TREE-STRUCTURED PROBABILISTIC MODEL OF MONOPHONIC WRITTEN MUSIC
BASED ON THE GENERATIVE THEORY OF TONAL MUSIC
Eita Nakamura1 , Masatoshi Hamanaka2 , Keiji Hirata3 , Kazuyoshi Yoshii1

ABSTRACTThis paper presents a probabilistic formulation of music language
modelling based on the generative theory of tonal music (GTTM)
named probabilistic GTTM (PGTTM). GTTM is a well-known music theory that describes the tree structure of written music in analogy with the phrase structure grammar of natural language. To develop a computational music language model incorporating GTTM
and a machine-learning framework for data-driven music grammar
induction, we construct a generative model of monophonic music
based on probabilistic context-free grammar, in which the time-spantree proposed in GTTM corresponds to the parse tree. Applying
the techniques of natural language processing, we also derive supervised and unsupervised learning algorithms based on the maximallikelihood estimation, and a Bayesian inference algorithm based on
the Gibbs sampling. Despite the conceptual simplicity of the model,
we found that the model automatically acquires music grammar fromdata and reproduces time-span trees of written music as accurately
as an analyser that required elaborate manual parameter tuning.
Index Terms— Statistical music language model, GTTM,
PCFG, time-span tree analysis, statistical grammar induction
1. INTRODUCTION
Music transcription is a challenging topic in music processing. For
this, acoustic modeling of musical sounds has been widely studied [1]. However, this is not sufficient to remove all ambiguitiesof transcription due to acoustic variations, and using prior knowledge on the output musical score is in order. The same situation
happens in speech recognition, where the knowledge on linguistics
is incorporated in the language model. The purpose of this study is
to construct a model that can capture the grammar of written music.
Musical pieces have a hierarchical structure. Notes are grouped
into motives, which are grouped into phrases, which are grouped intomusical sentences (or passages), which are grouped into sections,etc. In addition, some notes are more salient than adjacent notes.
This structure of music resembles the phrase structure of natural language [2], which can be represented as a syntax tree. The generative
theory of tonal music (GTTM) [3] attempts to model such a hierarchical structure of music. GTTM proposes to use a tree representation called the time-span tree to describe the relative importance ofnotes. The time-span tree can be interpreted as a simplification or
reduction process of a musical passage (Fig. 1). GTTM also proposes rules to derive a reasonable time-span tree for a given musicalpassage, which combine musical knowledge such as counterpoint
theory [4], harmonic theory [5], and Schenkerian analysis [6].
This work is partially supported by JSPS KAKENHI Nos. 24220006,
26700020, 26280089, and 15K16054, and JST CREST.

› levelTime-span tree

w level˙ level#& # c œ œ œ œjœ. œ ˙#& # c w

w

w

w

&
&
&

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c

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c

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Generation (this study)

Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan
2
Graduate School of Medicine, Kyoto University, Kyoto 606-8507, Japan
3
Future University Hakodate, Hakodate 041-8655, Japan

Reduction (GTTM)

