Proc. of the 11th International Symposium on CMMR, Plymouth, UK, June 16-19, 2015

σGTTM III: Learning based Time-span Tree
Generator based on PCFG
Masatoshi Hamanaka1 , Keiji Hirata2 , and Satoshi Tojo3
1
Kyoto University,
hamanaka@kuhp.kyoto-u.ac.jp,
2
Future University Hakodate,hirata@fun.ac.jp,
3
JAIST,
tojo@jaist.ac.jp,
http://gttm.jp/

Abstract. We propose an automatic analyzer for acquiring a time-spantree based on the generative theory of tonal music (GTTM). Althoughanalyzer based on GTTM was previously proposed, it requires manually tweaking the 46 adjustable parameters on a computer screen in order to analyze them properly. We reformalized the time-span reduction
in GTTM based on a statistical model called probabilistic context-freegrammar, which enables us to acquire the most probabilistic time-span
tree. We applied leave-one-out cross validation using three hundred sets
training data, which revealed that our analyzer outperformed the previous one.
Keywords: A generative theory of tonal music (GTTM), probabilistic
context-free grammar (PCFG), time-span tree.

1

Introduction

This paper describes a method for automatic generation of a time-span tree
based on the generative theory of tonal music (GTTM) [1]. The main advantage
of our method is that it is based on probabilistic context-free grammar (PCFG)[2], and we can therefore acquire a model that enables us to generate a time-spantree by statistically learning training data that has been analyzed manually by
a musicologist.
Generally, a piece of music will have more than one interpretation, and dealing with such ambiguity is a major obstacle when implementing a music theory
on a computer. A statistical model is suitable for constructing a music analyzer that allows such ambiguity. In other words, when we introduce a statisticalmodel in a musical analyzer, we can compare the likelihood of one interpretation to other interpretations. One such probabilistic model we introduce here is
PCFG, which is used for syntactic analysis of natural language.
PCFG consists of multiple rules, and each rule has a probability. The probability of a sentence is derived from multiplications of probabilities of applied303

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Masatoshi Hamanaka, Keiji Hirata, and Satoshi Tojo

rules for generat-ing the sentence. Finding a rule set that can generate the sentence is called ”parsing,” and a tree structure that indicates the parse result iscalled a ”parse tree.”
The time-span tree is a result of analyzing GTTM, which is a binary tree ineach leaf connecting a note lined in time order. Each middle node between roots
and leaves has labels that indicate which note is salient between two notes thatconnect directly to the node. Therefore, a time-span tree can acquire a reduction
melody by omitting non-salient notes.
In this study, we regard the time-span tree as a parse tree of a melody.
We can generate a melody by using PCFG. Then, the reduction process is aninverse problem of the generation. We ﬁrst set 645 PCFG rules and learned
the generation probabilities of the rules by using training data that had been
analyzed manually by a musicologist. Then we calculated the total probability ofeach parse tree of all the well-formed time-span trees and selected the maximum
one as the most appropriate time-span tree to solve the inverse problem.
We applied leave-one-out cross validation using 300 sets of training data. The
results indicated an average accuracy of 0.76, which outperformed the previous
GTTM analyzer.

2

Related work

We brieﬂy look back at the history of cognitive music theory. The implicationrealization model (I-R model) proposed by Eugene Narmour abstracts and expresses music according to symbol sequences from information from a musicalscore [3, 4]. Recently, the IRM has been implemented on computer and can beused to acquire the chain structures of I-R model from a score [5]. In contrast,
Schenkerian analysis is used to analyze the deeper structures called ʠ Urlinie ʡ(fundamental line) and ʠ Ursatz ʡ (fundamental structure) from the music surface [6]. Short segments of music can be analyzed using Schenkerian analysis
on a computer [7]. Another approach constructs a music theory for computer
implementation [8, 9].
The main advantage of analysis by GTTM is that it can acquire the tree
structures called time-span and prolongation trees. The time-span or prolongation tree provides a summarization of a piece of music, which can be used as therepresentation of an abstraction, resulting in a music retrieval system [10]. It canalso be used for performance rendering [11] and reproducing music [12]. Additionally, the time-span tree can be used for melody prediction [13] and melody
morphing [14].
Some other studies have applied PCFG to analyze music, for example, chord
analysis for jazz [15], analysis of the metrical structure [16], and automatic transcription [17]. These studies [Granroth 2012,Tanji 2008,Kameoka 2012] show theusefulness of using PCFG in musical analysis. Actually, the GTTM [1] has aconcept of generation that is also expressed in the title. However, no one has
yet regarded a time-span tree as a parse tree or estimated the most probabilistic
time-span tree.

