Hélène Papadopoulos

Geoffroy Peeters
email: peeters@ircam.fr

LARGE-SCALE STUDY OF CHORD ESTIMATION ALGORITHMS BASED ON CHROMA REPRESENTATION AND HMM

This paper deals with the automatic estimation of chord progression over time of an audio file. From the audio signal, a set of chroma vectors representing the pitch content of the file over time is extracted. From these observations the chord progression is then estimated using hidden Markov models. Several methods are proposed that allow taking into account music theory, perception of key and presence of higher harmonics of pitch notes. The proposed methods are then compared to existing algorithms. A large-scale evaluation on 110 hand-labeled songs from the Beatles allows concluding on improvement over the state of the art.

INTRODUCTION

In Western tonal music, chord progression determines the harmonic structure of a piece of music. Analysis of chord progression therefore plays a crucial role in the understanding of this music. Music classification, music retrieval or in general all applications based on music content analysis benefit from the knowledge of chord progression.

However, manual labeling of each chord of a music piece is a difficult and tedious task, even for a well-trained person. Considering the recent availability of large music collections (online music catalogs, mobile devices) the automatic estimation of chord progression has become a major challenge.

In this paper, we present and compare several methods for estimating chord progression directly from acoustic signals of musical recordings. The methods are evaluated and compared to state of the art algorithms on a relatively large dataset (110 songs from the Beatles).

BACKGROUND

Since their introduction in 1999, Pitch Class Profiles (Fujishima [START_REF] Fujishima | Real-time chord recognition of musical sound: A system using common lisp music[END_REF]) or Chroma-based representation (Wakefield [START_REF] Wakefield | Mathematical representation of joint time-chroma distribution[END_REF]) have become common features to automatically estimate chords or musical key from audio recordings ( [START_REF] Sheh | Chord segmentation and recognition using em-trained hmm[END_REF], [START_REF] Bello | A robust mid-level representation for harmonic content in music signal[END_REF], [START_REF] Harte | Automatic chord identification using a quantised chromagram[END_REF], [START_REF] Peeters | Chroma-based estimation of tonality from audio-signal analysis[END_REF], [START_REF] Peeters | Musical key estimation of audio signal based on hmm modeling of chroma vectors[END_REF], [START_REF] Gomez | Tonal description of polyphonic audio for music content processings[END_REF]). PCP/chroma vectors represent the intensity of the twelve semitones of the pitch classes.

Fujishima [START_REF] Fujishima | Real-time chord recognition of musical sound: A system using common lisp music[END_REF] uses this representation to derive a large set of chords using either a nearest-neighbor or a weighted sum method. This system is successfully evaluated but only using synthetic sounds. The first system evaluated on natural sounds (whole pieces of music of commercial recordings) is the one by Sheh and Ellis [START_REF] Sheh | Chord segmentation and recognition using em-trained hmm[END_REF]. Their system for chord segmentation/recognition relies on hidden Markov models (HMM) trained by EM algorithm. Another approach by Harte and Sandler [START_REF] Harte | Automatic chord identification using a quantised chromagram[END_REF] estimates chords using simple bit masks 1 compared to chroma features. Bello and Pickens [START_REF] Bello | A robust mid-level representation for harmonic content in music signal[END_REF] use an approach similar to [START_REF] Sheh | Chord segmentation and recognition using em-trained hmm[END_REF] but introduce musical knowledge in the hidden Markov model. They show improvement over [START_REF] Sheh | Chord segmentation and recognition using em-trained hmm[END_REF] using their system. It should be noticed that many other studies on chord estimation based on symbolic representation have been performed. In this paper we will refer to the recent studies made by [START_REF] Noland | Key estimation using a hidden markov model[END_REF].