1˙œ . œj ˙‘Left-type’ and ‘right-type’ production rule
S

VP

NP
DT

Parse tree

N

V

DT

1

& wG

Principal

NP
N

The man saw the dog

& ˙.G

3/4

& wG

Subordinate Subordinate

œE

1/4

& ˙E

1/2

1

Principal

˙G

1/2

Fig. 1. Example of time-span tree and the reduction and generationof a musical passage (top). The two types of production rules in theprobabilistic GTTM (bottom).
The computational formulation of GTTM has been studied in
the last decades [7–10] because the prescribed rules of GTTM are
not quantitatively or computationally formalised. Based on parameterisation of the rules in GTTM, an automatic time-span tree analyser(ATTA) was implemented [7]. The adjustment of the 46 parameters
remained as a problem. Recently a probabilistic method for timespan tree analysis called σGTTM III is proposed, which can learnsome parameters from labelled music data [10]. There are other notable studies on automatic music analysers [11, 12]. Although thesestudies provide grouping structure analysis, metrical analysis, andreduction analysis, they are not a generative model and cannot bedirectly applied to music transcription [1] or automatic music generation and arrangement [13]. Moreover, these studies have not taken
advantage of the developed machine-learning techniques in natural
language processing (NLP) [14–17].
To solve these problems, we formulate a probabilistic generative model of music based on GTTM named the probabilistic GTTM
(PGTTM). We interpret that the time-span tree represents a production process of musical notes and formulate PGTTM with probabilistic context-free grammar (PCFG) [18, 19]. In contrast to the phrase
structure grammar of linguistics where an intermediate node corresponds to grammatical categories such as ‘NP’ and ‘VP’, a node of
the time-span tree represents a musical note (Fig. 1). In addition,
two types of production rules must be distinguished to indicate therelative importance of children notes, and there are constraints on
the pitches and note values (lengths) to consistently describe the reduction/generation process.
There are other works that applied PCFG models for music. The
idea of using PCFG for the probabilistic formulation of GTTM isalready explored in σGTTM III [10], which is not fully formulated

as a generative model of musical notes and requires other analysing
tools [7, 8] to obtain the time-span trees. A two-dimensional treestructure model was proposed [20] and applied for multi-pitch analysis. Our study shares the same interests in unsupervised learning ofmusic language model. A PCFG model for harmonic analysis was
proposed in [21]. The main contribution of this study is to provide
an explicit construction of a statistical music language model that
integrates GTTM and a machine-learning framework for inducing
music grammar from data. In this study, we confine ourselves tomonophonic music.
We evaluate the proposed PGTTM in terms of the accuracy oftime-span tree analysis. We examine the performance of the modelwith both supervised and unsupervised learning, and compare the results with previously proposed analysers. We found that the PGTTM
performs as accurately as ATTA. This is encouraging as PGTTM isconceptually simple and does not require elaborate parameter tuning. Error analyses show that whereas local tree structures were relatively well estimated with the PGTTM, the higher-level structures
were poorly estimated. The result suggests the importance of introducing latent grammatical categories into the PGTTM. Ideas forfurther refinements are given at the end.
2. PROBABILISTIC GTTM (PGTTM)
This section explains the proposed model.
2.1. Basic model
Each musical note is represented as a pair (p, r) of pitch p and notevalue r. We can use either spelled pitches or integral pitches, andthe set of possible pitches is denoted by Ωp . A rest is represented
by a symbol ‘R’, which we also include in Ωp . We define the notevalue as the score-written length of the note relative to a whole note(a quarter note has a r = 1/4, a dotted half note has a r = 3/4,
etc.). The set of possible note values is denoted by Ωr . Then amonophonic passage can be represented as a sequence (pn , rn )N
n=1 ,where N is the number of notes and rests.
A PCFG model [18] is represented with the set of terminals ΩT ,the set of non-terminals ΩN which contains the start symbol S, andthe set of production rules R. In the Chomsky normal form, a production rule A → α (A ∈ ΩN , α ∈ ΩN ×ΩN ∪ ΩT ) is associatedwith a probability value P (A → α). The symbol A and the symbols
in α in a production rule will be called the parent and the children,
respectively. Given a sequence of non-terminals w = w1 · · · wN(wn ∈ ΩT ), a set of production rules to derive w from S is called aderivation of w, which can be represented as a parse tree (Fig. 1).
In the probabilistic formulation of GTTM, we need the following modifications and extensions of this ordinary PCFG. First, theparent and the children of a production rule should be represented
as musical notes. An exception is the most top production of notesfrom the start symbol, which will be explained in the next paragraph.
Second, to describe the structure of the principal note vs. the subordinate note, we distinguish two types of production rules; one has theprincipal note on the left and the other has it on the right. Then a production rule can be written in the form (p, r) → s(pL , rL )(pR , rR )where p, pL , pR ∈ Ωp , r, rL , rR ∈ Ωr , and s = L, R indicates that
(ps , rs ) is the principal note. Finally, since the time span must be
conserved in producing a note, we must have r = rL + rR , and thus
we can write rR = r − rL . The fact that the principal note has thesame pitch as the parent note requires p = ps (Fig. 1).
Let us now define the PGTTM. The production process begins
with the start symbol S with a note value rS , which corresponds tothe total note value of a musical passage. The production rules areeither of the form (S, rS ) → s(pL , rL )(pR , rS −rL ) or (p, r) →s(pL , rL )(pR , r−rL ) where p, pL , pR ∈ Ωp , r, rL ∈ Ωr , ands = L, R. The associated probabilities are written as P (S, rS ) →s(pL , rL )(pR , rS −rL ) and P (p, r) → s(pL , rL )(pR , r−rL ) ,which satisfy the following normalisation conditions:X
P (S, rS ) → s(pL , rL )(pR , rS −rL ) = 1, (1)s,pL ,pR ,rLX
P (p, r) → s(pL , rL )(pR , r−rL ) = 1.