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σGTTM III: Learning based Time-span Tree Generator based on PCFG

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We constructed four types of GTTM analyzers: ATTA, FATTA, σGTTM,
and σGTTMII. We extended the original theory of GTTM with a full externalization and parameterization and proposed a machine-executable extensionof the GTTM called exGTTM [18]. The externalization includes introducing analgorithm to generate a hierarchical structure of the time-span tree in a mixed
top-down and bottom-up manner, and the parameterization includes introducing a parameter for controlling the priorities of rules in order to avoid conﬂict
among the rules, as well as parameters for controlling the shape of the hierarchical time-span tree. We implemented exGTTM on a computer called the ATTA
(automatic time-span tree analyzer), which can output multiple analysis resultsby conﬁguring the parameters.
Although the ATTA has adjustable parameters for controlling the weight orpriority of each rule, these parameters have to be set manually. This takes a longtime because ﬁnding the optimal values of the settings themselves takes a longtime. We discuss the problem of ATTA in detail in 3.3.
The FATTA (full-automatic time-span tree analyzer) can automatically estimate the optimal parameters by introducing a feedback loop from higher-levelstructures to lower-level structures on the basis of the stability of the time-spantree [19]. The FATTA can output only one analysis result without manual conﬁguration. However, the FATTA performance is not good enough for analyzingtime-span trees.
We also developed σGTTM, a system that can detect the local groupingboundaries in GTTM analysis, by combining GTTM with statistical learning[20]. The σGTTM system statistically learns the priority of GTTM rules from100 sets of score and grouping structure data analyzed by a musicologist; it does
this by using a decision tree. Its performance, however, is insuﬃcient because
it can construct only one decision tree from 100 data sets and cannot output
multiple results.
The σGTTM II system assumes that a piece of music has multiple interpretations, and thus, it constructs multiple decision trees (each corresponding toan interpretation) by iteratively clustering the training data and training thedecision trees. The performance of the σGTTM II system outperformed both
the ATTA and σGTTM systems [21]. However, σGTTM and σGTTM II are
only suitable for grouping structures and cannot acquire time-span trees.

3Time-span reduction and its implementation problem

We use the grouping and metrical structures of music to derive a time-span
tree. The grouping structure is intended to formalize the intuitive belief that
tonal music is organized into groups that are in turn composed of subgroups.
These groups are graphically presented as several levels of arcs below a music
staﬀ. The metrical structure describes the rhythmic hierarchy of the piece byidentifying the position of strong beats at diﬀerent levels such as those of aquarter note, half note, a measure, two measures, and four measures. Strong
beats are illustrated as several levels of ”dots” below the musical staﬀ. The

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Masatoshi Hamanaka, Keiji Hirata, and Satoshi Tojo

time-span tree is a binary tree, which is a hierarchical structure describing therelative structural importance of notes that diﬀerentiate the essential parts ofthe melody from the ornamentation (Fig. 1). For example, the left side of Fig. 2
depicts a simple melody and its tree. The time-span (designated as ¡—¿) isrepresented by a single note, called a head, which is designated here as ”C4.”
In the tree, the essential notes are connected to a branch nearer to the root of
the tree. In contrast, the ornamentation notes are connected to the leaves of
the tree. In a separation, we hereafter call the branch ”primary” and the leaf
”secondary” (Fig. 3).

Time-span Tree

Metrical StructureGrouping Structure

Fig. 1. Time-span tree, metrical structure, and grouping structure.

C4

head

C4

                             



<---->

Instantiating

Abstracting

Fig. 2. Subsumption relation of melodies.

Parent me-spanPrimary (salient) branchSecondary (nonsalient) branchSecondaryme-spanPrimaryme-spanFig. 3. Primary and secondary time-span trees.

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σGTTM III: Learning based Time-span Tree Generator based on PCFG

3.1

5

Time-span segmentation

Before the time-span reduction, the time-span segmentation divides the entire
piece into hierarchical time-spans. We show the division procedure in Fig. 4,
which involves the following steps:
1. Regard all of the resultant groups of grouping analysis as time-spans.
2. Divide a time-span into two at the strongest beat when a time-span in thelowest level includes more than one note.
3. Repeat 2 recursively.
In [1], there are two rules for time-span segmentation, which are called segmentation rule 1 and segmentation rule 2. The former corresponds to the ﬁrst
item and the latter to the second item.

Fig. 4. Time-span segmentation.

