Rhythm Transcription of MIDI Performances Basedon Hierarchical Bayesian Modelling of Repetition
and Modification of Musical Note Patterns
Eita Nakamura, Katsutoshi Itoyama, Kazuyoshi Yoshii
Kyoto University, Kyoto 606-8501, Japan

I. I NTRODUCTION
Music transcription is a fundamental problem in music information processing, requiring the extraction of pitch andrhythm information from music audio signals. Many studies
on acoustic modelling of musical sounds have been carried out
for extracting pitch information [1,2]. Rhythm transcription (orquantisation), on the other hand, has been addressed in the aimof recognising score-written lengths (or note values) of musicalnotes [3–8]. Using prior knowledge on music scores is crucial
for music/rhythm transcription, like speech recognition [9],and machine-learning techniques have been studied in efforts
to construct music ‘language’ models. Hidden Markov models(HMMs) have been widely used to learn a ‘generic’ language
model with a dataset of musical pieces [5–7, 10, 11].
Since musical pieces typically have repetitive structure consisting of a limited number of note patterns, more accurate language models could be obtained by inferring compact grammars fitting individual pieces. Finding repeated patterns inmusic is a topic of computational music structure analysis, andmethods based on similarity matrices [12, 13], data compression [14], or Markov Oracle [15] have been studied. Once therepetitive structure of a piece is known, note-pattern models[6,8] could be used to learn the piece’s grammar. Conversely, if
used note patterns are given, we could infer the repetitive structure more accurately. To solve this chicken-and-egg problem,

Probability

Generic model (base measure)

Dirichlet process

Abstract—This paper presents a method of rhythm transcription (i.e., automatic recognition of note values in music performance signals) based on a Bayesian music language model that
describes the repetitive structure of musical notes. Conventionally, music language models for music transcription are trainedwith a dataset of musical pieces. Because typical musical pieces
have repetitions consisting of a limited number of note patterns,
better models fitting individual pieces could be obtained by inducing compact grammars. The main challenges are inducing
appropriate grammar for a score that is observed indirectly
through a performance and capturing incomplete repetitions,which can be represented as repetitions with modifications. We
propose a hierarchical Bayesian model in which the generationof a language model is described with a Dirichlet process andthe production of musical notes is described with a hierarchicalhidden Markov model (HMM) that incorporates the process ofmodifying note patterns. We derive an efficient algorithm basedon Gibbs sampling for simultaneously inferring from a performance signal the score and the individual language model behind
it. Evaluations showed that the proposed model outperformed
previously studied HMM-based models.

& œ œœ
(a)Score data

Learningœœ& œ œNote pattern& œ œModel for Repetition
piece 1
12 345Model forpiece 2&

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& 42 œ œ œ œ œœ œœ œ œ

Modiﬁcation (Syncopation & note insertion)
Score(b) (piece1)& 42 œ œ œ œ œ œ œœ œ œ œ œ œ
Tempo

12 345(c)

MIDI performance signal(sec)Fig. 1. Overview of the proposed model describing the generating process ofmusic score with incomplete repetitions and the process of music performance.

