A FEEDBACK FRAMEWORK FOR IMPROVED CHORD RECOGNITIONBASED ON NMF-BASED APPROXIMATE NOTE TRANSCRIPTION
Satoshi Maruo1
1

Kazuyoshi Yoshii1

Katsutoshi Itoyama1

Matthias Mauch2

Masataka Goto3

Graduate School of Informatics, Kyoto University 2 Queen Mary University of London
3
National Institute of Advanced Industrial Science and Technology (AIST)
Approximate note transcription (Bayesian NMF)

ABSTRACTThis paper presents a feedback framework that can improve chord
recognition for music audio signals by performing approximate
note transcription with Bayesian non-negative matrix factorization(NMF) using prior knowledge on chords. Although the names
and note compositions of chords are intrinsically linked with eachother (e.g., C major chords are highly likely to include C, E, and G
notes, and those notes are highly likely to be in C major chords),
chord recognition and note transcription (multipitch analysis) havebeen studied independently. To solve this chicken-and-egg problem, our framework iterates chord recognition and approximate notetranscription using each other’s results. More specifically, we first
perform approximate note transcription based on Bayesian NMF thatforces basis spectra to respectively correspond to different semitonelevel pitches covering the whole range. We then execute chord
recognition based on Bayesian hidden Markov models (HMMs) thatuse chroma features obtained from the activation patterns of thosepitches. To improve note transcription, we again perform Bayesian
NMF that encourages certain kinds of pitches in each chord region
to be activated. Experimental results showed that our feedback
framework gradually improved the accuracy of chord recognition.

Basisharmonic
spectraPriorsActivation patterns (piano-roll representation)
Feature extraction

Chord
recognition(HMM)

Chroma vectors
Feature classification
D

A

Bm

F#m

G

D

G

A

Fig. 1. Overview of our feedback framework combining chord
recognition with approximate note transcription.
The main limitation of most conventional methods of chord
recognition is that they extract shallow chroma vectors from music
audio signals without performing any note transcription or multipitch analysis. Since the names and note compositions of chords
are intrinsically linked with each other (e.g., a C major chord ishighly likely to include musical notes with pitch classes C, E, andG, and vice versa), a few studies have focused on the connection
between chord recognition and note transcription. Sumi et al. [10]improved chord recognition by using a multipitch analysis method
called PreFEst [11] to estimate the pitch trajectory of a bass line
and using that trajectory as a clue when estimating the root notesof chords. Mauch and Dixon [12] calculated reliable chroma vectors insensitive to the energy of overtones by using approximate
note transcription based on nonnegative least squares (NNLS), andRaczyǹski et al. [13] improved note transcription by consideringa sequence of chords as prior knowledge about pitch distributions.
Note that those studies have dealt with only one-way dependency
between chord recognition and note transcription.
In this paper we propose a feedback framework that can improve chord recognition by focusing on the mutual dependency ofchord recognition and note transcription (Fig. 1). More specifically,a target music audio signal is first transcribed into a piano-roll representation via Bayesian nonnegative matrix factorization (NMF) thatforces basis spectra to have harmonic structures corresponding todifferent semitone-level pitches. Overtone-insensitive chroma vectors are obtained from the piano roll as in [12], and then a chord
sequence is estimated by using Bayesian HMMs. A key feature of
our framework is to again perform Bayesian NMF that encourages
particular kinds of pitches (e.g., C, E, and G) to be activated in eachchord region (e.g., C major). This feedback enables us to calculate
more reliable chroma vectors from the refined piano roll.

Index Terms— Chord recognition, note transcription, Bayesianinference, nonnegative matrix factorization (NMF), hidden Markov
model (HMM).
1. INTRODUCTION
Automatic chord recognition for music audio signals is one of themost fundamental tasks in the field of music information processing [1, 2], in part because the chord patterns used in musical pieces
are clues useful in composer identification [3] and genre classification [4]. And because chord patterns are closely related to the musical mood, automatic chord recognition is indispensable for finding
users’ favorite pieces in large music collections [5].
Conventional methods of chord recognition generally consist of
two parts: extraction of acoustic feature vectors and classification of
those vectors. The most popular way of feature extraction is to calculate a 12-dimensional chroma vector at each frame or short segment(e.g., half beat) [6]. The chroma vector represents an energy distribution over the twelve traditional pitch classes C, C#, D, D#, ..., B. The
chroma vectors extracted from a region of a C major chord, for example, tend to take large values in the dimensions corresponding topitch classes C, E, and G. A standard way of feature classification is
to use a hidden Markov model (HMM) that represents the transition
probabilities between adjacent chords and the emission probabilities
of chroma vectors for each kind of chords [7–9].
This study was partially supported by JSPS KAKENHI 26700020,
24220006, 24700168 and CREST OngaCREST project.

