ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) | 978-1-7281-7605-5/20/$31.00 ©2021 IEEE | DOI: 10.1109/ICASSP39728.2021.9414249

STATISTICAL CORRECTION OF TRANSCRIBED MELODY NOTES BASED ON
PROBABILISTIC INTEGRATION OF A MUSIC LANGUAGE MODEL AND
A TRANSCRIPTION ERROR MODEL
Yuki Hiramatsu

Go Shibata

Ryo Nishikimi

Eita Nakamura

Kazuyoshi Yoshii

Graduate School of Informatics, Kyoto University, Japan
Singing voice spectrogram

ABSTRACTThis paper describes a statistical post-processing method for automatic singing transcription that corrects pitch and rhythm errors included in a transcribed note sequence. Although the performance offrame-level pitch estimation has been improved drastically by deeplearning techniques, note-level transcription of singing voice is still
an open problem. Inspired by the standard framework of statisticalmachine translation, we formulate a hierarchical generative model ofa transcribed note sequence that consists of a music language model
describing the pitch and onset transitions of a true note sequence anda transcription error model describing the addition of deletion, insertion, and substitution errors to the true sequence. Because the lengthof the true sequence might be different from that of the observed
transcribed sequence, the most likely sequences with possible different lengths are estimated with Viterbi decoding and the most likely
length is then selected with a sophisticated language model based ona long short-term memory (LSTM) network. The experimental results show that the proposed method can correct musically unnaturaltranscription errors.

Automatic singing transcription
Transcribed note sequence

Key estimationC

Error correction(Markov language model + Transcription errror model)

Candidate note sequences

Candidate selection(LSTM language model)

-10
-5
-20

Fig. 1. The overview of our statistical error correction method basedon candidate estimation and selection.
In this paper, we propose a statistical error correction methodthat estimates multiple candidates of a true note sequence from atranscribed sequence including errors and then selects the most likely
one (Fig. 1). More specifically, we formulate a hidden Markov model(HMM) that consists of a music language model representing thegenerative process of a true note sequence and a transcription error
model representing that of a transcribed note sequence from the true
sequence. Note that the lengths of the true and transcribed sequences
might be different because the transcription error model represents
the basic editing operations (i.e., insertion, deletion, and substitution) for a true note sequence. Given a transcribed note sequence, themost likely key is estimated and the most likely true note sequenceswith specified lengths are then estimated with Viterbi decoding. Finally, the note sequence with the most likely length is selected witha sophisticated language model based on a long short-term memory(LSTM) network [6–8].
The main contribution of this paper is to build a statistical error
correction framework based on a music language model and a transcription error model for AST in the same way as the statistical machine translation framework based on a target language model and atarget-to-source back-translation model [9, 10]. Another noticeable
contribution is to propose a hierarchical HMM as the transcription
error model for dealing with the editing operations from the probabilistic generative point of view. Our model consists of an insertiondeletion model for alignment between true and transcribed note sequences with different lengths and a substitution model for modification of the pitches and onset positions of notes. Our method isthus more sophisticated than the basic HMM-based post-processing
method that can correct only substitution errors.

Index Terms— Singing transcription, music language models,statistical modeling, symbolic music processing
1. INTRODUCTION
Automatic singing transcription (AST) refers to estimating a symbolic musical score from singing voice and has been considered
to be an important task from the technical and practical points ofview [1–4]. In AST, pitch errors, rhythm errors, and extra/missingnote errors are unavoidable because the singing voice has complicated pitch trajectories. In this study, we tackle a new research topic
that aims to correct such errors included in musical scores estimatedby an AST method.
To reduce transcription errors in AST, music language models that represent a probability distribution of musical scores haveoften been used with audio transcription models [1, 2, 5]. Given
that a more typical musical score has a higher generative probability, AST methods using both language and transcription models
are expected to improve the musical naturalness of estimated scores.
Such language models, for example, include a Markov model representing the transitions of semitone-level pitches [1] and a metrical
Markov model representing the transitions of metrical onset positions [2]. Nonetheless, the estimated score still includes a number ofmusically-unnatural errors because language models are considered
to have limited impact when they are used in combination with transcription models. This calls for a post-processing step that performs
error correction in the purely symbolic domain.

2. RELATED WORKThis section reviews the use of language models for music and speech
applications. Melody style conversion aims to change only the style

This work is supported in part by JST ACCEL No. JPM-JAC1602 and
JSPS KAKENHI Nos. 16H01744, 19K20340, 19H04137, and 20K21813.

