symbol denoting a particular note, or a rest, or a sym isis symbol denoting agiven particular note, aarest, rest, oo sym symbol denoting aparticular note, rest, sym is aaasymbol denoting note, or a[4] or isgiven defined as follows : Equ giv of the form r),ofwhere the of form r musical is (sym, amusical r),rhythms,” where number, r musical is and aof rational number, and musical rhythms,” Rhythm perception and producmerge is then defined astwo follows :oror given two of rhythms,” Rhythm perception and producmerge is then defined as :particular given two rhythmic merge is then defined asmerge follows :athen rhythmic ofrational musical rhythms,” Rhythm perception and producmerge is then defined asmerge follows two rhythmic of Rhythm perception and producrhythms,” Rhythm perception and producmerge is:then then defined follows :an given isfollows defined asas follows :Chiang, given ofmusical musical rhythms,” Rhythm perception and producof rhythms,” Rhythm perception and produc[4] L. Huang and D. L. H i (sym, i “Towards Proc. iomatic and more compact solutions. This simplification number the symbol values itincorresponds to,and (either a note, chord, This problem very similar to we addressed and Computation Music: 5th International Contion IOI(on), defined as tion, follows :pp. IOI(on) on chord... chord... chord... chord... chord... , a)IOI(interl = i =tion, i+1 tion, pp. 239–244, 2000. values merge( ), in = pp. 239–244, 2000. values merge( ,)athe )one = IOI(interleave(onset( values and ,is1 merge( ,aand = IOI(interleave(onset( tion, pp. 239–244, 2000. tion, 239–244, 2000. and merge( ,values ) to IOI(interleave(onset( ),IOI( ons pp. 239–244, 2000. and merge( =)IOI(int IOI(in values and ,2 ,merge( ,1in =Int. tion, pp. 239–244, 2000. tion, pp. 239–244, 2000. 1 1 11values 111 1 1rhyth2 , merge( 1) 2 2 ,and 1 , the 1very 2 ,the 121one 1 Proc. of 9th Worksh Proc 1 1 1 1 one 21 12the 1 ), o 1of 21we 1list 1Encodi 2 1= 2 12 2 Let onset( ) be the list Let onset( onsets corresponding ) be of onsets corresponding to rhythin Music This problem is very similar This problem to the is similar addressed to in we addressed Figure 4. Workflow to obtain notations corresponding to a 7. REFERENCES 7. REFERENCES 7. REFERENCES 7. REFERENCES 7. REFERENCES Assoc notations of this merged rhythmic value to obtain RTs cor5.1 Problem statement notations ofstatement this merged rhythmic value to obtain RTs cor5.1 Problem statement notations of this merged rhythmic value to obtain RTs cor5.1 Problem notations of this merged rhythmic value to obtain RTs cornotations of this merged rhythmic value to obtain RTs corProblem statement roblem statement a ference, rest, etc).MCM We thus consider extended rhythmic values partLecture 4.To All we have toin dotwo now is toof define theonsets, merge oper5.1 Problem statement 5.1 statement ser. Notes Artificial oni . on the Lilypond translation of RTs generTo interleave two series of we define the follo step isProblem performed interleave two series onsets, we define the following To interleave two series ofof onsets, we define the followi To interleave two series onsets, we define the follow To interleave series of onsets, we define the followin Rhythmic values. 