978-1-4673-6997-8/15/$31.00 ©2015 IEEE

Observed spectrogram

Priors196

ICASSP 2015

Original spectrogram

2. RELATED WORKThis section introduces related work on chord recognition in termsof feature extraction and classification.

Chromagram obtained by MIR toolbox

2.1. Feature extraction

Activation patterns obtained by NMF

The basic method of calculating the chroma vector is to accumulate
the energy of frequency bins corresponding to each pitch class [6].
Since this method cannot distinguish the energy of fundamental frequencies (F0s) from that of harmonic partials, the obtained
chroma vector does not precisely represent the energy distribution
over the twelve pitch classes. Several methods for reducing thepower of overtones before calculating chroma vectors have been
proposed. Lee [14], for example, proposed a harmonic productspectrum (HPS), whereas Mauch and Dixon [12] used approximate note transcription based on nonnegative least squares (NNLS).
Saito et al. [15] used specmurt analysis for roughly extracting thepower of F0s by assuming a common harmonic structure, andUeda et al. [16] reduced the power of percussive components byusing a harmonic/percussive sound separation (HPSS) method.
Several variants of chroma vectors have been proposed. Müller
and Ewert [17] proposed a CRP chromagram that is insensitive tothe pitches and timbres of music audio signals. Ni et al. [18] proposed a loudness-based chromagram that takes into account the factthat perception of loudness is not linearly proportional to the poweror amplitude spectrum, and Harte et al. [19] proposed a method of
tonal centroid transformation that converts a 12-dimensional chroma
vector into a 6-dimensional tonal-centroid vector.

Chromagram calculated from the above

Activation patterns obtained by NMF with chord priors

Chromagram calculated from the above

Fig. 2. Comparison of three variants of chromagrams: The bottom chromagram shows the clearest distributions over the 12 pitchclasses.
be a sequence of the corresponding chord labels. Both X and Z aregiven as the training data. For simplicity, we assume that the training
data consists of only one musical piece having the same length as
the sequence X̂ (the extension to dealing with multiple pieces isstraightforward).
The chord label is represented as a combination of a root note
(C, C#, · · · , B) and a type (“maj” or “min”). In addition, the special
symbol N is used for representing “no chord.” Let K be the size ofthe chord vocabulary (K = 25). Since the main focus of this paper is
to show the effectiveness of mutually combining chord recognition
with note transcription, we tackle the essential part of chord recognition by posing the following assumptions:2.2. Feature classification
HMMs can be used to model a vocabulary of chords as latent states;
decoding such an HMM then corresponds to transcribing the optimal
chord sequence. An important extension of this approach is to estimate chords and keys at the same time. Lee and Slaney [20] trained
24 key-specific HMMs corresponding to the 24 major/minor keys
and selected the best model with high probability for a given audio
signal. To deal with key changes, some studies tried to take into
account the transition between adjacent keys [12,21,22]. Deep neural networks (DNNs) and recurrent neural networks (RNNs), whichhave significantly improved the accuracy of speech recognition, haverecently been used for chord recognition [23,24]. This approach canunify feature extraction and classification into the same network.

• Correct beat positions are given in advance. This assumption
is not critical because many promising methods of beat tracking have recently been proposed [25, 26].
• Chord boundaries are on beat positions (quarter-note level) or
half-beat positions (eight-note level). This assumption holds
true for the vast majority of popular music.
• Only two types of chords, “maj” and “min” chords, are taken
into account. Other types of chords (e.g., “maj7” and “dim”)
are forcibly categorized into those two types as in [27].
• The key of a musical piece remains the same from the beginning to the end (i. e., key changes are not taken into account).

3. PROPOSED METHODThis section describes the proposed method of chord recognition
based on approximate note transcription. Our method consistsof Bayesian NMF-based feature extraction (inspired by [12]) andBayesian HMM-based feature classification (inspired by [20]).

To perform reliable chord recognition, the tuning of each musical piece should be adjusted in advance as in [28].
3.2. Chord recognition based on Bayesian HMM
We explain our method of chord recognition that represents the generative process of chroma vectors by using Bayesian HMMs.

