Proceedings of the Sound and Music Computing Conference 2013, SMC 2013, Stockholm, Sweden

ACOUSTIC SCORE FOLLOWING TO MUSICAL PERFORMANCEWITH ERRORS AND ARBITRARY REPEATS AND SKIPS
FOR AUTOMATIC ACCOMPANIMENTTomohiko Nakamura, Eita Nakamura† and Shigeki Sagayama†
Graduate School of Information Science and Technology, The University of Tokyo
7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan{nakamura, enakamura, sagayama}@hil.t.u-tokyo.ac.jp
ABSTRACT

performances and their acoustic signals, score following
is a challenging task in musical signal processing and has
been a field of research since [1, 2] and further exploredin [3–12] (see [13] for a review).
Particularly during music practice, performers often repeat and/or skip sections for correcting errors or for practicing specific sections again and again, and it is desirable
to handle such repeats/skips for application of score following in practical situations. In [5, 6, 12], score following algorithms allowing repeats/skips from and to specificscore positions were studied. Although there are performers’ tendencies on from and to where repeats/skips occur,
estimation of the specific score positions is generically difficult, especially in practical situations where scores areprepared in musical instrument digital interface (MIDI)
data or performances by various performers are necessary
to be dealt with. Therefore it is attractive to have score following algorithms which can handle arbitrary repeats/skips
from and to any score positions.
Allowing arbitrary repeats/skips leads to a large search
space and results in two problems: (i) large computational
complexity and (ii) a risk of lowering score-following accuracy. As we later discuss in detail, simply-generalizedversions of algorithms in [3, 5, 6] are difficult to work inreal time for practical scores with O(1000) to O(10000)
notes, 1 and it is unavoidable to reduce the computational
complexity.
Statistical approach to score following has advantages inhandling acoustic variation of musical performances andwas used in many previous works [13]. In this approach,
one can either estimate the score position first and thetempo [3, 4], or estimate simultaneously the score position
and the tempo [9,10,12]. Since the search space is too large
in the latter case when dealing with arbitrary repeats/skips,
we adopt the former method.
In the following, we discuss certain hidden Markov models (HMMs) for musical performance, which explicitly
models performance errors and arbitrary repeats/skips. We
show, when assuming independence of transition probabilities from and to where repeats/skips are made, the computational complexity can be reduced significantly, enabling
us to construct acoustic score-following algorithms whichhandle arbitrary repeats/skips and work in real time. We
experimentally evaluate the performance of the proposed
algorithms for human performances in practice and also

We discuss acoustic score-following algorithms for monophonic musical performances with arbitrary repeats andskips as well as performance errors, particularly focusing
on reducing the computational complexity. Repeats/skips
are often made arbitrarily during musical practice, and it
is desirable to deal with arbitrary repeats/skips for wide
application of score following. Allowing arbitrary repeats/skips in performance models demands reducing thecomputational complexity for score following. We showthat for certain hidden Markov models which assume independence of transition probabilities from and to where
repeats/skips are made, the computational complexity canbe reduced from O(M 2 ) down to O(M ) for the number ofnotes M , and construct score-following algorithms basedon the models. We experimentally show that the proposed
algorithms work in real time with practical scores (up to
about 10000 notes) and can catch up with the performancesin around 3.8 s after repeats/skips.
1. INTRODUCTION
Audio score following is the real-time alignment of acoustic signal of musical performance to the performance score,and has wide application such as automatic accompaniment, automatic score page turning and automatic captioning to music videos. It is particularly essential for automatic accompaniment, which synchronizes the accompaniment automatically to human performances in real timeand helps music performers and lovers practice ensemble
music by themselves.
Human performances have tempo fluctuation due to performers’ physical limitation and their expression of musical ideas. Musical performers, both amateurs and professionals, also make performance errors such as pitch errors and note insertions and deletions. In addition to these,
acoustic signals of musical performances are of complex
nature because of possible noise and acoustic variation ofmusical instruments. According to these features of human† Presently with National Institute of Informatics

2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo, 101-8430, Japan
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Creative Commons Attribution 3.0 Unported License,

et

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stricted use, distribution, and reproduction in any medium, provided the original

1 For example, there are around 1900 notes in the first movement of
the clarinet part in Mozart’s Clarinet Quintet.

author and source are credited.