2015, 2 6 2 3 6 2 3 6 3 Association for Computati Asso mic value onset( mic ) is value a list defined . onset( as follows ) is a list : for defined as follows : for 2 6 3 2 6 3 May 2015. [On 53–64 part 4. All we have to part do now 4. All is to we define have to the do merge now is operto define the merge operhythm notations of value [ 1/5 2/15 1/15 1/5 1/15 2/15 1/5 ] (schema-05.tx polyrhythm. hythm notations of value [ 1/5 2/15 1/15 1/5 1/15 2/15 1/5 ] (schema-05.tx hythm notations of value [ 1/5 2/15 1/15 1/5 1/15 2/15 1/5 ] (schema-05.tx hythm notations of value [ 1/5 2/15 1/15 1/5 1/15 2/15 1/5 ] (schema-05.tx hythm notations of value [ 1/5 2/15 1/15 1/5 1/15 2/15 1/5 ] (schema-05.tx hythm notations value [1/5 1/5 2/15 1/15 1/5 1/15 2/15 1/5 ](schema-05.tx hythm notations ofof value [Jacquemard, 2/15 1/5 1/15 2/15 1/5 1 1 11 1 11),11(G, 11) responding to the merged A schema of this 1 number, 19vol. 1 responding to to the merged polyrhythm. schema of this 1 1=1),), 1(A, 1 1 1(C, responding totothe the merged polyrhythm. A schema ofof this responding themerged merged polyrhythm. Aschema schema this responding polyrhythm. of this [2] A. Ycart, F.of J. Bresson, and S. the form r), where r(schema-05.tx is a9rational [2] A. Ycart, F.J. Jacquemard, J.S.1/15 Bresson, and S. ator. Staworko, [2] A.reconstitute Ycart, F. Jacquemard, Bresson, and Staworko, [2] A. Ycart, F. Jacquemard, J. Bresson, and S. P P A. Ycart, F. Jacquemard, J. Bresson, and S. Staworko, [2] A.Ycart, Ycart, F. Jacquemard, J.(sym, Bresson, S.]Staworko, [2] A. F. Jacquemard, J. Bresson, and S. D. M. Tom Collins and A. Eds., Example 91= =), (E, Example If1and (E, ) ), and 3Staworko, 5 Staworko, Inpolyrhythm. orderAtoA be able to[2] be able to the merged Example If =Volk, = (C, ), (E, ),If(C, (G, )(C, and =1(G, Example 9Staworko, If (C, ), 1(E, ), 1(G, ) and (A, ), (C, ), 3and 5 Intelligence, 3 3 5 3 3 3 5 3 5 3 3 3 3 3 5 3 3 5 5 5 5 3 333 3 5 5 5 3 3 3 5 5 5 3 5 3 3 3 3 3 3 5 5 5 333 5 3 3 5 555 5 5 5 3 333 3 3 3 5 5 5 5 3 3 3 5 3 333 5 555 333 3 3 3 5 33 3 3 555 3 3 3 3 553 3 5 53 35 3 35 5 3 3 5 5 5 55 33 33 55 5 33 5 55 33 3 5 5 5 33 3 3 555 5 3 5 35 55 3 333 555 3 55 5 5 555 3 3 333 3 3 555 5 3 3 3 3 3 5 3 3 3 5 5 3 3 5 5 5 5 5 5 55 5 5 5 3 3 5 5 5 3 3 3 5 5 5 3 3 i 5 3 3 3 i 1 j=0 j 1 1 1 2 6 3 1 2 2 3 3 333 3 Music engraving by LilyPond 2.18.2—www.lilypond.org Music by LilyPond 2.18.2—www.lilypond.org Music engraving byengraving LilyPond 2.18.2—www.lilypond.org Music by LilyPond 2.18.2—www.lilypond.org engraving by LilyPond 2.18.2—www.lilypond.org Music Music engraving byengraving LilyPond 2.18.2—www.lilypond.org Music engraving LilyPond 2.18.2—www.lilypond.org Music engraving byby LilyPond 2.18.2—www.lilypond.org i i+1 i 3 5 555 5 333 3 555 3 5 333 3 3 5 333 555 3 3 5 333 3 5 Music by LilyPond 2.18.2—www.lilypond.org Music engraving byengraving LilyPond 2.18.2—www.lilypond.org Music engraving 2.18.2—www.lilypond.org Music engraving byLilyPond LilyPond 2.18.2—www.lilypond.org Music engraving bybyLilyPond 2.18.2—www.lilypond.org Music by LilyPond 2.18.2—www.lilypond.org Music engraving byengraving LilyPond 2.18.2—www.lilypond.org Music by LilyPond 2.18.2—www.lilypond.org engraving by LilyPond 2.18.2—www.lilypond.org Music Music engraving byengraving LilyPond 2.18.2—www.lilypond.org Music engraving LilyPond 2.18.2—www.lilypond.org Music engraving byby LilyPond 2.18.2—www.lilypond.org Music by LilyPond 2.18.2—www.lilypond.org Music engraving byengraving LilyPond 2.18.2—www.lilypond.org Music engraving 2.18.2—www.lilypond.org Music engraving byLilyPond LilyPond 2.18.2—www.lilypond.org Music engraving bybyLilyPond 2.18.2—www.lilypond.org Intelligence, D. M. Tom Collins and A. Volk, Eds., vol. In order to be to be able to reconstitute the D. M. Tom Collins A. Volk, Eds., vol.and Tom andand A.D. Volk, Eds., vol. In to be able toable be able to reconstitute