The methods proposed in this paper start from the abovementioned approaches. We systematically evaluate them and propose improvements to chord estimation systems. As most previous methods, the signal observations are the chroma features and the chord progression is represented using a hidden Markov model. Various ways of constructing the HMM are studied using either music theory, results from cognitive studies, smoothed training, multivariate Gaussian models or normalized-correlation.

Another major contribution of this paper is the use of harmonic extension of the PCP (Harmonic Pitch Class Profiles) in the case of chord estimation. Actually, most previous methods operate a direct mapping between the PCP/chroma values and the pitch of a note, i.e. a C note is represented by a single non-zero value in the chroma vector. In other words, the assumption is made that what we observe in the spectrum is directly the pitch of the notes. This is not true: each note produces a set of harmonics and thus a mixture of non-zero values in the chroma vector. Therefore, values at pitch classes other than those of the notes will occur in the chroma vectors. For this reason, we propose to consider the presence of the harmonics in the model's parameters.

Harmonic Pitch Class Profiles (HPCP).

To deal with this, in the case of key estimation, Gomez [START_REF] Gomez | Tonal description of polyphonic audio for music content processings[END_REF] proposes to take into account the harmonics of the notes using a theoretical spectral envelope. Izmirli measures the contribution of the harmonics on a piano database [START_REF] Izmirli | Template based key finding from audio[END_REF]. Peeters proposes in [START_REF] Peeters | Chroma-based estimation of tonality from audio-signal analysis[END_REF] the use of a Harmonic Peak Subtraction function which reduces the influence of the higher harmonics of each pitch.

In what follows, we rely on the model presented in [START_REF] Gomez | Tonal description of polyphonic audio for music content processings[END_REF]. This model extends the Pitch Class Profiles (PCPs) to the Harmonic Pitch Class Profiles (HPCPs). For this, a theoretical amplitude is attributed to each harmonic composing the spectrum of a note with an empirical decay factor set to 0.6 so that this contribution decreases with the frequency. The contribution for the first 6 harmonics of a note is given in Table 1. Therefore, higher harmonics contribute to the pitch class of their fundamental frequencies.

Tab. 1. First 6 harmonics of a note and given amplitudes

Organization of the paper. The paper is organized as follows. In section 3.1 and 3.2, we detail the extraction of the chroma vectors from the audio signal. In section 3.3, we study several approaches to estimate the chords from the succession of chroma vectors over time using HMM and we describe in particular various configurations of the observation probabilities (section 3.3.2) and transition probabilities (section 3.3.3). In section 4, we evaluate our system and compare it to current existing systems.

SYSTEM

3.1. Pre-processings

Parameters

We work directly on the audio signal. In our analysis, the signal is down-sampled to 11025Hz, converted to mono and converted to the frequency domain by a DFT using a Blackman window of length 0.48s with 12.5% overlap. Because of frequency resolution limits (the frequency distance between adjacent semitone pitches becomes small in low frequencies), we only consider frequencies above 60Hz. The upper limit is set to 1kHz because the fundamentals and harmonics of the music notes in popular music are usually stronger than the non-harmonic components up to 1kHz [START_REF] Namunu | Automatic Structure Detection for Popular Music[END_REF]. This choice is also supported by the fact that the mapping operated between the energy of the harmonics and the notes is only valid for the lowest harmonics, hence the lowest part of the spectrum.

Tuning

The energy peaks in the spectrogram will be mapped to the chroma vectors. It is therefore important that the peak frequencies correspond as close as possible to usual pitch values (262.6, 277.2, 293.7, ... Hz). Since the instruments may have been tuned according to a reference pitch different from the standard A4 = 440Hz, it is necessary to estimate the tuning of the track. Here, the tuning is estimated using the method proposed by Peeters in [START_REF] Peeters | Musical key estimation of audio signal based on hmm modeling of chroma vectors[END_REF]. The signal is then re-sampled so that the rest of the system can be based on a tuning of 440Hz.