978-1-7281-7605-5/21/$31.00 ©2021 IEEE

(AST)

256

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Extended pitch classes

Length and Key*+

True sequence )

!"

!#

!$

!(

!%

!&

xon ∈ {0, · · · , 15} indicating a 16th-note-level relative position ineach bar, and zn is defined in the same way. In this paper, the input
sequence is assumed to have the time signature of 4/4 and include no
key changes. The extended pitch-class sequence {xpn }N
n=1 is computed from a MIDI note number sequence {x̃pn }N
n=1 as follows:!'

xp1 = x̃1 mod 36,
xpn = xpn−1 + (x̃pn − x̃pn−1 )



!%

!"




!#






!'

!$




!(


















(1)
(2)

This representation is convenient to capture the relative pitch dynamics of a sung melody within three octaves.

!&

mod 36.



16th-note-level
relative positions

3.2. Model formulation

Fig. 2. The music language model that represents the generative
probability of a true note sequence Z given a key S and a length M .

We formulate a joint model of a transcribed note sequence X and atrue note sequence Z. Let M and S be the latent length and key ofZ, where a key S ∈ {0, . . . , 11} indicates {C, C#, . . . , B}. Let
Y = {yn }N
n=1 be a latent index sequence that aligns Z with X,
where yn ∈ {1, · · · , M } indicates that xn is derived from zyn , i.e.,N
ZY , {zyn }N
n=1 corresponds to {xn }n=1 one by one. Using theselatent variables, the full probabilistic model is given byof a note sequence while preserving the original content. Inspired by
the statistical machine translation framework [9], a statistical melody
style conversion method was proposed by integrating music language models of individual styles and conversion models between
different styles, where these models can be learned from existing
melodies in an unsupervised manner [11]. The language model of atarget style is based on a Markov model that represents the generative process of a target-style note sequence. The conversion model
between target and source styles represents the back-translation process of a source-style note sequence from a target-style sequence.
Given a source-style sequence, the most likely target-style sequencecan be estimated with Viterbi decoding.
In this study, we take the same approach to error correction in
AST. Specifically, we integrate a music language model representing the generative process of a true note sequence and a transcription
error model representing the generative process of a transcribed sequence from a true sequence. While the conversion model proposedin [11] does not allow the number of notes to be changed, i.e., deals
with only substitutions, our transcription error model deals with insertions, deletions, and substitutions.
In automatic speech recognition (ASR), rescoring of N-best candidates estimated by an ASR method has often been used [12–14],
where a BERT-based language model can be used for computing thegenerative probabilities of candidate word sequences [12]. Such a
post-processing method based on a complicated yet powerful language model is useful, especially when the language model is hardto integrate with the inference process of an ASR system. In thisstudy, we take the same approach to error correction in AST. Specifically, true sequence candidates with different lengths are estimatedwith a Markov language model and the best candidate is selectedwith an LSTM language model.

p(X, Y, Z, S, M ) = p(X, Y|Z, M )p(Z|S, M )p(S, M ),(3)

where p(X, Y|Z, M ) and p(Z|S, M ) are the transcription error model and the music language model, respectively, and p(S, M ) is aprior distribution on the key S and the length M . Note that we assume p(X, Y|Z, S, M ) = p(X, Y|Z, M ) in (3).
3.2.1. Music language model
The music language model p(Z|S, M ) in (3) gives the generative
probability of a true note sequence Z given a key S and a length M(Fig. 2). It is based on a standard autoregressive model as follows:p(Z|S, M ) = p(z1 |S)MYp(zm |z1:m−1 , S),

(4)

m=2

where the notation i:j indicates a set of indices from i to j. In thispaper, (4) is implemented as a first-order Markov model or an LSTM
model.
In the first-order Markov model, each term of (4) is represented
with a categorical distribution as follows:S

p(z1 |S) = Categorical(z1 |π S ),p(zm |z1:m−1 , S) = p(zm |zm−1 , S)

(5)

= Categorical(zm |φS
zm−1 ),

(6)

35,15
S
,{π(z
p ,z o ) }z p =−1,z o =0 is a set of the initial probabilities

where πover the possible combinations of 37 pitches (including rest) and 16
35,15
S
onset positions under the key S, φS
z ,{φ(z p ,z o ),(ẑ p ,ẑ o ) }ẑ p =−1,ẑ o =0p
is a set of the transition probabilities from a note z = (z , z o ) under
the key S. For standard notes z = (z p , z o ) with z p ≥ 0, we assume
the tonic invariance (transposition symmetry) as follows:

3. PROPOSED METHOD
We formulate a hierarchical generative model of a transcribed notesequence consisting of a Markov language model and a transcription
error model. Given a transcribed sequence, we estimate a key withthe language model and infer true sequence candidates with different
lengths via Viterbi decoding. Finally, we select the best candidate byrescoring the candidates with the LSTM language model.