the merged sequence of `Intelligence, =Intelligence, [o , .merged .D.. ,M. op+1 ] Collins the rhythmic value In to order to able betoable toorder beto able to reconstitute the In order to able be toorder be able reconstitute the merged Intelligence, D. M. Tom Collins andA. A.VV Intelligence, M. Tom Collins In order be be able reconstitute merged In order beable able tobe be able toonsets reconstitute the In totobe tothe able tomerged reconstitute the merged 1merged To notate polyrhythms, we thus gorhythmic around the of Representations and Term Rewriting,” in Mathem Representations and Term Rewriting,” Mathematics ahave list of onsets on, we call the inverse transformaRepresentations and Term Rewriting,” in Mathemati Representations Term Rewriting,” in Mathema Representations and Term Rewriting,” in Mathematic Given aGiven list of onsets on, we call the inverse 9110, Oscar Bandtlow and Elaine Given arhythmic list ofof onsets on, we call the inverse Given alist list onsets on, we call the inverse transformaGiven aproblem of onsets on, we call the inverse transforma9110, Oscar Bandtlow andin Elaine Chew. 9110, Oscar Bandtlow and Elaine Chew. London, rhythmic value, we also have attribute to each 9110, Oscar Bandtlow and Elaine Chew. London, 9110, Oscar Bandtlow and Elaine Chew. London, value, we also have to attribute to each rational value, we also have to attribute to each rational rhythmic value, we also have to attribute to each rational 9110, Oscar Bandtlow and Elaine ChC 9110, Oscar Bandtlow and Elaine Che rhythmic value, we also to attribute to each rational rhythmic value, we also have attribute toeach each rational rhythmic value, we also have attribute to rational σ = [d .transforma,to dtransformaby drational 1 , . .toto p ] defined i = oi+1 − oi for all 1 ≤ i ≤ and Computation in Music: 5th International and Computation in Jun. Music: 5th International Contion IOI(on), defined as follows :note, IOI(on) = on and Computation in Music: 5th International Co and Computation inJun. Music: 5th International Cp and Computation in Music: 5th International Con IOI(on), defined as follows =i(either on United Kingdom: Jun. 2015 tion IOI(on), defined as follows :itnote, = on tion IOI(on), defined asfollows follows : IOI(on) = on tion IOI(on), defined IOI(on) on United Kingdom: Springer, Jun. 2015, p. 12 United Kingdom: Springer, 2015, p.Springer, 12. [Online]. number the symbol corresponds to (either aa note, a chord, United Kingdom: Springer, 2015, p.Springer, 12. [Online]. merging trees by going through the time domain. We first United Kingdom: Springer, Jun. 2015, p. 12. [Online]. number the symbol it (either corresponds ai+1 note, chord, i(either number the symbol it corresponds to aaIOI(on) chord, number the symbol it corresponds to (either a:corresponds a chord, United Kingdom: Springer, Jun. 2015, itoa i+1 United Kingdom: Jun. 2015, p. number thetion symbol it corresponds to (either ait:note, chord, i= i+1 ito i+1 number theas symbol itIOI(on) note, chord, number the symbol corresponds to (either aai+1 note, aachord, p.consider Finally, ference, MCM 2015, ser. Lecture Notes in Art ference, MCM 2015, ser. Lecture Notes in Artificial on . ference, MCM 2015, ser. Lecture Notes in Artific ference, MCM 2015, ser. Lecture Notes in Artific ference, MCM 2015, ser. Lecture Notes in Artifici on . Available: https://hal.inria.fr/hal-0113 on . on . on . Available: https://hal.inria.fr/hal-01138642 Available: https://hal.inria.fr/hal-01138642 a rest, etc). We thus consider extended rhythmic values