Chromagram computation

The second stage of our analysis is the extraction of a sequence of observation vectors. The signal is converted from the frequency domain to the chroma domain. Chroma vectors are related to our perception of pitch [START_REF] Wakefield | Mathematical representation of joint time-chroma distribution[END_REF]. They represent the intensity of the 12 semitone pitch classes over time. The temporal sequence of chroma vectors over time is known as chromagram. We compute the chromagram using the method proposed by Peeters in [START_REF] Peeters | Chroma-based estimation of tonality from audio-signal analysis[END_REF].

The chromagram is computed in three steps. First, the values of the DFT are mapped to a semitone pitch spectrum using the mapping function:

n(f k ) = 12 log 2 ( f k 440 ) + 69, n ∈ + (1)
where f k are the frequencies of the Fourier transform and n correspond to the semitone pitch scale values. Then, the semitone pitch spectrum is smoothed over time using a median filtering 2 . This provides a reduction of transients and noise in the signal. This smoothing allows us to significantly improve the overall results.

Finally, after this smoothing, the semitone pitches n are mapped to the the semitone pitch classes c within the mapping function:

c(n) = mod (n, 12) (2)
We obtain a sequence of 12-dimensional vectors that are suitable feature vectors for our analysis.

Chord estimation from the chroma vectors using hidden Markov models

We describe here several methods to estimate the chord progression over time of an audio signal. All these methods are based on hidden Markov models (HMMs) [START_REF] Rabiner | A tutorial on hidden markov model and selected applications in speech[END_REF]. The various methods differ in the way observation probabilities and transition probabilities are computed.

We consider an ergodic 24-states HMM, each state representing a single chord. Our chord lexicon is composed of 24 Major and minor triads (C Major, C# Major, . . . , B Major, C minor, . . . , B minor). Each state in the model generates an observation vector, the chroma feature, with some probability. This is defined by the observation probabilities. In part 3.3.2, we study three approaches to define these probabilities. The first one (Method 1) learns these probabilities by training a Gaussian model on chord-normalized chroma vectors. The second one (Method 2) does not use the training set but defines probabilities based only on music theory, considering the presence of higher harmonics (using the HPCPs). The third one (Method 3) is close to Method 2 but defines probabilities based on a normalized-correlation measure rather than a Gaussian model.

In music pieces, the transitions between chords result from musical rules that should be reflected in the state transition matrix. This is in fact the main reason why the problem is modeled using a Markov model. In part 3. In what follows, we denote by π and T , the initial state distribution and state transition probability distribution.

Given the observations, we estimate the most likely chord sequence over time in a maximum likelihood sense.

Initial state distribution

Since we do not know a priori which chord the piece begins with, we initialize π at 1 24 for each of the 24 states. This choice has also been made in [START_REF] Bello | A robust mid-level representation for harmonic content in music signal[END_REF].

Observation symbol probability distribution

Method 1: Modeling by a multivariate Gaussian trained on a labeled dataset

With this method, the observation distribution is modeled by 24 (one for each state) single 12-dimensional multivariate Gaussian distributions defined by their mean vectors µ i and covariance matrices Σ i , with i denoting the i th state.

In [START_REF] Sheh | Chord segmentation and recognition using em-trained hmm[END_REF], the model is trained using the standard expectation maximization (EM) algorithm for HMM parameters estimation. The parameters µ and Σ are initialized with random values. According to [START_REF] Bello | A robust mid-level representation for harmonic content in music signal[END_REF], on the one hand, the template for a chord is almost universal and should not change from song to song. On the other hand, it is unlikely that every chord of the lexicon will be present in the training dataset. That is why it is proposed to selectively train the model, disallowing adjustments of µ and Σ while π and T are updated. We also believe that any reasonably sized training set will be insufficient to appropriately estimate the model's parameters. Indeed, since the number of observations in the dataset will likely differ among the 24 possible chords, training directly the model on the dataset may lead to overfit the model to a specific type of music (that means learning the characteristics of the dataset).