3.2

Time-span reduction

As a result, a music piece is formed into a binary tree, at each node of which themore important branch extends upward as a head note. The selection of a headat each node is hierarchically computed from the lower level. Therefore, heads
are selected from leaves to root branches.
Nine rules are deﬁned for the time-span reduction preference rules that indicate superiority of one tree over another. These rules consist of local rules andbroad rules.
For example, both TSRPR1 and TSRPR5 are rules related to the metricalstructure. However, TSRPR1 is a local rule, and TSRPR5 is a more global rule.

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Masatoshi Hamanaka, Keiji Hirata, and Satoshi Tojo

TSRPR1 (Metrical Position) Of the possible choice for head of a timespan tree T, prefer a choice that is in a relatively strong metrical position.
TSRPR5 (Metrical Stability) In choosing the head of a time-span T,
prefer a choice that results in a more stable choice of metrical structure.
The biggest problem when implementing time-span reduction on computeris that there is little information on how to combine local and broad rules andhow to construct hierarchical time span trees.

3.3

Problems of time-span reduction in ATTA

In the ATTA we introduced adjustable parameters for control the strength ofeach rule of time-span preference rule in order to overcome the problem in 3.2.
A hierarchical time-span tree is constructed by iterating the calculation of theplausibility of the head for the current heads and choosing the next level heads(Fig. 5).

Dtimespan[i]Dtimespan

[i]

(i)

(i)

..
.

D

timespan(i)[i]Fig. 5. Selecting the next-level heads in the time-span reduction.

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σGTTM III: Learning based Time-span Tree Generator based on PCFG

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However, the performance of time-span reduction in ATTA was not suﬃcient. We investigated some pieces of music for which the time-span reduction
performance was not very good, and the results indicated that many of themhad a weak beat becomes a head (primary) near the leaves of the tree and astrong beat becomes a head near the root of the tree (Fig. 6). In other words,
the ATTA cannot perform well in songs where the important rules are diﬀerent
depending on the branching which is near the leaves or the root of the tree.
We can consider two ways of solving this problem. The ﬁrst one is to introduce
a more adjustable parameter that enables us to control the strength of rules at
each level of the time-span. However, this idea is not practical because it isdiﬃcult to adjust each parameter manually. The second way is described in thenext section.

Strong beat becomes a head
Weak beat becomes a head##& # 24 œœ œ #œ œ

œ. œ œ

œ

œ.

œ
J

œ.

œ
J

œ

œ #œ œ #œ

œ. œ œ œœ œ œ œœœ ‰J

Fig. 6. Example of a piece that is not good performance.

4

σGTTMIII: Learning based time-span tree generator

The time-span tree can extract an abstracted melody by reducing ornamentation
notes. An example of time-span reduction is shown in Fig. 7. The time-span tree
in the ﬁgure is from melody A, which embodies the results of GTTM analyses.
We can obtain an abstracted melody B by slicing the tree in the middle andomitting notes that are connected to branches under line B. In the same manner,
if we slice the tree higher up at line C, we can get a more abstracted melody C.
If we apply this reduction process in the inverse direction, we can use it as a
generation process as follows (Fig. 8).

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Proc. of the 11th International Symposium on CMMR, Plymouth, UK, June 16-19, 20158

Masatoshi Hamanaka, Keiji Hirata, and Satoshi Tojo

Fig. 7. Time-span reduction.

b
&bb w
b w
&bb
˙
b
&bb ˙
œ
b
&bb œ
œ œ
b
œ œ
&bb œ œ œ œ œ
œ

w
w
˙
œ

˙
œ

œ œ œ

œ

œ

œ œ œ

œ œ

Fig. 8. Generation process of note sequence.

1. Identify the note in which the time-span (length) is the same as the whole
piece of music.
2. Separate the time-span of the note and make a primary note and a secondarynote.
3. Repeat 2 recursively to the leaves of the tree.
By expressing the above generation process using a probabilistic model, we canacquire the most probabilistic time-span tree.
4.1

Training data

The training data we used for the probabilistic model was a musical structural
database based on GTTM that we constructed [1].
We collected 300 8-bar-long monophonic classical music pieces that included
notes, rests, slurs, accents, and articulations entered manually using music notation software called ”Finale” [23]. We exported the MusicXML by using aplugin called ”Dolet.” The 300 whole pieces and the 8 bars were selected by amusicologist.

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We asked a musicology expert to manually analyze the score data faithfully
with regard to the GTTM by using the manual editor in the GTTM analysis tool(Fig. 9) in order to assist in editing the grouping structure, metrical structure,and time-span tree. Three other experts crosschecked these manually produced
results.
The analyzer and database can be downloaded at http://www.gttm.jp/

Fig. 9. Interactive GTTM analyzer.