S0p ,z o ) = π(|z p −S| ,z o ) ,
π(z
36
0
φS
(z p ,z o ),(ẑ p ,ẑ o ) = φ(|z p −S|36 ,z o ),(|ẑ p −S|36 ,ẑ o ) ,3.1. Problem specificationpo
Our goal is to estimate a true note sequence Z={zm =(zm
, zm
)}M
m=1
p
o
from an erroneous transcribed sequence X = {xn = (xn , xn )}N
n=1 ,
where M is the length of Z, N is the length of X, each note xn
is represented by a pair of xpn ∈ {−1, 0, · · · , 35} indicating an
extended pitch class in three octaves (−1 indicates the rest) and

(7)
(8)

where |i − j|k is the modulus of |i − j| with respect to k.
In the LSTM model, each term of (4), i.e., the categorical distribution of zm , is recursively predicted at each time step m by using
an LSTM network. In this paper, the LSTM network is trained fromexisting melody note sequences that are transposed into the C major or A minor key. To evaluate the generative probability of a notesequence Z under an arbitrary key S, Z is thus transposed to the C
major or A minor key and fed to the LSTM network.

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True sequence )

!,

!-

!.

!/

!1

!0

Index sequence *

1

2

2

3

5

7

Inserted and deletedsequence )*

!"#

!"$

!"%

!"&

!"(

!"'

Transcribed sequence 2 +,

+-

+.

+/

+1

+0

p(yn = m|yn−1 = m, M ) = ηins ,p(yn = m|yn−1 = m − 2, M ) = ηdel .

!3

When a decent AST method is used, it is natural to enforce the sequential alignment (prohibit the cross alignment) between Z and X
and more than two successive notes are unlikely to be deleted atonce. In addition to the left-to-right property of this Markov model,
we thus consider the following constraints:

Insertion
Deletion

Substitution

y1 ∈ {1, 2},yn − yn−1 ∈ {0, 1, 2},yN ∈ {M − 1, M }.

Fig. 3. The transcription error model consisting of the insertiondeletion model that aligns a true sequence Z with a transcribed sequence X through an index sequence Y and the substitution modelthat modifies an aligned sequence ZY = {zyn }N
n=1 to X.

The music language models are trained using existing melody notesequences including no errors with key annotations. These sequences
can be transposed into C major or A minor. The Markov language
model is trained by maximum likelihood estimation and the LSTM
language model is trained such that the sequential predictive probability of the next note given the history of notes is maximized.
The probabilities of the transcription error model χp , χo , ηins ,and ηdel are calculated from the alignment obtained by the methodof [15] between transcribed and true note sequences. We set therest-to-note substitution probability 1 − χp−1,−1 and the note-to-rest
substitution probabilities {χpzp ,−1 }35
z p =0 to zero because these probabilities cannot be calculated from the alignment where the rests are
ignored and these substitutions can be represented as the combinations of the deletion, insertion, and note-to-note substitutions.

The transcription error model p(X, Y|Z, M ) in (3) gives the generative probability of a transcribed note sequence X and an index
sequence Y from a true sequence Z and a length M (Fig. 3). It isdefined as a latent variable model given by(9)

where p(X|Y, Z) is the substitution model that represents the generative probability of a transcribed sequence X from a sequence of thesame length ZY specified by a true sequence Z with an alignment
Y, and p(Y|M ) is the insertion-deletion model that represents theprobability distribution over all possible alignments between two sequences of lengths M and N (|X| = |Y| = N and |Z| = M ).
Assuming that pitch and onset substitution errors happen independently at any position of the aligned sequence ZY , the substitution model p(X|Y, Z) in (9) is factorized as follows:p(X|Y, Z) =N
Y

3.4. Error correction as Viterbi decoding
We first estimate the key S assuming that the key of X is the sameas the key of Z as follows:p(xn |zyn )S ∗ = arg max pM arkov (Z = X|S, M = |X|).