In order to learn the observation distribution for each of the 24 possible chords, we propose to first learn the model for the C Major chord and the C minor chord and then map the two trained models to all possible chords by circular permutation. A similar approach was proposed in [START_REF] Peeters | Musical key estimation of audio signal based on hmm modeling of chroma vectors[END_REF] in the case of key estimation. We proceed as follows:

1. All the chroma vectors of the labeled training dataset are mapped to a root-note of C using circular permutation.

2. The mean vector and the covariance matrix for the C Major (C minor) chord are computed from all C Major (C minor) chroma vectors.

3. The mean vectors and covariance matrices for all chords are obtained from the two trained models by circular permutation.

The mean vectors for the C Major and C minor chords trained on the dataset presented in 4.1 are represented in the left part of Figure 1. Note that in this case we do not make any assumption on the signal (instrumentation, harmonics, etc.) and we do not introduce any musical knowledge. In what follows, we will call this method "Method 1".

Method 2: Modeling by a multivariate Gaussian based on music theory considering the presence of higher harmonics

In this case, the observation distribution does not rely on any training on a given dataset. As in [START_REF] Bello | A robust mid-level representation for harmonic content in music signal[END_REF], the observation distribution relies directly on music theory; however a major difference with [START_REF] Bello | A robust mid-level representation for harmonic content in music signal[END_REF] is that we consider the presence of the higher harmonics of the theoretical notes in the construction of the multivariate Gaussian models (by modifying the parameters µ and Σ). This consideration allows us to significantly improve the results over the method proposed in [START_REF] Bello | A robust mid-level representation for harmonic content in music signal[END_REF].

In [START_REF] Bello | A robust mid-level representation for harmonic content in music signal[END_REF], the mean vectors and covariance matrices reflect musical knowledge. The mean vectors are 12-dimensional vectors with 1 if the note belongs to the chord and 0 otherwise. For instance, for a C Major chord (C-E-G), the mean vector will be 100010010000 (see middle-left part of Figure 1). In the covariance matrices, pitch which comprise the triad are more correlated than pitch which do not belong to the triad. The covariance between pitches which comprise the triad is thus given a non-zero value. The value is attributed with respect to music theory and empirical evidence from Krumhansl work [START_REF] Krumhansl | Cognitive Foundations of Musical Pitch[END_REF], that is to say that the dominant (fifth degree) is more important than the mediant (third degree) in characterizing the root of a triad 3 .

In this paper, we propose to take into account the contribution of the higher harmonics of the theoretical notes into the Gaussian parameters. We do this in the following way. Mean vectors: For each note of a chord, we add the contribution of the harmonics in the mean vectors. The amplitude contribution of the h th harmonic of a note is similar to the one proposed by [START_REF] Gomez | Tonal description of polyphonic audio for music content processings[END_REF]: 0.6 h-1 . Table 2 indicates the considered harmonics and the corresponding amplitudes for the C Major and the C minor templates. We represent the corresponding mean vectors for C Major and minor (in the case of 4 harmonics) in the middle-right part of Figure 1.