we need a framework for simultaneously inducing grammar
and analysing music data.
This inter-relation between grammar induction and data analysis has been gathering attention in the field of natural language processing (NLP), and Bayesian methods have been
extensively developed [16, 17]. A method for simultaneously
learning word vocabulary and segmenting sentences into words[18] is analogous to our problem (characters vs. notes andwords vs. note patterns). Recent studies have focused on the
problem of modification to repeated patterns (words) [19, 20].
Modelling modification is necessary to identify incomplete
repetitions, which are common in music [21].
In this paper, we propose a hierarchical Bayesian model for
simultaneously inferring from a performance signal the score
and the individual language model behind it (Fig. 1). (a) First a
language model based on a hierarchical HMM of note patterns
is generated, then (b) musical notes in the score are generated
by the language model, and finally (c) a performance signal
is generated based on the score. Compact grammar models
are represented as outputs of a Dirichlet process [22] witha small concentration parameter. Incomplete repetitions are
represented as repetitions with modifications and describedwith a probabilistic model for modification of note patterns.
The process of music performance is described with a tempo
fluctuation model used in studies on score-performance matching [23,24]. In this study we focus on the rhythmic aspect andconfine ourselves to monophonic music.
The main contribution of this study is its treatment of incomplete repetitions of note patterns. Bayesian grammar induction for music has been addressed in previous studies using probabilistic context-free grammar (PCFG) models using
note patterns without modification [8] or note-wise productions without note patterns [25, 26]. The problem of simultaneously inducing grammar and segmenting text data withmodification has not been addressed in NLP. We evaluatedthe model by comparing its accuracy of rhythm transcription
with that of previously studied HMM-based models.
II. R ELATED W ORKPreviously studied statistical music language models forrhythm/music transcription are reviewed in this section (Fig. 2).
The output of a language model is a sequence of notes x1:N =x1 · · · xN where N denotes the number of notes. (Similar
notations will be used throughout this paper.) Since we arefocusing on the rhythmic aspect, xn represents the note valueof the n-th note. The note value is defined as the score-written
note length relative to a whole note (a quarter note has an
x = 1/4, a dotted half note has an x = 3/4, etc.).

2&4 œ

Note-patternmodel (Sec. II-B)

Note-level model(Sec. II-A)
Metrical model(Sec. II-C)

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Fig. 2. Three different representations of a music score in previously proposedmodels [5–8].

describes notes. Each note zm is indicated by the pair (k, `),and the transition probability is given asP zm =(k 0 , `0 )|zm−1 =(k, `) = δ`L πkk0 δ`0 1 + δkk0 δ`0 (`+1) ,where δ denotes Kronecker’s delta. A similar model based on
PCFG has also been proposed [8].
An additional advantage of the note-pattern model is that it
can incorporate the logical constraints for triplet notes etc. On
the other hand, a problem of this model is the treatment ofsyncopations. Since a syncopated note lies across a bar boundary, which is typically a boundary of note patterns, syncopatednotes cannot be described with the above note-pattern model.

A. Note-Level Markov Model
Markov models of musical notes have been proposed inearly studies [6]. In the first-order model the probability of
x1:N is given as a product of transition probabilities:
NY
P (x1:N ) =P (xn |xn−1 ),(1)n=1where, with an abuse of notation, P (x1 |x0 ) ≡ P (x1 ) signifies
the initial probability. We can extend it to a p-th-order modelby replacing P (xn |xn−1 ) with P (xn |xn−1 , . . . , xn−p+1 ).
A problem with this note-level model is that certain logicalconstraints on the sequence of note values cannot be incorporated in the model. For example, triplet notes must appear in
triplets or in pairs with a double triplet note. This constraint
cannot be described with a note-level Markov model of any
order. Nor can metrical structure be incorporated in the model.
B. Note-Pattern ModelIn most music, including classical and popular music, musical notes have metrical structure, and note-pattern models
incorporating metrical structure have been proposed in previous studies [6, 8]. In the note-pattern Markov model [6], notepatterns with a fixed time span (e.g., a bar) are considered as
the state space. We notate a note pattern (a string of notes)
with Bk = zk,1 · · · zk,L (k = 1, . . . , K), where k indexes theset of K note patterns and zk,` (` = 1, . . . , L) denotes the`-th note in note pattern k. The probability of the sequence ofpatterns w1 · · · wI (wi ∈ {Bk }Kk=1 ) is given as a product oftransition probabilities πkk0 = P (wi = Bk0 |wi−1 = Bk ) andan initial probability πk0 = P (w1 = Bk ).
The sequence of notes, denoted by z1:M , is obtained byconcatenating the generated note patterns w1:I , and its probability is described with a hierarchical Markov model [27]:The upper level describes note patterns and the lower level