3.1. Problem specification
The goal of chord recognition is to transcribe a target music signal
into a sequence of chord labels. Let X̂ = {x̂1 , x̂2 , · · · , x̂T } be asequence of chroma vectors extracted from the target signal and Ẑ ={ẑ1 , ẑ2 , · · · , ẑT } be a sequence of the corresponding chord labels,
where T is the length of those sequences. The chord boundaries are
determined by detecting chord change positions from Ẑ. We aim toconvert X̂ to Ẑ by using some kind of classifiers.
To train a statistical classifier, we use a chord-label-annotatedmusic signal. Let X = {x1 , x2 , · · · , xT } be a sequence of chroma
vectors extracted from the music signal and Z = {z1 , z2 , · · · , zT }3.2.1. Feature extraction
We propose a method that extracts robust chroma features from temporal activation patterns of musical notes in a way similar to thatin [12]. A key difference is that our method uses Bayesian NMF
instead of NNLS because it enables us to take prior knowledge on
harmonic structures and chord labels into account in a principledmanner (see Section 3.3).
At each frame t, we calculate a 12-dimensional chroma vector xt from the activation patterns of different pitches obtained197

form as the prior distribution:

by Bayesian NMF (Fig. 2). In this paper the “frame” indicates ashort half-beat segment. More specifically, we focus on 60 different
pitches from C2 to B6 (MIDI note numbers from 36 to 95) in fiveoctaves. To calculate the value of each dimension of xt , we accumulate the amplitude values of five pitches (e.g., C2, C3, C4, C5,
and C6) corresponding to the same pitch class (e.g., C) at frame t.
Finally, each dimension of chroma vectors is standardized over all
frames such that the mean and variance of the dimension are equal
to 0 and 1.

p(ϕk |Z) = D(ϕk |α0 + nk ),where nk is a K-dimensional vector in which each element nkk′ isthe number of transitions from chord k to chord k′ in Z.
Since the true posterior p(π, µ, Λ|X, Z) cannot be computed
analytically, we use a variational Bayesian (VB) method [29] thatapproximates it as a factorized distribution as follows:p(π, µ, Λ|X, Z) ≈ q(π)q(µ, Λ),We propose a method that classifies chroma vectors by using 24 keyspecific HMMs that respectively correspond to 24 major/minor keys
in a way similar to that in [20]. A key difference is to formulate
Bayesian HMMs for avoiding overfitting to the training data. Each
HMM encodes the generative process of the observed data X via thelatent variables Z under a condition that a particular key is assumed.
For a set of model parameters, Θ (= {ϕ, π, µ, Λ} explained later),the joint probability over X and Z is given byT∏

q(πk ) = D(πk |γ0 + mk ),(8)−1
q(µkl , Λkl ) = N (µkl |ukl , (βkl Λkl ) )W(Λkl |Wkl , νkl ). (9)To optimize these parameters, we alternate VB-E and VB-M steps as
in the expectation-maximization (EM) algorithm. Because of space
limitation, we omit the updating formula (see Ch. 10 in [29]).
We utilize the circular characteristics of the twelve pitch classes
to make the maximum use of X and Z for model training [16, 30].

p(xt |zt )p(zt |zt−1 ), (1)t=1• Training p(ϕ|Z): The key underlying each chord zt is transposed to C by shifting the root note of zt . We thus have totrain chord transitions for only two keys (C major and C minor). The other 22 key-specific HMMs can be obtained bypermuting the elements of ϕ.

where p(zt |zt−1 ) is the transition probability indicating how likely
the previous chord zt−1 is to make a transition to the current chordzn and p(xt |zt ) is the emission probability indicating how likely thechroma vector xt is to be generated from the chord zn . p(z1 |z0 ) =p(z1 ) is the initial state probability. More specifically, p(zt |zt−1 ) isrepresented as a discrete distribution as follows:′p(zt = k |zt−1 = k, ϕk ) = ϕkk′ ,• Training p(π, µ, Λ|X, Z): The root note of each chord zt isshifted to C by circular-shifting of the elements of the chroma
vector xt . We thus have to independently train GMMs for
only two different “types” of chords (C “maj” and C “min”).
The other 22 GMMs can be obtained by shifting the elementsof the Gaussian parameters. Note that the 24 GMMs areshared over all the 24 key-specific HMMs.