N
Yp(xpn |zypn )p(xon |zyon ),∗and theGiven the key S ∗ , we then estimate the index sequence YM∗
true sequence ZM for all possible M as follows:

(10)

n=1

∗
YM
, Z∗M = arg max p(X, Y, Z|S ∗ , M ).

where p(xpn |zypn ) and p(xon |zyon ) are the pitch and onset substitution
probabilities, respectively. For xp , z p ≥ 0, we assume that the pitchand onset substitution probabilities only depend on the differencesof the pitches and the onset positions respectively as follows:

p

(22)

S

n=1

=

(19)
(20)
(21)

3.3. Model training

3.2.2. Transcription error modelp(X, Y|Z, M ) = p(X|Y, Z)p(Y|M ),

(17)
(18)

p(xpn = xp |zypn = z p ) = χp|xp −zp |16 ,(11)p(xon = xo |zyon = z o ) = χo|xo −zo |36 ,(12)p(xpn = −1|zypn = −1) = χp−1,−1 ,p(xpn = −1|zypn = z p ) = χpzp ,−1 ,(13)

We finally select the most likely true note sequence Z∗M ∗ as follows:1
M ∗ = arg max
log pLST M (Z∗M |S ∗ , M ),(24)
M
M
where we use the LSTM language model.
To solve (23), we interpret p(X, Y, Z) as an ordinary HMM ofN
length N consisting of X = {xn }N
n=1 and ZY = {zyn }n=1 as
observed and latent variables, respectively, as follows:(14)

p(X, Y, Z) = p(X|Y, Z)p(Y, Z),o{χ∆p }35∆p =0 and χ

where χ == {χ∆o }15∆o =0 are the pitchand onset substitution probabilities, respectively, and χp−1,−1 and{χpzp ,−1 }35
z p =0 are hyperparameters related to the rest note.

NYp(yn |yn−1 , M ).

z1p(zyn+1 , yn+1 |zyn , yn ) =(yn+1 = yn ),δzyn+1 ,zyn p(yn+1 |yn )
p(zyn+1 |zyn )p(yn+1 |yn )(yn+1 = yn + 1),
max
{p(z
|z
)p(z
|z
y
y
+1
y
+1
y

n
n
n )}p(yn+1 |yn )n+1
zyn +1(yn+1 = yn + 2).

(15)

n=2

This model is parameterized by an insertion probability ηins and adeletion probability ηdel . For m ≤ M , the initial and transition
probabilities are defined as follows:p(y1 = 2|M ) = ηdel ,(25)

where p(X|Y, Z) is given by (10) and p(Y, Z) is factorized as(p(z1 )p(y1 )(y1 = 1),p(zy1 , y1 ) =max{p(z1 )p(z2 |z1 )}p(y1 ) (y1 = 2),

The insertion-deletion model p(Y|M ) in (9) represents the conditional probability of an index sequence Y given the true length M .
It is defined as a first-order left-to-right Markov model as follows:p(Y|M ) = p(y1 |M )

(23)

Y,Z

The most likely sequence Z∗Y can be estimated efficiently with Viterbi decoding.

(16)

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Table 1. Evaluation results [%] (lower E∗ and higher Rdn are better).
Ep
EeEm
Eon Eof f Eall Rdn
Transcribed
8.48 17.4 10.9 39.0
31.4
21.4 95.9
Corrected (LSTM)
9.43 15.5 15.0 41.7
31.7
22.7 97.5
Corrected (Markov) 9.95 17.7 12.7 42.0
31.8
22.8 97.3
Corrected (oracle)
9.15 13.3 14.6 39.8
29.4
21.2 97.4
Ground truth
97.1

Table 2. Cross entropies [bits/note] (lower is better).
Markov LSTM
Transcribed
5.89
5.62
Corrected (LSTM)
4.50
4.51
Ground truth
4.18
4.63
ismir2021_011

             

 

Ground-truth

4. EVALUATION
We report experiments conducted to evaluate transcription accuracy
and musical naturalness of the corrected sequences.