(2)s,pL ,pR ,rL
There is a constraint that P (p, r) → s(pL , rL )(pR , r−rL ) = 0
unless p = ps as explained above, and we also assume that rests
must not be a principal note so that the probability is zero unlessp ∈ Ωp \{R}. A set of production rules that yields a given musicalpassage is called a time-span tree.
The production probabilities strongly depend on the key of themusical passage, and this dependence can be theoretically describedby first considering a different PGTTM for each of 24 keys (major
keys and minor keys on 12 pitch classes) and then constructing their
mixture model. In the following we consider a situation that the key
is known in advance, so all passages can be transposed to C major or
C minor.
2.2. Simplifications and refinements
Simplifications of the PGTTM described in Section 2.1 are necessary
if we are to derive computationally tractable inference algorithms.
First, we assume that the production rule probabilities are independent for pitch and note values. This means that the production ruleprobability is factorised as follows:P (p, r) → s(pL , rL )(pR , r−rL )= P s (s)P p (p → spL pR )P r (r → srL (r−rL ))
(3)
P
s
p
P
(s)
=
1,
P
(p
→
sp
p
)
=
1,
and
where
L R
pL ,pRsP
rrL P (r → srL (r−rL )) = 1, and similarly for the production rules from S. Second, we represent the pitch with an
integral pitch class, and thus Ωp = {C,C#,· · · ,B,R} (where
C#=D[ etc.). We will use the following notations: φs = P s (s),θsppL pR = P p (p → spL pR ), ρsrrL = P r (r → srL (r−rL )),and Θ = (φs , θsppL pR , ρsrrL )s,p,pL ,pR ,r,rL . Finally, we assume
that the probabilities for note values are scale invariant so thatP r (r → srL (r−rL )) is a function of rL /r and not separately dependent of r and rL . The probability of a time-span tree T is givenasYc((p,r)→s(pL ,pR ,rL );T )P (T |Θ) =φs θsppL pR ρsrrL
P