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Proceedings of the Sound and Music Computing Conference 2013, SMC 2013, Stockholm, Sweden

Figure 2. Representation of errors and repeats/skips inthe performance model. Deletion (green arrows) is represented by a transition to the state after the next. Insertion(purple arrows) is described as a self transition, and substitution (orange objects) is represented by emission of CQF
spectrum of incorrect pitch. Repeat/skip is expressed as atransition to a remote state (red arrows).

Figure 1. The performance HMM consists of states corresponding to notes, and the state emits acoustic features ofthe performed note.
examine whether there is any significant lowering of scorefollowing accuracy. We confine ourselves to monophonicperformances for the sake of simplicity.

2.2 Feature Extraction from Acoustic Signal
The variation in acoustic signals of the performance islarge even within the same pitch. For score following,
therefore, features are preferred to be sensitive to pitch
information and less sensitive to timbre and volume. As
stated in [8], this requirement is matched by the normalized
output of constant-Q filters (CQFs) with central frequencies at semitone intervals (CQF spectrum). For shorter
calculation time, the CQF spectrum was calculated with afast frame-wise algorithm [14]. Since a spectrum changes
significantly at the onset time and is otherwise stationary,
spectral flux is employed to distinguish successive notes ofthe same pitch [15].

2. HMM-BASED PERFORMANCE MODEL
2.1 HMM for Score Following
We regard score following as an inverse problem of estimating score positions from acoustic signals by modelinghuman performances. The human performance without errors and repeats/skips can be seen as a process of making atransition to the next note, and emitting an acoustic featureof the performed note. By associating the notes on scorewith hidden states, the performance is also interpreted as astate transition sequence. The performance often includes
changes in tempos and note durations because of physical limitation and musical expression, and acoustic signalsof the performance include noise and acoustic variations.
These state transitions and emission of acoustic features
are described as a stochastic process [3]. Assuming thatthe transitions depend only on the current state, the performance is represented by an HMM as shown in Fig. 1.
A performance with insertion/deletion errors are also described by an HMM [3]. Insertion is represented by a self
transition and deletion is represented by a transition to thestate after the next as shown in Fig. 2. These are described
as
(ins)

Ai,i = ai + (1 − ai )Ai(del), Ai,i+2 = (1 − ai )Ai

2.3 Emission Probability
As shown in Fig. 2, substitution is represented by emissionof CQF spectrum of incorrect pitch, and the corresponding probability is described as a mixture weight of a Gaussian mixture model for emission probability. The emission
probability bi (yt ) at the i-th state of a CQF spectrum yt attime t is thus!
bi (yt ) =ωk (i)N (yt |µk , Σk )(3)
k∈K

where N (·|µk , Σk ) denotes a multidimensional normal
distribution with mean µk and covariance matrix Σk , K
is the set of all pitches, and ωk (i) stands for the mixture
weight.

. (1)Here, {Ai,j }M
i,j=1 is the state-transition probability matrix,
and the durational self-transition probability ai is determined by matching the expected staying time with the duration di of the i-th note, which yields
di =

∞
!

k=1

kak−1
(1 − ai ) =i1.
1 − ai3. MODELING OF ARBITRARY REPEATS/SKIPS
AND THE COMPUTATIONAL COMPLEXITY
3.1 Topology of the Performance HMM
As discussed in Sec. 2, a performance with insertion/deletion/substitution errors is represented by left-toright transitions to neighboring states and emission of
acoustic features of incorrect pitch. On the contrary, repeats/skips from and to arbitrary notes are represented by(2)

These errors are expressed as transitions to neighboringstates and the HMM topology is left-to-right.

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Proceedings of the Sound and Music Computing Conference 2013, SMC 2013, Stockholm, Swedentransitions from each state to all the states, including remote ones (two examples are shown in Fig. 2.). Therefore,
the topology of the performance model with arbitrary repeats/skips, which generalizes the models in [3, 5, 6], iscomplex, resulting in a large search space.
3.2 Computational Complexity of Score Following

Figure 3. Representation of repeats/skips in the proposed
performance model with the pause state (a blue disk) corresponding to pause sections at repeats/skips. Those areexpressed as two-step transitions via the pause state (redarrows).

The score position is estimated by calculating the mostprobable state given the CQF spectrums up to the time ofestimation. In equations,ŝt = argmax p(st |y1:t ) = argmax p(y1:t , st )
st

st

(4)

Substituting (6) into (5), we have⎡i!
αt (i) = bi (yt ) ⎣αt−1 (j)Bj,i

where st and ŝt denote the state random variable at time ttand its estimated value, and y1:t = {yτ }τ =1 stands for the
CQF spectrum sequence. The second equation is derived
from the Bayes’ theorem.
(4) can be solved by applying the online forward algorithm, and its update rule is described as

j=i−2

+

M
%!
j=1

αt (i) = bi (yt )M!