C. Metrical Model
Another type of model that incorporates the metrical structure is the metrical (grid) model [5,7], in which musical notes
are represented by their onset beat positions in a bar. Let sn
denote the beat position of the n-th note and let G denotethe time span of a bar. The generation of musical notes is
described with a Markov model on a grid of beat positions,and thus the probability of the sequence s1:N +1 is given as a
product of transition probabilities P (sn |sn−1 ) as in Eq. (1).
The note value of the n-th note is given as(
sn+1 − sn ,(sn+1 > sn );xn =(2)
G + sn+1 − sn , (sn+1 ≤ sn ),i.e., sn+1 is interpreted as a beat position in the next bar if it
is smaller than or equal to sn .
Although the original metrical model limits note values tothe length of a bar, it is possible to extend the model todescribe larger note values. This can be done by introducing a
discrete variable jn = 0, 1, . . . , J−1 for each note n (for somepositive integer J), which describes how may bar lines are
passed between note onset n−1 and n. In other words, the notevalue of the n-th note is now given as xn = jn+1 G+sn+1 −sn .
By taking the pair (sn , jn ) as a state variable, we can extend
the model to accommodate note values up to JG.
The metrical model is advantageous in the treatment of syncopations. In Eq. (2) the n-th note is syncopated if sn+1 6= 0
and sn+1 ≤ sn . On the other hand, its disadvantage is thedifficulty of modifying or extending the model. For example, itis necessary to construct different models for different metres.
III. P ROPOSED M ODELThe proposed model consists of two components; a language model and a performance model. Details of these mod-els and an inference algorithm are explained in this section.
A. Language Model
To describe incomplete repetitions of note patterns, we extend the model in Sec. II-B in two directions; integration ofmodification of note patterns and Bayesian extension based on
the Dirichlet process to describe compact grammar.
1) Modification of Note Patterns: The output of our language model, denoted by x1:N , is obtained by modifying thenote sequence generated by the note-pattern model in Sec. II-B,
denoted by z1:M in the following (Fig. 3). We consider note
insertions and syncopations, which are typical modificationsof note patterns in music practice. The total note values of thecorresponding notes are unchanged after these modifications.
Note insertions are represented by divided notes and thus
described with a probability of the form P (y1 · · · yQ |zm ) where
y1 , . . . , yQ are notes produced from zm by insertions, which
satisfy y1 +· · ·+yQ = zm . We use the symbol Q as the numberof notes in an insertion pattern and q(= 1, . . . , Q) as an index
of a note in that pattern. Let Ch = y1 · · · yQ (h = 1, . . . , H)
denote an insertion pattern (including the unchanged case)and φ(k`)h denote the probability P Ch |zm = (k, `) . The
sequence of notes after note insertions will be denoted byy1:N , which is specified by an applied insertion pattern hm
for each zm . This process of inserting notes can be integrated
in the basic model as yet another lower-level Markov model.
Syncopations can be regarded as simultaneous deviations of
the last note of a note pattern and the first note of the next
pattern (Fig. 3). Let x1:N represent a score with syncopations,obtained from y1:N . Syncopations can be parameterised withthe degree of syncopation s, so that notes are modified as
yn → xn = yn + s,

yn+1 → xn+1 = yn+1 − s,(3)where xn+1 must be the first note of a note pattern1 . The
parameter s can take either positive or negative values. A
positive (negative) s represents a suspension (anticipation).
Syncopations can be integrated in the model by extending thestate space of the basic model, wi , to a pair (wi , si ). Using
the notation θs = P (s), the transition probability is extended
as P (wi = Bk0 , si |wi−1 = Bk , si−1 ) = πkk0 θsi , assuming
independence of the probabilities of the component variables.
Extension of the initial probability is similar.
In summary, a score x1:N generated by the language modelis specified by stochastic variables w1:I , s1:I , and h1:M . Thismeans that each note xn is specified with a set of indices(k, `, h, q, s) and the language model can be described as aMarkov model with the following transition probability:P xn = (k 0 , `0 , h0 , q 0 , s0 )|xn−1 = (k, `, h, q, s)= δqQ φ(k0 `0 )h0 δq0 1 δ`L πkk0 θs0 δ`0 1 + δkk0 δss0 δ(`+1)`0+ δhh0 δ(q+1)q0 δkk0 δ``0 δss0 .