s,p,pL ,pR ,r,rL

where c((p, r) → s(pL , pR , rL ); T ) is the number of times that the
production rule (p, r) → s(pL , rL )(pR , r−rL ) appears in T .
It has been noted that the choice of the principal note is influenced by the magnitude relation of metrical weights of the relevant
note pair (Rule TSRPR 1 in [3]). Here, the metrical weight of a note
indicates the relative strength of the metrical (beat) position of its
onset [22]. To incorporate this preference rule, we allow that theprobability P s depends on the relative metrical weights of the relevant note pair. If we represent the metrical weights of note (pL , rL )and (pR , rR ) as ωL and ωR , P s take different values depending onwhether ωL /ωR is less than, equal to, or greater than unity.
2.3. Bayesian formulation of the PGTTM
As we explain in Sec. 3, we can derive an unsupervised learningalgorithm for PGTTM based on the maximal-likelihood principle.
An alternative way of unsupervised learning is Bayesian inference[14]. With the Bayesian formulation, we can develop a Monte-Carlomethod for inference that can learn parameters without getting stuck
in local optima.
For Bayesian inference, we introduce a prior distribution for the
probabilistic parameters of the PGTTM. Since the production-ruleprobabilities obey categorical distributions, Dirichlet distributions
are their natural priors. Let η = (ηs )s , λsp = (λspL pR )pL ,pR , andνsr = (νsrrL )rL denote the Dirichlet parameters for φ = (φs )s ,
θsp = (θsppL pR )pL ,pR , and ρsr = (ρsrrL )rL , respectively. Thenthe prior distributions are given as PDir (φ|η) and similarly for θsp
and ρsr , where PDir denotes the Dirichlet distribution.
3. INFERENCE ALGORITHMS FOR PGTTM
Basic inference algorithms for PCFG have been well-developed.
With some modifications, we can derive inference algorithms for the
PGTTM. This section summarises the results.
3.1. CYK-Viterbi, inside, and outside algorithms
Given a musical passage W = (pn , rn )N
n=1 , the most probable timespan tree can be obtained with the CYK-Viterbi algorithm, which isa dynamic programming. We will use the notation Wn = (pn , rn ),Wnm = Wn · · · Wm , and rnm = rn + · · · + rm . We introduce avariable γnmp = maxT P (T ) for 1 ≤ n ≤ m ≤ N and p ∈Ωp where T denotes a time-span tree with parent (p, rnm ) and yieldTWnm (we express this as (p, rnm ) =⇒ Wnm ). This can be calculated
recursively as
n
n+` 
)
P (p, rnn+` ) → s(pL , rnn+k−1 )(pR , rn+kγn(n+`)p = max
LRk,s,p ,po(4)· γn(n+k−1)pL γ(n+k)(n+`)pR
for all p ∈ Ωp with the initial values γnnp = δp,Wn (δ denotes Kronecker’s delta). Then the probability of the most probable time-spantree is given as γ1N S . In the calculation of Eq. (4), the argumentsthat yield the maximal probability are also memorised as k̂n(n+`)p ,ŝn(n+`)p , etc. Once all γnmp have been computed, we can obtainthe most probable time-span tree by expanding the tree according tothese arguments starting from (S, r1N ), as in the back-tracking procedure of the Viterbi algorithm for hidden Markov model.
To learn parameters of the PGTTM, we calculate the inside variables βnmp (W ) and outside variables αnmp (W ) defined asXβnmp (W ) =P (T ),(5)Tm )=T :(p,rn⇒Wnm

X

αnmp (W ) =
T

N )=
N
T :(S,r1⇒W1n−1 (p,rnm )Wm+1

P (T ).

(6)

3.2. EM algorithm for maximal-likelihood estimation
Given a set of (unlabelled) data W , the probability parameters
can be estimated by the maximal-likelihood principle: Θ̂ =argmaxΘ P (W |Θ). The expectation-maximisation (EM) algorithm
can be derived by the following iterative minimisation [23]:Θnew = argmax Q(Θ, Θold )ΘX= argmax
P (T |W, Θold )ln P (T, W |Θ).
Θ

(8)

T

By differentiating Q(Θ, Θold ) with respect to φs , θsppL pR , andρsrrL , the following updating equations are derived:
X

1
φnew
= φ
C (p, r) → s(pL , pR , rL ); Θold ,s
G p,p ,p ,r,r
L

R

L


1 X
new
θspp
= θ
C (p, r) → s(pL , pR , rL ); Θold ,L pR
Gsp r,rLρnewsrrL =X1
C (p, r) → s(pL , pR , rL ); ΘoldGρsr p,p ,pLR

where G’s are normalisation constants and we have definedC (p, r) → s(pL , pR , rL ); Θold
X P (T |Θold )c((p, r) → s(pL , pR , rL ); T )=.
P (W |Θold )
T ∈T

(9)

W

(TW denotes the set of all possible time-span trees of W .) The quantity in Eq. (9) is given as
old
old
φold
s θsppL pR ρsrrL