4.2

PCFG model for generating time-span tree

We introduce here a probabilistic context-free grammar (PCFG), which we used
to construct a probabilistic generation model of a melody. The PCFG has multiple generation rules with probabilities that can represent the separation ofprimary and secondary notes in each node of a time-span tree.
The PCFG of G can be deﬁned by a quintuple:
G = {T, M, S, R, P }(1)

where T is the set of terminal symbols, M is the set of non-terminal symbols, S is
the start symbol, R is the set of production rules, and P is the set of probabilities
on generation rules.
T : terminal symbols. Notes are the terminal symbols.
M : non-terminal symbols. Time-spans are the non-terminal symbols.
S : start symbol. The start symbol of PCFG for generating a time-span tree
is the length of an entire piece of music without any rests, which means thelength of the time-span of the whole piece.

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R: production rules. There are two kinds of production rules. The ﬁrst kind
involves a time-span from which two time-spans are generated. We call thisrule the time-span separation rule. The sum of the length of two time-spansis the same as the length of the original time-span. There are several ways
to generate two time-spans from one time-span. The second kind of rule
involves a time-span to generate a note, which we call the note generation
rule. The length of the original time-span and the length of the generated
note are the same (Fig. 10).
P : probabilities of generation rules. Each production rule has a probability. For example, the probability of generating a time-span that has thelength of 32nd generate 32nd note is almost 1.00. On the other hand, theprobability of generating a time span that has the length of a double note
generate double note is almost 0.00 because there are many other production
rules for a double note (Fig. 10).

Producon Rules
Time-span
Separaon rules

Probability       

 


..
.

Note generaon
rules

0.35



0.12
..
.

                     

0.62

..
.

..
.

 

 
..
.

                    ..
.
                   
..
.

0.01
..
.

0.44
..
.

0.99
..
.

Fig. 10. Example of production rules and its probabilities.

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σGTTM III: Learning based Time-span Tree Generator based on PCFG

4.3

11

Design rules of PCFG for generating time-span tree

We have so far only discussed the length of a note in the basic PCFG model
described in 4.2. Other relevant information is as follows.
Primary and secondary. When two time-spans are generated according to
the generation rules, one is assigned as a primary time-span and the otheris assigned as a secondary time-span.
Pitch. When the primary and secondary time-spans are generated, the primary
pitch will inherited and second pitch will generated.
Order of time-spans. There are two orders when primary and secondary timespans are generated. In one, the primary time-span is generated before thesecondary time-span. In the other, the secondary time-span is generated before the primary time-span.
Numbers of dots. As described in 3.2, the metrical structure strongly aﬀects
the generation of the time-span. Therefore, the numbers of dots in the primary and secondary time-spans should be included in the model.
Use of the above information in the naive implementation of the rules of
PCFG results in too many rules, and the probability of most of them is zero
because we have limited numbers of training datasets. We solve this problem ofspace training data by abstracting the rules as follows.
3 types of pitch change. We graded the changes in pitch of primary and secondary time-spans as up, down, or the same.
7 types of duration ratio. We graded the length ratio of primary and secondary time-spans as one closest to the 4 times, 3 times, 2 times, 1 time,
1/2 time, 1/3 time, and 1/4 time.
2 orders of time-spans. One order is that the primary time-span is before thesecondary; the other is that the secondary time-span is before the primary.
3 types of numbers of dots. We graded the dots of primary and secondary
time-spans in three groups: primary has many dots, secondary has many
dots, and primary and secondary have the same number of dots.
5 types of head time-span length. We graded the length of the time-span
before separating it into primary and secondary as the one closest to sixteenth, eighth, quarter, half, whole, and double time.
We established 645 PCFG rules in total, which consist of 630 (=3x7x2x3x5)
time-span separation rules and 15 (=3x5) note generation rules.
4.4

Generation of time-span tree by using PCFG

The probability of each PCFG rule was achieved using supervised learning by
counting 19,296 nodes of 300 time-span trees in our database. Then we wereable to obtain the most probabilistic time-span tree by calculating the total
probability of each parse tree of all the well-formed time-span trees and selecting
the maximum one (Fig. 11).

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Generating all the well-formed time-span trees requires substantial computing time because the combinations of trees increase exponentially when the numbers of notes increase. To reduce the computing time, we parallelized eight processes for generating time-span trees for each piece of music. By using the PCFG,
we needed three weeks to acquire plausible time-span trees of 100 pieces of music in the GTTM database using a PC cluster (16 machines of Intel Xeon E52430@2.00 GHz 12core). The computing time depended on the musical pieces;the longest was two weeks, and the shortest one was two minutes.