Corrected

4.1. Experimental conditions
For evaluation, we conducted 5-fold cross validation using 60 songs
with the time signature of 4/4 and no key transposition taken fromthe RWC Music Database [16]. The note sequences were estimatedby a convolutional neural network followed by an LSTM network.
For each transcribed sequence X of length N , {[rN ] : r ∈ {r̄ −0.1, . . . , r̄ − 0.02, r̄, r̄ + 0.02, . . . , r̄ + 0.1}} were considered as thecandidates of the true length M , where r̄ was the average of ratios oftrue lengths to transcribed lengths. The optimal length was selectedby using the LSTM language model or the Markov language model.
We trained these music language models by using the melody scoresof 206 Beatles songs and 328 J-pop songs.
To evaluate the proposed method, we calculated the pitch error rate Ep , the extra note rate Ee , the missing note rate Em , theonset-time error rate Eon , the offset-time error rate Eof f , and theoverall error rate Eall [17] by comparing transcribed and correctedsequences with the ground-truth sequences. The musical naturalness
was evaluated in terms of the rate of diatonic notes Rdn because themajority of notes should be on a scale. We considered C major scale{C, D, E, F, G, A, B}, C harmonic minor scale {C, D, Eb, F, G,
Ab, B}, and the other 22 transposed scales. Because detailed key
annotations were unavailable, we used as Rdn the maximum of the
diatonic note rates computed for all scales.



 
  


 



 
  


 

 
  
             Transcribed

Pitch correction


 

 



 



 
 






 



 

  
 

Extra note correction

Fig. 4. The proposed method corrected a pitch error and an extra
icassp_074
note error (RWC-MDB-P-2001
No.11).
        
Ground-truth        Transcribed                                    Corrected

Rhythm error




 

      

Extra note correction  

Rhythm error








Fig. 5. The proposed method corrected extra note errors, but added
icassp2021_080
rhythm errors (RWC-MDB-P-2001
No.74).
        
Ground-truth     Transcribed  Corrected

        

  





                  

Pitch error  



 








            

Rhythm correction and extra note correction

Fig. 6. The proposed method corrected a rhythm error and an extra
note error, but added a pitch error (RWC-MDB-P-2001 No.80).

4.2. Experimental results
The experimental results are listed in Table 1. Among the 60 songs,
the proposed method reduced the overall error rate Eall for 15 songs.
The overall error rate Eall of the proposed method with the LSTM
language model was lower than that with the Markov language model
only. This indicates the effectiveness of the LSTM language modelin the rescoring step. If the length M was selected for each song sothat the overall error rate of the song was minimized (oracle condition), Eall of the corrected sequences was reduced to 21.2%, which
was lower than that of the transcribed sequences 21.4%. This showed
the potential of the proposed method for improving the transcription
accuracy. The diatonic note rate Rdn of the corrected sequences washigher than that of the transcribed sequences and was closer to thatof the ground-truth sequences.
The cross-entropies per note obtained by the Markov and LSTM
language models were shown in Table 2. The cross-entropy per noteof the corrected sequences was lower than that of the transcribedsequences. The higher Rdn and the lower cross-entropy of correctedsequences indicate that the proposed method successfully improved
the musically naturalness in exchange for the slight decrease of thetranscription accuracy (trade-off problem). Three examples of error
correction with the proposed method are shown in Figs. 4, 5, and 6.
In Fig. 4, the proposed method corrected a pitch error and an extra
note error. In Fig. 5, the proposed method deleted extra successivenotes of the same pitch and the corrected sequence has a musicallynatural rhythm. In Fig. 6, the pitches of the corrected sequence wereon a scale and were considered to be musically coherent although
the proposed method made a pitch error.
5. CONCLUSION
This paper presented a statistical error correction method as a postprocessing step of AST. Our method is based on an HMM that consists of a Markov language model that generates a true sequence anda transcription error model that generates an erroneous transcribedsequence from a true sequence. Given a transcribed sequence, we estimate candidate note sequences via Viterbi decoding for all possible
lengths and choose the optimal length based on an LSTM language
model. We found that the corrected sequences were more musicallynatural than the transcribed sequences and the proposed method had
potential for improving the transcription accuracy.
In future work, we plan to integrate the LSTM language modelwith the transcription error model. We believe that a post-processing
correction method based on a powerful music language model is effective for AST because transcribed sequences are expected to bemusically natural from the practical points of view1 .
1 The demo page: https://music-lsmtse.github.io

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[3] A. Laaksonen, “Automatic melody transcription based on
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