(4)2) Dirichlet Prior: The parameters of the language model,
πk = (πkk0 )k0 , π 0 = (πk0 )k , φ(k`) = (φ(k`)h )h , and θ =(θs )s , characterise the statistical properties of music. Because
used note patterns and types of modifications vary among
1 The variable s introduced here has no relations with sn used in Sec. II-C.

Note-patterns
w/o modiﬁcation

Generatedmusic score

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⊕ Note insertion
Note-patterns
w/ inserted notes⊕ Syncopation

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Fig. 3. Model for modification of note patterns.

musical pieces, different values of the parameters are considered for individual pieces. In the Bayesian framework, theseparameters are regarded as being generated from prior models.
The Dirichlet process can serve as a prior model that cancontrol the sparseness of the generated distributions. In thecase of finite distributions, the Dirichlet process for a discrete
distribution π is described with a base distribution ω and aconcentration parameter α as follows:π ∼ DP(α, ω) = Dir(αω),(5)

where Dir(·) denotes the Dirichlet distribution. Distributions
obtained in this way satisfy E[π] = ω, and for small α mostcomponents of π tend to be zero. We put such Dirichlet priors
for πk and π 0 :π 0 ∼ Dir(αω 0 ).

πk ∼ Dir(αωk ),(6)0

The hyperparameters ωk and ω , when learned from a database,are interpreted as a generic model (Fig. 1), or they can be set
to uniform distributions in an unsupervised learning setting.
When concentration parameter α is small, a compact grammar
is induced; i.e., a small number of note patterns will be used
for each piece. We also put Dirichlet priors for φ(k`) and θ:φ(k`) ∼ Dir(ξ),θ ∼ Dir(λ).

(7)

B. Performance ModelIn the setup for rhythm transcription, the score is observed
indirectly through a performance. The performance signal isspecified with a sequence of onset times t1:N +1 , or equivalently, a sequence of inter-onset intervals d1:N where dn =tn+1 − tn (n = 1, . . . , N ). A performance model gives theprobability P (d1:N |x1:N ) of a performance given a score.
The performance model we use is based on a linear dynamical system proposed for score-performance matching [23,24].
The model describes two sources of temporal fluctuations inmusic performance. One is the fluctuation in onset time due
to human motor noise and the other is the variation in tempos.
Tempo is considered as a latent variable vn , which represents
the ratio dn /xn up to the noise of onset time, and its variation is described with a Markov process. Assuming that thesources for tempo variation and the noise in onset time areboth Gaussian, the performance model is given asvn |vn−1 ∼ N (vn−1 , σv2 ),dn |vn , xn ∼ N (vn xn , σt2 ), (8)where σv (σt ) is the standard deviation for tempo variation(motor noise). The complete-data probability for the performance model is given asNY
P (d1:N , v1:N |x1:N ) =P (dn |vn , xn )P (vn |vn−1 ), (9)n=1where P (v1 |v0 ) ≡ P (v1 ) signifies the initial probability for
tempo. In practice, values of the tempo variable is discretised
in a range typically used for music practice to enable inference. By combining Eqs. (4) and (9), the proposed model is ahierarchical HMM [27] with the latent variable Zn ≡ (xn , vn ).