Tab. 2. The first 6 harmonics and their amplitude for a C Major (C minor) triad

Covariance matrices: [START_REF] Bello | A robust mid-level representation for harmonic content in music signal[END_REF] only considers the correlation between the chroma vectors corresponding to the pitch of the notes belonging to a given chord. In our method, we also consider the correlation between the harmonics of each note. For example, for a C Major chord (C-E-G), D is the 3 rd harmonic of G. Hence, we attribute a non-zero value to the covariance between D and G. As in [START_REF] Bello | A robust mid-level representation for harmonic content in music signal[END_REF], the values we use are heuristic but we still respect the rule that the dominant is more important than the mediant in characterizing the root of a triad 4 . The covariance matrices we propose for a C Major and a C minor chord are represented in Figure 2 above the covariance matrices proposed in [START_REF] Bello | A robust mid-level representation for harmonic content in music signal[END_REF]. In what follows, we will call this method "Method 2". 3 In [START_REF] Bello | A robust mid-level representation for harmonic content in music signal[END_REF], the covariance of the tonic with the dominant and of the dominant with the mediant is set to 0.8. The covariance of the tonic with the mediant is set to 0.6. Since we both use songs from the Beatles to evaluate our system, we will use the same values when testing method [START_REF] Bello | A robust mid-level representation for harmonic content in music signal[END_REF]. 4 The covariance of the tonic with the dominant is set to 0.6; the covariance of the dominant with the mediant is set to 0.5; the covariance of the tonic with the mediant is set to 0.3; the covariance of a note with its second harmonic is set to 0.1; the other non-zero values are set to 0.05. The matrix needs to be positive, semi-definite, so we set the non-triad diagonal members to 0.1. In this method, the observation probabilities are not modeled by a multivariate Gaussian distribution. They are obtained by computing the correlation between the observation vectors and a set of chord templates.

(a) C C# D D# E F F# G G# A A# B CM C#M DM D#M EM FM F#M GM G#M AM A#M BM (b) C C# D D# E F F# G G# A A# B CM C#M DM D#M EM FM F#M GM G#M AM A#M BM (c) C C# D D# E F F# G G# A A# B CM C#M DM D#M EM FM F#M GM G#M AM A#M BM 0 0.2 0.4 0.6 0.8 1 (d) C C# D D# E F F# G G# A A# B CM C#M DM D#M EM FM F#M GM G#M AM A#M BM
Chord templates: The chord templates are the theoretical chroma vectors corresponding to the 24 Major and minor triads. A chord template is a 12-dimensional vector which contains the theoretical amplitude values of the notes and their harmonics composing a chord. We consider 24 chord templates corresponding to the 24 Major and minor triads. The amplitude of a note in the template is non-zero if the note belongs to the considered chord (fundamental or harmonic). As in the case of the mean vectors in Method 2, we attribute an amplitude of 0.6 h-1 to the harmonic h. In section 4, we will compare the system results without considering any harmonic (nbh = 1), with 4 harmonics (nbh = 4) and with 6 harmonics (nbh = 6). In the right part of Figure 1, we represent the chord templates of C Major and C minor when considering 6 harmonics.

Observation probabilities: For each chroma vector, we compute the correlation between the vector and the 24 chroma templates. We obtain 24 values P (c i ), i ∈ [1, 24], normalized so that i P (c i ) = 1. We now have 24 "pseudoprobabilities" which are used as observation probabilities in the HMM. In what follows, we will call this method "Method 3".

State transition probability distribution

Method A: Theoretical approach using the doubly-nested circle of fifths This method has been proposed by [START_REF] Bello | A robust mid-level representation for harmonic content in music signal[END_REF]. The transition probability between two chords is derived from musical knowledge: their distance in the doubly-nested circle of fifths (see Figure 3). The doubly-nested circle of fifths depicts relationships among the 12 equal-tempered pitch classes comprising the chromatic scale. Although we do not know which state is going to follow another one, musical rules allow us to make hypotheses that are more probable than others. For instance, especially in popular western music, an A major chord is more likely to be followed by a F# minor or D major chord than by a G# Major chord. The corresponding state transition matrix is represented in the left part of Fig- Krumhansl studies the proximity between the various musical keys using correlations between key profiles obtained from perceptual tests. These correlations have been used by [START_REF] Noland | Key estimation using a hidden markov model[END_REF] to derive a key transition matrix in the context of local key estimation. In our experiments, we have achieved good results for chord estimation using the key transition matrix from [START_REF] Noland | Key estimation using a hidden markov model[END_REF] as a chord transition matrix (see middle part of Figure 4).