αt−1 (j)Aj,i⎤(7)αt−1 (j)Cj Di ⎦ .

Since the sum in parentheses in the second term on theright-hand side is independent of i, it is sufficient to calculate this once at each estimation. The computational complexity of the sum is O(M ) and that of the rest of (7) isO(M ). Thus, we can reduce the computational complexity
required for the estimation from O(M 2 ) down to O(M ).
We obtain a similar model by focusing on a silent pause
which is often made at repeats/skips before resuming performance. Such a pause can be represented by an additional state (the pause state). Since the repeats/skips are described as two-step transitions via the pause state as shownin Fig. 3, the tendencies of pausing and resuming the performances can be expressed as the transitions probabilities
to the pause state and those from the pause state. In equations, the transition matrix of the model is

(5)

j=1

where αt (i):=p(y1:t , st =i) is the forward variable. Here,the initial value α1 (i)=bi (y1 )πi is calculated with the initial distribution πi . (5) indicates that the computational
complexity for score following is O(M 2 ) since there are
M summations over M states. As shown in Sec. 4, theO(M 2 ) complexity is too large for the score follower towork in real time for scores with a number of notes larger
than a few hundreds, and therefore it is crucial to reducethe complexity for processing practical scores.
3.3 Algorithms for Reducing Computational
Complexity

Ãj,i = Bj,i , Ãj,N = Cj , ÃN,i = (1 − ÃN,N )Di

In order to reduce the computational complexity, someconstraints on the state-transition probability matrix arenecessary. In this section, we propose two models and algorithms reduced the complexity to linear orders.
Human performers probably perform with their tendencies of pausing before repeats/skips and resuming after
them. We can represent the tendencies at each state as
the probabilities of pausing and resuming, or Cj and Di .
The distribution of where human performers resume is also
probably dependent on where they pause. However, allowing the dependence results in O(M 2 ) computational complexity as shown in Sec. 3.2, and thus we assume that the
distribution of where human performers resume is independent of where they pause. With this assumption, thetransition matrix can be written as
Aj,i = Bj,i + Cj Di&(8)for i, j ∈ [1, M ] where the N -th state is the pause state and
N =M + 1.
Naively, the computational complexity for updating theforward variable in the model is O(N 2 ). However, since
the transition probabilities to the note states except for
those from neighboring notes and the pause state are zero,
the complexity for updating the forward variable for thenote states is reduced to O(M ). For the pause state, wemust deal with transitions from all the states, and the complexity for calculating its forward variable is O(N ). Therefore, the overall computational complexity is reduced toO(N ) ≃ O(M ).
While the above discussion of computational complexity is based on the forward algorithm, a similar discussion
is valid for the Viterbi algorithm. With a slight modification, the discussion can also be generalized for Mealy-type
emission probabilities of the form similar to Aj,i in (6).

(6)

where Bj,i is a band matrix with bandwidth three representing the straight performance and deletion/insertioner"rors."Note that the normalization
conditions
A
=1
i j,i
"
and i Di =1 yield Cj =1 − i Bj,i .

3.4 Comparison of the Two Models
The two models discussed in the previous section has asimilar structure as seen in (6) and (8). In both models,301

Proceedings of the Sound and Music Computing Conference 2013, SMC 2013, Stockholm, Sweden

one can describe tendencies of performance on the distributions of notes to which repeats/skips are made. Both themodels rely on the independence of the distribution fromthe notes before them. The difference is the explicit modeling of the pause state in the latter model. In actual performances, silent pauses at repeats/skips often exist andtheir duration is long to some extent. Therefore, the lattermodel is expected to be more suited for score following.
However, since quantitative comparison of both the models is difficult, we provide experiments for evaluating theperformances of the models in Sec. 4.

Real time factor

100

Conventional
Proposed

10

1
0.1
0.01
100

4. EVALUATION OF COMPUTATIONALCOMPLEXITY AND SCORE-FOLLOWINGPERFORMANCE

1000

10000

The number of notes
Figure 4. Real time factor (RTF) and its standard deviationof score following with the various number of notes in theperformance score. The red points represent RTFs of theconventional algorithm, and the blue ones represent RTFs
of the proposed algorithm with the pause state.