Estimated
Scaling
Substitution

C. Inference
Our goal is to simultaneously infer the latent variables Z =Z1:N and the parameters of the language model, Θ = (πk , π 0 ,
φz , θ), given the observed performance signal D = d1:Nand the hyperparameters Λ = (ωk , ω 0 , α, ξ, λ). In the modellearning step, the parameters Θ are estimated by maximising
the posterior P (Θ|D, Λ). In the transcription step, the latentvariables are estimated by maximising the posterior P (Z|D, Θ)∝ P (Z, D|Θ), which can be done by the standard Viterbi
algorithm. Since direct maximisation of P (Θ|D, Λ) is difficult,
we use a Gibbs sampling method that yields asymptoticallyexact inference. In this method, samples are drawn from thejoint distribution P (Z, Θ|D, Λ), from which samples from the
distribution P (Θ|D, Λ) can be obtained instantly.
The Gibbs sampling method is based on alternating samplings of the parameters from the probabilities P (Θ|Z, D, Λ)and the latent variables from the probabilities P (Z|Θ, D, Λ).
In the former sampling, model parameters are sampled fromposterior Dirichlet distributions. For example, the transition
probability πk is sampled as
πk |Z, Λ ∼ Dir αωk + fk (Z) ,(10)where fkk0 (Z) is the number of times that transition Bk →Bk0
appears in x1:N . Other parameters can be sampled similarly.
We can use the forward filtering-backward sampling method
to draw samples from P (Z|Θ, D, Λ). After computing theforward variables αn (Zn ) = P (Zn , d1:n |Θ) by the forward
algorithm, the latent variables are sampled iteratively as
P (Zn |Zn+1:N , D, Θ) ∝ πZn Zn+1 P (dn+1 |Zn , Zn+1 )αn (Zn )with an initial draw of ZN from P (ZN |d1:N , Θ) ∝ αN (ZN ).
Note that the latent variables x1:N and v1:N are highly correlated in P (Z|D, Λ) and should be sampled jointly.
Because the number of states of the language models canbe O(104 ) and the product state space with the tempo variable
has even more states, the computational cost of the forward
algorithm can be impractical. This can be understood fromthe quadratic time complexity of the forward algorithm: With
Ns states the time complexity of the forward algorithm isO(N Ns2 ). To solve this problem, we can use particle filtering
for the approximate calculation of the forward variables. WithNp particles the time complexity is reduced to O(N Ns Np ).
IV. E VALUATION
A. Setup
We evaluated the proposed model by comparing its accuracyof rhythm transcription with that of previously studied models
based on HMMs. A database of MIDI performances of 30
Japanese popular songs by various artists was prepared by theauthors (the durations of the pieces ranged from about 15 sec

True
Fig. 4. Example of scaling operations and substitutions to recover the correcttranscription from an estimated result.

to 50 sec). All normal, dotted, and triplet note values ranging
from the whole note to the 16th note were used as candidate
note values. For the proposed model, all note patterns of halfnote length consisting of these candidate note values and thefollowing note patterns of whole-note lengths were used: (1),
(3/4, 1/4), and (1/4, 3/4). All possible pairs of those notevalues were used as note insertion patterns. The degree ofsyncopation was also taken from the candidate note values
and their negative values and zero.
We tested the model in three different learning conditions,supervised, semi-supervised, and unsupervised, and also in
two cases of with or without modifications to note patterns(total of six cases). The supervised learning condition without
modifications is equivalent to the original note pattern model in[6], for which (πk )k and π 0 were trained with the melodies of
100 songs in the RWC popular music database [28]. For semisupervised learning (ωk )k , ω 0 , and λ were trained with thesame dataset. For unsupervised learning, all ωk and ω 0 were
set to uniform distributions and λs = 10 if s = 0 and 0.05 otherwise. Other hyperparameters were set as α = 700, ξh = 0.1
for the non-insertion case and 0.01 otherwise, σt = 0.02 s,
and σv = 0.06 seconds per quarter note [24].
For comparison, we implemented the note-level HMM using
note bigrams [6] and the metrical HMM [5]. These models
were also trained with the RWC database, and we set J = 2
for the metrical model. The performance model was the sameas that for the proposed model.
We used as an evaluation measure the rhythm correction
ratio, i.e., the ratio of the smallest number of edit operations
needed to correct the estimated result to the number of notesin the data. In addition to note-wise correction (substitution),
the scaling operation applied for a subsequence of note values
was included (Fig. 4). This is because there is arbitrarinessin choosing the unit of note values: For example, a quarter
note played in a tempo of 60 BPM has the same duration
as a half note played in a tempo of 120 BPM. Although thedetails must be omitted for the lack of space, the smallest
number of necessary edit operations Ne can be calculated bya dynamic programming similar to that used in computationof the Levenshtein distance. The rhythm correction ratio R isthen given as R = Ne /N .
B. Results
Results are shown in TABLE I. For the proposed model,
incorporation of the modification model of note patterns im-