(2)

where ϕk = [ϕk1 , · · · , ϕkK ] is a set of transition probabilities thatsum to unity (1 ≤ k ≤ K). Note that the transition probabilities ϕheavily depend on the key. p(xt |zt ), on the other hand, is represented as a Gaussian mixture model (GMM) as follows:Tp(xt |zt = k, πk , µkl , Λkl ) =L∑πkl N (xt |µkl , Λ−1
kl ),3.2.4. Prediction
Using a trained HMM, we aim to obtain the optimal Ẑ maximizing
a joint predictive probability given by∫p(X̂, Ẑ|X, Z) = p(X̂, Ẑ|Θ)p(Θ|X, Z)dΘ.
(10)

(3)

l=1

where L is the number of Gaussians, πk = [πk1 , · · · , πkL ]T is aset of mixing weights, and µkl and Λkl are respectively the mean
vector and precision matrix of the l-th component Gaussian. This
GMM represents a probability distribution over chroma vectors ofchord k (1 ≤ k ≤ K). To complete the Bayesian formulation, we
put conjugate priors on unknown parameters Θ as follows:p(ϕk ) = D(ϕk |α0 ), p(πk ) = D(πk |γ0 ),p(µkl , Λkl ) = N (µkl |u0 , (β0 Λk )−1 )W(Λkl |W0 , ν0 ),(7)

where two factors q(π) and q(µ, Λ) are iteratively optimized suchthat the Kullback-Leibler divergence between p(π, µ, Λ|X, Z) and
q(π)q(µ, Λ) is minimized. As a result, q(π) and q(µ, Λ) are foundto have the same forms as the corresponding priors:

3.2.2. Model formulationp(X, Z|Θ) = p(X|Z, Θ)p(Z|Θ) =(6)

An approximate solution could be obtained without integral computation by simply getting the optimal Θ̂ that maximizes p(Θ|X, Z)
and then finding the optimal Ẑ that maximizes p(X̂, Ẑ|Θ̂) by using
the Viterbi algorithm [31]. Instead, since p(X̂, Ẑ|X, Z) is more robust to outliers, we approximate it as a predictive HMM that consistsof predictive transition and emission distributions (see Eqs. (2) and(3) for comparison) obtained by marginalizing out Θ as follows:

(4)
(5)

where D and W indicate the Dirichlet and Wishart distributionsand ∗0 ’s are hyperparameters given in advance; α0 and γ0 are
K-dimensional nonnegative vectors, u0 is a 12-dimensional vector, β0 is a nonnegative scalar, W0 is a 12 × 12 scale matrix,and ν0 is a degree of freedom. Using Eqs. (1)–(5), we obtain thecomplete probability over all random variables: p(X, Z, Θ) =p(X|Z, π, µ, Λ)p(Z|ϕ)p(ϕ)p(π)p(µ, Λ).

p(ẑt = k′ |ẑt−1 = k, Z) = E[ϕkk′ ],p(x̂t |ẑt = k, X, Z) =

L
∑

(11)

E[πkl ]St(x̂t |ukl , Vkl , νkl − 11), (12)l=1where St indicates the Student-t distribution and Vkl is given by

3.2.3. Training

Vkl =Given the training data X and Z, we aim to calculate a posterior distribution over the model parameters, p(ϕ, π, µ, Λ|X, Z), according to the Bayes rule. In this setting, the posterior can be factorizedas p(ϕ, π, µ, Λ|X, Z) = p(π, µ, Λ|X, Z)p(ϕ|Z). Using the conjugacy between Eq. (2) and Eq. (4), we obtain p(ϕ|Z) in the same(νkl − 11)βkl
Wkl .
1 + βkl(13)

To obtain the optimal Ẑ, we can use the Viterbi algorithm. This
decoding procedure is performed in parallel with respect to the 24
key-specific HMMs, and the HMM with the highest p(X̂, Ẑ|X, Z)
is selected for estimating the key of Ẑ.

198

0.3
0.2
0.1
0

C maj

C C# D D# E F F# G G# A A# B

0.3
0.2
0.1
0

C min

Table 1. Experimental results.
FeatureChroma NMF NMF with priors
Recognition rate (%)
69.0
71.6
71.9

C C# D D# E F F# G G# A A# B

Fig. 3. Pitch-class distributions indicating how likely each pitchclass is to be used in a C maj or C min chord.

4. EVALUATION
Here we describe our experiments evaluating the effectiveness of ourframework.

3.3. Approximate note transcription based on Bayesian NMF
Here we explain Bayesian NMF for note transcription and how to incorporate prior knowledge on harmonic structures and chord labels.