Method C: Trained approach using the EM algorithm

This approach uses the transition matrix provided by the training of the HMM using the Baum-Welsh (EM) algorithm, i.e. the system is trained using on the one side the succession of chroma vectors extracted from the audio signal and on the other side the corresponding chord labels. Although this approach is the usual one and the one used for example in [START_REF] Sheh | Chord segmentation and recognition using em-trained hmm[END_REF] [4], it did not provide satisfactory results in our evaluation.

Method D: Trained approach using the chord transcription

As opposed to the previous method, this approach is only based on symbolic information, i.e. the chord labels transcription of the training set. From the succession of transcribed chord labels over time, we derive an "annotation" transition matrix which is, as in the previous case, specific to the training set (in our case the Beatles corpus).

We want to learn from the training set the probabilities of transiting from one chord to another. We achieve this by counting the number of occurrences of each chord transition in the training set. Our goal is to construct a 24-dimensional matrix T that indexes all the chord transitions. However, be- cause the distribution of musical keys is not homogeneous in the training set, we are likely to favor specific chord transitions 5 , and therefore the transition matrix will be imbalanced. In order to face this problem, we only consider relative chord transitions (GM → CM transition is considered equivalent to CM → FM). We denote by T (i, j) the value of the transition matrix that represents the probability of transiting between chord i at time t -1 to chord j at time t. The indexes i, j ∈ [START_REF] Fujishima | Real-time chord recognition of musical sound: A system using common lisp music[END_REF][START_REF] Rabiner | A tutorial on hidden markov model and selected applications in speech[END_REF] represent the Major (M) chords, i, j ∈ [START_REF] Krumhansl | Cognitive Foundations of Musical Pitch[END_REF]24] the minor (m) chords. The matrix is therefore composed by four sub-matrices which represent transitions between M to M, m to m, M to m and m to M chords. These four cases are processed separately.

CMC#MDMD#MEM FMF#MGMG#MAMA#MBM cmc#mdmd#mem fm f#mgmg#mama#mbm CM C#M DM D#M EM FM F#M GM G#M AM A#M BM cm c#m dm d#m em fm f#m gm g#m am a#m bm CMC#MDMD#MEM FMF#MGMG#MAMA#MBM cmc#mdmd#mem fm f#m gmg#mama#mbm CM C#M DM D#M EM FM F#M GM G#M AM A#M BM cm c#m dm d#m em fm f#m gm g#
1. We first select from the training set all chord transitions belonging to a specific case (MM, mm, Mm, mM).

2. For each chord belonging to a given subset, we then compute the relative chord transitions. Each chord transition i → j is characterized by the equivalent transition from/to a root-note of C. We denote it by f (i, j).

3. We then form a 12-dimensional vector τ (k) by counting the number of relative chord transitions f (i, j) = k.

Using these vectors, we form the

T (1, k ∈ [1, 12]) (MM), T (13, k ∈ [13, 24]) (mm), T (1, k ∈ [13, 24]) (Mm), T (13, k ∈ [1, 12]) (mM).
5. The diagonal of the sub-matrices (self-transition) is processed in a separate way. We set the diagonal values to 1.1 max(τ (k))

6. The rest of the sub-matrices, are constructed by circular permutation.

7. We finally normalize the matrix T so that the sum of each row is equal to 1.

The resulting matrix trained on the dataset presented in section 4.1 is represented in the right part of Figure 4. It is interesting to observe the predominance (high transition values in the matrix) of typical transitions in the matrix, such as the II/V/I (transition between dm, GM and CM) which seems usual in this set of Beatles albums. However, the amount of transitions between Major and minor chords is much lower than the amount of transitions between two Major chords in this training set. The consequence of that, is a lower recognition rate for tracks with Major to minor chords.

Chord progression over time detection

In all cases (Method 1, 2, 3, A, B, C or D), the optimal succession of chords over time is found using the Viterbi decoding algorithm [START_REF] Gold | Speech and audio Signal Processing: Processing and Perception of Speech and Music[END_REF] which gives us the most likely path trough the HMM states given our sequence of chroma observations.