TABLE I
AVERAGE RHYTHM CORRECTION RATE R WITH STANDARD ERROR .
L OWER IS BETTER .

MIDI performance (input)

Model
Learning
Modification
R [%]1 Proposed
UnsupervisedX
12.8 ± 1.3
2
Unsupervised
16.5 ± 1.8
3
Semi-supervisedX
6.6 ± 1.0
4
Semi-supervised
17.7 ± 2.2
5SupervisedX
7.8 ± 1.2
6
Supervised
14.7 ± 1.8
7 Note-level HMM [6]Supervised
7.9 ± 1.4
8 Metrical HMM [5]Supervised
7.3 ± 1.3

&cproved the average rhythm correction rate in all learning conditions. By comparing the results for supervised learning andsemi-supervised learning with the modification model, we seethat the grammar induction for individual pieces indeed works
effectively. In the best case, i.e., the semi-supervised learning
condition with the modification model, the result outperformed
the previously studied models. With the modification model,
the unsupervised case was 5 points lower than the supervisedcase. An example result (Fig. 5) shows that the proposedmodel succeeded to capture the incomplete repetitive structurewith syncopations and note insertions.
There is still room for improving the performance of the
proposed model. One direction is removing the arbitrarinessin choosing the fixed length of note patterns by introducing
variable-length note-pattern model, which can be done with theuse of hierarchical Dirichlet process. Using pitch informationand modelling hierarchical repetitions would also be effective.
V. C ONCLUSION
We have developed a framework for simultaneously inferring the score and the individual language model behind it
from a performance signal. The proposed model has succeeded
to learn compact grammar and segment a piece into representative note patterns. We plan to extend the model to an infinite
vocabulary model, so that the representative note patterns ofvariable length can be automatically inferred. Another direction is an extension for polyphonic music, for which introducing voice structure is a key issue. The presented formulationof simultaneous grammar induction and data analysis would
be effective for realistic music transcription from audio signals
and also for NLP.
ACKNOWLEDGEMENTS
The authors thank Daichi Mochihashi and Ryo Nishikimi
for useful discussions. This work was partly supported by JST
OngaCREST Project, JSPS KAKENHI 24220006, 26700020,
26280089, 15K16054, 16H01744 and 16J05486, and Kayamori
Foundation. EN is supported by the JSPS research fellowship
program (PD).
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&

(sec)

Correct scoreœ œ œ œ œ œ œ œ œ œ œ #œ œ œ . œ œ œ œ œ œ œ œ œ œ œ œ œ œ . œ œ œ . œ œ œ . œ œJEstimated score (output)&œ œ œ œ œ œ œ œ œ œ œ #œ œ œ . œ œ œ œ œ œ œ œ œ œ œ œ œ œ . œ œ œ . œ œ œ . œ œJEstimated note patterns without modiﬁcation&œ œ œ œ œ œ œ œ œ œ œ œ. œ œ œ œ œ œ œ œ œ œ œ œ œ. j œ. j œ. œ
#œ
œ
œ
J
J

Fig. 5. Example result of rhythm transcription by the proposed model (semisupervised with modifications modelled). The third low shows the estimatedresult and the fourth low shows the result after removing modifications.
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