4.1. Experimental conditions

3.3.1. Model formulation and Bayesian inference
NMF has been the most popular choice for source separation andmultipitch analysis of music audio signals [32, 33]. It approximatesa nonnegative matrix (constant-Q spectrogram) X ∈ RM ×N as the
product of two nonnegative matrices W ∈ RM ×R and H ∈ RR×N
such that X ≈ W H, where W = [w1 , · · · , wR ] is a set of R basis
spectra over M frequency bins and H = [h1 , · · · , hR ]T is a set ofthe corresponding activation patterns over N frames. To evaluate theapproximation error, we use the Kullback-Leibler (KL) divergencein an element-wise manner. Minimizing the KL divergence is equivalent to maximizing the Poisson likelihood given by
(
)
R
(
)
∑
R
p Xmn {wrm , hrn }r=1 = P Xmn
wrm hrn . (14)We used the Beatles dataset [34], out of which 137 songs having
major scales were used for 5-fold cross validation. We thus trained
12 key-specific HMMs corresponding to the 12 major keys in our
experiments. The music audio signals were converted into constantQ spectrograms with a frequency interval of 33.3 cents and a timeinterval of 64 ms. The hyperparameters were given by L = 32,
α0 = 1, γ0 = 1, u0 = 0, β0 = ν0 = 12, W0 = I, R = 88,hwh
and aw
rm = brm = arn = brn = 1 if no prior knowledge was used.
To evaluate the performance of chord recognition, we calculated theduration-based matching rate between the estimation results and theground truth. For comparison, we tested a baseline method based on
Bayesian HMMs trained by using chroma vectors [35] extracted via
the MIR toolbox without any note transcription.

r=1

4.2. Experimental results

To formulate a Bayesian model of NMF, we put independent gamma
priors on individual elements of W and H as follows:wp(wrm ) = G(wrm |aw
rm , brm ),p(hrn ) = G(hrn |ahrn , bhrn ),As shown in Table 1, the experimental results indicated the effectiveness of mutually combining chord recognition with approximate
note transcription. We found that harmonic overtones were reduced
by NMF, and the obtained chromagrams showed clearer pitch-class
distributions than normal ones. Although the recognition rate wasfurther improved by using prior knowledge about chords, the improvement was not as much as we had expected. Since relatively
simple chords (e.g., triad), which often appear in Beatles dataset,originally showed sparse chromagrams, the effectiveness of NMF
was limited even if chord priors were used. The proposed method is
expected to work more effectively for musical pieces with complex
chords (e.g., seventh and ninth).
If chord recognition and approximate note transcription are iterated in a mutually-dependent manner, the recognition rate is expected to be further improved. The computational time, however, isalso increased in proportion to the number of iterations. To solve thisproblem, it is necessary to formulate a unified probabilistic model ofchord recognition and note transcription.

(15)
(16)

where a∗ and b∗ are the shape and rate hyperparameters.
Since the true posterior p(W , H|X) cannot be computed analytically, we also use a VB method that approximates it as a factorized distribution as follows:p(W , H|X) ≈ q(W )q(H).

(17)

As a result, optimal q(W ) and q(H) are found to have the sameforms of the prior distributions:∑∑wq(wrm ) = G(wrm |awrm +n E[hrn ]),n λmnk Xmn , brm +∑∑q(hrn ) = G(hrn |ahrn + m λmnk Xmn , bhrn + m E[wrm ]),where λmnk ∝∑exp(E[log wrm ] + E[log hrn ]) is an auxiliary variable such that k λmnk = 1.
3.3.2. Hyperparameter adjustment
We adjust p(W ) such that basis spectra have harmonic structures
corresponding to prefixed semitone-level pitches. We assume thatthe power of overtones decays exponentially with a decaying ratio of
0.5 in a typical harmonic structure. A hyperparameter vector aw
k =
w
T
[aw
over frequency bins is set to be proportional to
k1 , · · · , akM ]the typical harmonic structure. Note that the larger the vector aw
k isscaled, the stronger the impact of the prior is.
Similarly, we can adjust p(H) such that particular kinds of pitchclasses are encouraged to be activated in the region of each chord.
As shown in Fig. 3, we learn how likely each pitch class is to be
used in a C maj or C min chord by using music scores (MIDI files)
with chord annotations in which the musical notes of each chord
region are transposed such that the root note of the chord is C. A
hyperparameter ahrn is set to be proportional to the probability of thecorresponding pitch class.

5. CONCLUSION
This paper presented a feedback framework that combines chord
recognition based on Bayesian HMMs with approximate note transcription based on Bayesian NMF. Those models can make use ofeach other’s information. Experimental results showed the effectiveness of our framework, but there would be much room for significantly improving the chord recognition rate. To maximize the potential of the framework, we will try to appropriately adjust or automatically learn the hyperparameters of Bayesian NMF. On the other
hand, the improvement of note transcription could be expected in ourframework. We therefore plan to evaluate not only chord recognition
but also note transcription. To guarantee the convergence theoretically, we aim to formulate a unified probabilistic model of chord
recognition and note transcription that can be jointly optimized in a
principled manner.

199

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