N
N
X
X

m−1
X

m ,r−rδrnm ,r δrnk ,rL δrk+1L

n=1 m=n+1 k=n
old
old
old
· αnmp(W )βnkp(W )β(k+1)mp(W ) (10)LR

where β old (αold ) is the inside (outside) variable calculated withΘold . The case for production rules with parent (S, rS ) is similar.
3.3. Bayesian inference algorithm using Gibbs sampling
Given data W and hyperparameters Λ = (η, λsp , νsr ), the goalof the Bayesian learning is to estimate the posterior distribution of
the model parameters P (Θ|W, Λ), which generally cannot be solved
analytically. Fortunately we can develop a Monte-Carlo method todraw samples from P (T, Θ|W, Λ) (T is a time-span tree) via the
Gibbs sampling [14].
The Gibbs sampling method is based on alternating samplingsof the probabilities P (Θ|T, W, Λ) and P (T |Θ, W, Λ). The former
probability can be written as a product of Dirichlet distributions as
P (φ|T, W, Λ) = PDir (φ|η + f (T )) whereXfs (T ) =c (p, r) → s(pL , pR , rL ); T
(11)

If the dependency on W is self-evident, we simplify the notation as
βnmp and αnmp . These quantities can also be computed by dynamic
programming algorithms called the inside and outside algorithms.
For PGTTM, the inside algorithm has an update equation same as
Eq. (4) except
P with the maximisation maxk,s,pL ,pR replaced byp,pL ,pR ,r,rL
a summation k,s,pL ,pR . The outside algorithm is based on thefollowing recursive equation:and similarly for λsp and ηsr . Therefore Θ can be sampled fromXkαnmp =
P (q, rnk ) → s(p, rnm )(pR , rm+1
) αnkq β(m+1)kpR P (Θ|T, W, Λ) by sampling from the Dirichlet distributions.
Next, given a set of sampled parameters Θ, a time-span tree T
s,q,pR ,kX
can be sampled from P (T |Θ, W, Λ) = P (T |Θ, W ) by the folm
n−1
m 
+
P (q, rk ) → s(pL , rk )(p, rn ) αkmq βk(n−1)pL (7)lowing recursive sampling for each node. Each node is indicateds,q,pL ,k
as a triplet (n, m, p) (1 ≤ n ≤ m ≤ N , p ∈ Ωp ) meaning
with initial values α1N S = 1 and α1N p = 0 (p 6= S). In practicethat the symbol p spans the time span rnm . Starting from the top
the left-hand side of Eq. (7) can be computed more efficiently bynode (1, N, S), if there is a node (n, m, p) with n<m, it is exapplying the constraints explained below Eq. (2).
panded by sampling s, k, pL , pR from the distribution P (p, rnm ) →Table 1. Accuracies of time-span tree analyses.
Learning condition
Supervised (open)
Supervised (closed)
Unsupervised (EM)Unsupervised (Gibbs)
ATTA
σGTTM III

Accuracy (%)
44.1
44.9
32.3
31.4
44
76

Incorrect branches
Estimated& 42# nodes
4237
2548
1578
998
606
358
253

Accuracy (%)
60.8
43.6
35.0
22.8
13.2
3.9
3.6

Children matched (%)
80.4
52.7
45.8
35.6
21.9
8.9
9.1

& 42 œ œ œ œ œ œ œ

œœœœ

œœœ

œœœœ

œœœ

œœ

# œ œ œ j‰
œ

œœœœ

œœœ

œœœœ œœœœœ# œ œ œ j‰
œFig. 2. Example result of time-span tree analysis; an estimated timespan tree (top) and the correct time-span tree (bottom). Incorrectly
estimated branches are indicated with red broken lines.

s(pL , rnk−1 )(pR , rkm )|Θ, W , which is given asβn(k−1)pL βkmpRφs θsppL pR ρs(rm )(rk−1 )nnβnmpwhere β’s are the inside variables calculated with Θ.