4.1 Experimental Conditions
4.1.1 Overall Conditions
To evaluate our algorithms, we conducted three experiments. The first experiment examines quantitatively
whether the proposed algorithms works in real time withthe practical scores, the second one evaluates the performance of the proposed algorithms in following repeats/skips, and the third one evaluates score-following accuracy and examines whether there is a lowering of accuracy in modeling arbitrary repeats/skips for performances
without repeats/skips.
In all the experiments, we used acoustic signals of monophonic performances at 16 kHz sampling rate and thescores were prepared in MIDI format. CQF spectrums
were extracted by using 128 ms frames with a 20 ms hopsize, and the emission probabilities of the performancemodels were trained by clarinet performances in RWC
musical instrument sound database [16]. The parametersof the proposed algorithm without the pause state were
(ins)
(del)
set as π=[1, 0, 0, · · · , 0]⊤ , Ai =Ai = exp(−500),
Ci = exp(−1000), and Di =1/M for i∈[1, M ]. For theother proposed algorithm, the parameters were set as
ÃN,N =0.98 in addition to the above. The probabilities
of making errors of semitone, whole tone and perfect 12th
were set as 0.001, 0.001, and 0.0001, respectively.

sical and popular music pieces and nursery rhymes, partially from RWC music database [16]. 43 repeats/skips and
45 insertion/deletion/substitution errors were made naturally in the performances, and the ranges of repeats/skips
were distributed from 0.1 s to 85 s in score time (0 bars to
43 bars). The performer did not waited the score follower’s
catching up with his performance. As evaluation measures,the detection rate of repeats/skips and the following time
were employed. The following time is defined as the timeinterval (in units of seconds and notes) between the repeator skip and the time when the score follower caught up
with the performance within a range of ∆ ms.
We compared the proposed algorithms with the algorithm
without modeling of repeats/skips which corresponds tothe previous work [3]. While Cano et al. used slightly
different acoustic features of pitch and energy, CQF spectrums were employed as acoustic features in this experiment. The difference does not result in lowering the scorefollowing accuracy, and rather improves it as stated in [8],and we believe that our choice of the acoustic features isadequate.

4.1.2 Condition on the First Experiment4.1.4 Condition on the Third Experiment

Since the computational complexity mainly depends on thenumber of notes, and not on pitches and durations, artificially prepared scores with various numbers of notes wereused in the first experiment. The machine had an Intel Core
2 Duo P9400 2.40 GHz with 6 MB of cache and 2 GB
of RAM, and the operating system was Ubuntu 12.04LTS.
The evaluation measure was the real time factor (RTF) defined as the ratio of the processing time and the hop-size,
which is less than one if and only if the algorithms work inreal time.

In the third experiment, a sufficient amount of real performances could not be prepared, and we used monophonic acoustic signals converted from MIDI signals. For
the MIDI signals, the melody parts of 112 popular musicpieces and royalty-free ones without repeats/skips in RWC
music database were employed [16]. Evaluation measures
were the piecewise precision rate and the overall precision
rate used in the MIREX contest [17]. The piecewise precision rate (PPR) is the average of detection rates of notes
in each piece, and the overall precision rate (OPR) is thedetection rate of notes in all pieces.

4.1.3 Condition on the Second Experiment

4.2 Results and Discussions

In the second experiment, for evaluating the scorefollowing performance under practical situations, we usedacoustic signals of 14 recorded performances (total 1687 s)
by an amateur clarinet performer during his musical practice. Seven different songs were performed including clas-

4.2.1 First Experiment
The result of the first experiment is shown in Fig. 4, where
the RTF was averaged over 95 calculations for each con-302

Following time [notes]

Following time [s]Proceedings of the Sound and Music Computing Conference 2013, SMC 2013, Stockholm, Sweden

18
16
14
12
10
8
6
4
2
0
w/
pause

no
pause

no
repeat/skip(a) Following time in second.

25
20
15
10
5
0
w/
pause

no
pause

no
repeat/skip(b) Following time in notes.

Figure 5. Following time (both (a) in second and (b) in notes) of the proposed algorithm with the pause state (w/ pause),that without the pause state (no pause) and the conventional algorithm (no repeat/skip) in left-to-right fashion.
detected only eight repeats/skips, and followed those with
11 ± 3 s and 17 ± 8 notes delay. It is obvious that modeling repeats/skips significantly improves the performancein following repeats/skips.
The algorithm with the pause state had 11 undetected repeats/skips. Some of the undetected repeats/skips werecaused by the existence of similar sections and phrases
such as choruses in popular music. Others happened in
the cases where only a few notes were performed between
the repeats/skips. Such scores and performances are generally difficult to follow both for computers and humans.
Because human accompanists would need comparable following time, the proposed algorithms are applicable topractical use.