Producon Rules of PCFG
Producon Rules
Time-span
Separaon rules Probability      

 





                     

..
.
..
.

 

 

œ

œ

œ

œ œ œ œ . œ œ œ œ œ œ . œ #œ œ œ œ œ œ œ œ

p

œ

œ
J ‰

j
œ
œ
b
& b b bbb 42 œ œ n œ œ œ œ œ n œ œ b œ œ œ n œ œ œ œ œ œ œ œ œ œ œ œ œ n œ œ œ œ œ n œ œ b œ œ œ n œ œ œ œ œ œ œ œ œ ‰

0.44
..
.

0.99

..
.

œ

œ œ œ . œ #œ œ œ œ œ . œ #œ œ œ œ œ . œ œ œ œ œ #œ..
.

                    ..
.
                   

Input melody

& 24

0.01

..
.

Generang phasejœœb& b b bbb 42 œ œ n œ œ œ œ œ n œ œ b œ œ œ n œ œ œ œ œ œ œ œ œ œ œ œ œ n œ œ œ œ œ n œ œ b œ œ œ n œ œ œ œ œ œ œ œ œ ‰

0.62

..
.

Learning phase

300 pieces

0.12

..
.

Note generaon
rules

GTTM databese

0.35

Learning probabilies..
.

Calculate thetotal probability

PC-cluster
0.01
0.04
0.07

Generate all
well-formed
me-spantrees

Most probabilisc me-span tree

0.25

ATTA (metrical strucuture analysis)..
σGTTMII (grouping structure analysis).
grouping andmetrical structures

0.10
0.08
0.11

œ

œ

œ

œ

Input grouping and metrical structures(experiment in Sec. 5)

Fig. 11. Overview of М GTTMIII.

5

Experimental results

We evaluated the performance of σGTTMIII by leave-one-out cross validation inorder to compare its accuracy with that of ATTA, a previously developed timespan tree analyzer [18]. We used the same 100 pieces of music as used to evaluate
ATTA, and they were numbered from 1 to 100 in the 300-piece database. Whencalculating accuracy, we did not consider the possibility that a low-level error

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would propagate up to a higher level; we counted wrong answers without regard
to the diﬀerences in time-span levels.
Accuracy =Numbers of matched node in the time-span tree
Numbers of node in the time-span tree

(2)

The results of our experiments are given in Table 1. The ATTA has two kindsof results because the accuracy of ATTA varies depending on its adjustableparameters.
It took us an average of about 10 min per piece to ﬁnd the plausible tuning forthe set of parameter. The σGTTMIII outperformed ATTA in average accuracy inboth the baseline and after tuning the parameters. The σGTTMIII outperformed
the baseline performance of ATTA for all the pieces. After the parameters weretuned, the ATTA outperformed σGTTMIII in a few of the pieces.
Table 1. Accuracies of ATTA and М GTTMIII.
Baseline

ATTA withperformance conﬁgured σGTTMIII

Melodies
1. Moments Musicaux
2. Wiegenlied
3. Traumerei
4. Sinfonie Nr.9 d moll Op.125 4.Satz An die Freude
5. The Nutcracker Suite Op.71a No.8 Waltz of the Flowers

Total (100 melodies)6of ATTAparameters

0.71
0.54
0.50
0.22
0.42
..
.
0.44

0.84
0.69
0.63
0.48
0.91
..
.
0.60

0.88
0.78
0.84
0.68
0.72
..
.
0.76

Conclusion

We described σGTTMIII, a time-span tree generator based on PCFG. We set
645 rules for generating time-spans and after carrying out supervised learning of
the rules of PCFG, the σGTTMIII outperformed ATTA, the previous time-spantree analyzer, in average accuracy.
Some applications such as melody morphing or melody summarization require the use of a time-span tree [10–13]. However, such applications are notpractical because the accuracy of the previous time-span analyzer is not suﬃcient and requires manual parameter tuning.
We plan to construct an application for automatic melody morphing or summarization. We also plan to achieve un-supervised learning of rules to improve
the accuracy.

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Acknowledgments. This work was supported in part by JSPS KAKENHI
Grant Numbers 23500145, 25330434, and 25700036 and PRESTO, JST.

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3. Narmour E: The Analysis and Cognition of Basic Melodic Structure. University of
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5. Yazawa, S., Terasawa, H., Hamanaka, M., Hirata, K., and Tojo, S.: Analysis thechain structures of Implication-Realization Model in a melody, IPSJ SIG Technical
Report, 2010-MUS-87, No. 1 (2002) (in Japanese).
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