œœœManually annotated

Table 2. Results of error analysis.
Height
1
2
3
4
5
6
7≥

œœœœ

(12)

4. EVALUATION
4.1. Evaluation setup
To test PGTTM, we implemented a time-span tree analyser basedon the CYK-Viterbi algorithm and evaluated the analyser with a
database of manually labelled time-span trees. The database consisted of 300 musical passages with time-span trees analysed by amusic expert [9].
To test its ability to learn the parameters, we evaluated the
PGTTM in different learning conditions. The first two conditions
were supervised learning, one the open condition (piece-wise crossvalidation) in which the training data did not include the test data
and the other the closed condition in which the test data was also
used for learning. To avoid zero frequencies, we uniformly added
a 0.1 count for all frequencies. The next conditions were unsupervised learning, one based on the EM algorithm (Sec. 3.2) andthe other based on the Gibbs sampling method (Sec. 3.3). For the
EM algorithm, the initial parameter values were randomly chosen.
For the Gibbs sampling method, all hyperparameters were set as
0.1 and after sampling, the parameter set that yielded the maximal
likelihood was chosen and applied EM iterations before it was used
for the time-span tree analysis.
An estimated time-span tree was compared with the reference inthe database and the accuracy was calculated in the following way.
First a node of an estimated time-span tree was defined as matched
if it had a corresponding node with the same parent and the samechildren in the reference tree. Then the accuracy was defined as theratio of the number of matched nodes to the total number of nodes.
4.2. Evaluation results and error analysis
The results in Table 1 show that the accuracies for the PGTTM were
nearly equal to that for ATTA. For reference, the accuracy is also
shown for σGTTM III, which uses the true group boundary of notesin the annotated data and thus cannot be fairly compared to the otheralgorithms. The fact that the open and closed tests for supervised
learning had nearly equal accuracies indicates that there was little
data sparseness and the accuracy would not improve much with alarger data size. The EM algorithm and the Gibbs sampling had
similar accuracies that are lower than the case of supervised learning.

A detailed analysis of accuracies are listed in Table 2 where
nodes are classified according to the height, which is defined as the
maximal distance from descendant leaves, and the individual accuracies are shown. Because we had similar results, we indicate only thecase for the open supervised learning. In the table, the proportion ofnodes that has a corresponding node with the same children but notnecessarily with the same parent in the reference tree is also shown.
The discrepancy between this value and the accuracy indicates the
proportion of nodes whose parent was incorrectly estimated. Thereis a clear tendency that the nodes with lower height have higher accuracies. This can also be confirmed in an example of estimatedtime-span trees in Fig. 2. The example illustrates the typical casewhere the cadential note is not correctly selected as the most important note of the passage. Such misidentification of the most important note causes cascading errors in the estimated time-span tree.
4.3. Discussion
It is encouraging that the PGTTM worked as accurately as ATTA
without elaborate manual tuning of parameters. However the results
also suggest refinements of the model are necessary to improve theaccuracy. First, as is the case with NLP, the production rule probabilities depend rather strongly on the context, and it would be important
to incorporate dependence between several notes in successions into
the model [16]. Another important issue is the chose of symbols
in the grammar. Since we have not essentially used non-terminals,
the production rules of PGTTM are the same for all positions andheights in the time-span tree. The fact that notes with higher metrical weight are often more important in the lower-height nodes but
the clauses are often on weak beats suggests that we need to extend
the model with latent symbols, which would improve the time-spantree analysis for nodes with higher heights. Symbol refinement used
for NLP [19, 24] can be an aid for this extension.
5. CONCLUSION
Based on GTTM, we have constructed a probabilistic tree structure
model of written music. We formulated the PGTTM based on an
extension of PCFG, for which both supervised and unsupervised
learning techniques can be applied. Despite the conceptually simple
construction of the model, the PGTTM produced musical syntactic
parsing as accurately as the previously proposed method with elaborate manual tuning of parameters. The results suggest further refinements of the grammatical model. For future work, we plan to apply
the model for music transcription and automatic music arrangement
and extend the model for polyphony.

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