Available: https://hal.inria.fr/hal-01138642 Available: https://hal.inria.fr/hal-01138642 a rest, etc). We thus extended rhythmic values i a rest, etc). We thus consider extended rhythmic values a rest, etc). We thus consider extended rhythmic values Available: https://hal.inria.fr/hal-011386 i Available: https://hal.inria.fr/hal-0113864 a rest, etc). We thus consider extended rhythmic values i i i a rest, etc). We thus consider extended rhythmic values a rest, etc). We thus consider extended rhythmic values convert the two RTs into their rhythmic values, from which Tom Intelligence, D. M. Collins and A. Volk, Eds Intelligence, D. D. M. Tom Collins andand A. Volk, Eds., vol. In order to be able to be able to reconstitute the merged Intelligence, D. M. Tom Collins and A. Volk, Eds., vv Intelligence, D.M. M.Tom Tom Collins and A.Volk, Volk, Eds., Intelligence, Collins A. Eds., vo In(sym, order to be able to be able reconstitute the merged In(sym, order toof be able to be able totoreconstitute reconstitute the merged Inorder order tobe be tobe able reconstitute the merged In to able to the of the (sym, r), where amerged rational number, and ofr), the form (sym, r), where rnumber, is number, and of the form (sym, r), where rrational isable atorational and of the form where rto is abe rational number, and of the form r), where rable isform aform number, and of the (sym, r), where rais isrrational ais rational number, and the form (sym, r), where rational number, and we obtain the corresponding onset lists. We then merge σ1rto σaeach = IOI onset(σ ) Oscar ∪Oscar onset(σ .and 2each 19110, 2) Oscar Bandtlow and Elaine Chew. Lo 9110, Oscar Bandtlow andand Elaine Chew. London, rhythmic value, we also to attribute to rational 9110, Bandtlow Elaine Chew. Londo 9110, Oscar Bandtlow and Elaine Chew. Lond 9110, Bandtlow Elaine Chew. London rhythmic value, we also have tohave attribute to each rational rhythmic value, we also have totoattribute attribute tok each rational rhythmic value, wealso also have attribute toeach rational rhythmic value, we have to rational the two onset lists into one onset list, from which we get United Kingdom: Springer, Jun. 2015, p. 12. [On 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. [Onlin United Kingdom: Springer, Jun. 2015, 12.[Online [Onlin United Kingdom: Springer, Jun. 2015, p.p.p.12. number thethe symbol it corresponds to (either a note, chord, number the symbol corresponds (either number the symbol corresponds (either anote, note, achord, chord, 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 [1] A. Klapuri et al., “Musical meter estimation and music transcription,” in Cambridge Music Processing Colloquium, 2003, pp. 40–45. [2] J. Bresson, C. Agon, and G. Assayag, “OpenMusic: visual programming environment for music composition, analysis and research,” in Proc. of the 19th ACM Int. Conf. on Multimedia. ACM, 2011, pp. 743–746. [3] A. T. Cemgil, P. Desain, and B. Kappen, “Rhythm quantization for transcription,” Computer Music Journal, vol. 24, no. 2, pp. 60–76, 2000. ! # 41 !! ! 3 3 5 # ! ! # !! ! ! ! 5 3 ! 3 ! ! ! 5 ! ! ! 5 5 ! ! 3 ! ! !! "" ! ! ! 3 ! ! 3 ! ! !" ! 5 ! !! ! 3 !! ! ! 3 ! 3 ! 5 ! ! ! 5 5 ! 5 ! ! 3 ! !" ! ! ! ! 3 Figure 7. Stravinsky, The Rite of Spring (Violin I) and 6 alternative notations. [4] H. C. Longuet-Higgins and C. S. Lee, “The rhythmic interpretation of monophonic music,” Music Perception: An Interdisciplinary Journal, vol. 1, no. 4, pp. 424–441, 1984. [5] J. Pressing and P. 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