Evaluation
w/ pause no pause no repeat/skip
MeasureDetection rate
32/43
29/43
8/43
of repeats/skips
Table 1. Detection rate of repeats/skips by the proposed
algorithm with the pause state (w/ pause), that without thepause state (no pause) and the conventional algorithm (norepeat/skip).
dition. Only the result of the proposed algorithm with thepause state is shown, since the result was similar for theother. The figure shows that in the proposed algorithm, the
RTF increases asymptotically in proportion to M and, inthe conventional algorithm, asymptotically in proportion toM 2 , which is consistent with the theoretical result in Section 3.3. The result shows that the score follower worked
in real time on the computer up to around 10000 notes, andthe conventional one up to around 300 notes. The conventional algorithm is difficult to handle the practical scoreswith over O(100) notes, and for those with 10000 notes,
the computation time is around 2 s, or ten times the hopsize. On the other hand, the computational time is reduced
to around 0.02 s, or one hundredth, by the proposed algorithms, and almost all the practical scores can be used. Although the detail of the upper bound of the number of notes
for real-time working may be changed on other computers
because of difference in processing power, the reduction of
the computational complexity by the proposed algorithms
always remains effective.

4.2.3 Third Experiment
In the third experiment, all the PPRs were 0.839 ± 0.009,
the OPRs by the algorithms except that without the pause
state were 30073/36051 and the other was 30070/36051.
There were only the slight difference between the proposed
algorithms and the conventional one in PPR and OPR, andthis result shows that the modeling of repeats/skips did not
lower the accuracy significantly.
4.3 Implementation to Automatic Accompaniment
We also implemented the proposed score-following algorithms to automatic accompaniment.
As an accompaniment playback module, a tempo estimation[18] and a playback speed conversion of acoustic signals of accompaniment [19] were employed.
Fig. 6 shows the accompaniment result to the performances with repeats by the algorithm with the pause
state, and videos for such performances are available
at http://hil.t.u-tokyo.ac.jp/˜nakamura/demo/automatic_accompaniment.html.

4.2.2 Second Experiment
In the second experiment, the algorithm with the pause
state detected 32 repeats/skips of 43, and its following timewas 3.9 ± 0.8 s (8.0 ± 1.5 notes) for ∆=500 ms as shownin Fig. 5 and Table 1. On the other hand, the algorithm
without the pause state detected 29 repeats/skips, and its
following time was 4.9 ± 1.0 s (10 ± 2 notes) for ∆=500
ms. As we conjectured in Sec. 3.4, the proposed algorithm
without the pause state followed repeats/skips later thanthat with the pause state. In contrast to those algorithms,
the conventional algorithm corresponding to the one in [3]5. CONCLUSION
We have proposed two score-following algorithms
for monophonic performances with both insertion/deletion/substitution errors and arbitrary repeats/skips. (i) Assuming the independence of transition

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0
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Figure 6. The automatic accompaniment result for a human performance with repeats by the algorithm with thepause state. The gray circle expresses a real onset, and the
red line represents the played accompaniment.
probabilities from and to where repeats/skips are made,we have shown that the computational complexity is
reduced from O(M 2 ) down to O(M ). (ii) Focusing
on a silent pause which are often made at repeats/skips
before resuming performance, we have revealed that thecomputational complexity is also reduced down to O(M )by explicit modeling of the existence of the pause. We
have experimentally shown that the proposed algorithms
work in real time for the practical scores up to 10000 notes
and can catch up with performances in around 3.8 s after
repeats/skips. The experiment has indicated that there is
not a significant lowering of the score-following accuracy
originating in modeling arbitrary repeats/skips.
As future works, an extension to polyphonic music is
important to enable the score followers to process more
scores and performances by other instruments as discussed
in [7, 17]. Using tempo information is important to improve the performance of the algorithms and to help us to
use beat information as discussed in [9–11].
Acknowledgments
We thank Naoya Ito for playing the clarinet, and Hirokazu Kameoka for useful discussions. This researchhas been partly supported by the Grant-in-Aid from JSPS(#23240021).
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