EVALUATION AND RESULTS

Test set and protocol

The system has been tested on a set of 110 hand-labeled files from the first eight albums of the Beatles. We have used this dataset since it allows a direct comparison to other publications. The chord annotations where kindly provided by C. Harte 6 . All the recordings are polyphonic, multi-instrumental songs containing drums and vocal parts. To our knowledge, it is the first attempt to evaluate a chord detection system on such a large dataset.

Since our chord lexicon only represents Major an minor triads, we have performed a mapping of complex chords in the annotation (such as Major and minor 6 ths , 7 ths , 9 ths , augmented and diminished chords) to their root triad. This point is important when analyzing the results.

Results

The results are indicated in Table 3.

• Res row corresponds to the exact recognition rate on all the frames (approximately 135.000 with the chosen parameters).

• Rct row represents the "close triads" recognition rate as discussed below. In this table, we compare the various methods for observation distribution and the choice of parameters:

• (Method 1) Gaussian observation distribution with training. For this method, the evaluation has been performed using a 8-folds cross-validation (each album was evaluated using the seven remaining albums as training data).

• (Method 2, nbh = 1) Gaussian observation distribution with music theory as proposed in [START_REF] Bello | A robust mid-level representation for harmonic content in music signal[END_REF].

• (Method 2, nbh = 4) Our proposal: Gaussian observation distribution with music theory considering the presence of four higher harmonics .

• (Method 3, nbh = 1, 4, 6) Our proposal: Observation distribution from correlation with templates combined with music theory considering the presence of one, four or six higher harmonics.

Note that we only present here the results obtained using method B for the transition matrix (see explanation below).

Tab. 3. Chord recognition rate using method 1, 2 and 3 for the observation distribution and method B (theoretical transition matrix based on correlation between key profiles) for the transition matrix. Res: exact chord recognition rate. Rcl: chord recognition rate including close triads. nbh: number of harmonics considered in the model

Analysis of results

Chord estimation method: The results obtained with the various methods are pretty close to each other. In our experiments, the best results were obtained with Method 3 (70.96%). Note that there was no training of the observation distribution in this case. Despite the fact that Method 1 uses training (and is therefore likely to fit very well to the characteristics of the Beatles), Method 2 with nbh = 4 (which does not use training at all) gives as high results 7 . Transition matrix: The best results were obtained using the theoretical transition matrix based on correlation between key profiles (Method B). The transition matrix based on the doubly-nested circle of fifths (Method A) gives slightly lower results. We do not present the results obtained with the two trained matrices (Methods C and D) because they are much lower for reasons explained in section 3.3.3 (overfitting to the dataset). Number of harmonics: Considering the presence of higher harmonics in the creation of the parameters clearly improves the results. For instance, for Method 3, considering 6 harmonics in the templates brings about 5% relative improvement. This is even clearer in the case of Method 2 where considering harmonics in the model's parameters brings about 12.5% relative improvement.

Discussion

Chords confusion due to ambiguous mapping: As it can be seen in Table 3, the standard deviation of the results is relatively high (up to 19%) independently from the chosen method. A deeper analysis of the results would show that the number of bad recognition comes from a reduced set of songs with partial or complex (non-triads) chords. For instance, for Love You To from Revolver, we obtain less than 3% of chords correctly identified. Provided annotation indicates that almost all the chords of this song but a few are Cmin(*b3) chords, i.e. a triad without the third note (C-G). In such case, it is difficult to make a decision between Major and minor chords in the absence of musical key information. Our system in fact has recognized for all cases Major chords which makes the recognition rate decrease. In future works, in order to avoid that, we plan to include information about the tonal context as it has been proposed in [START_REF] Shenoy | Key, Chord and Rhythm Tracking of Popular Music Recordings[END_REF] or [START_REF] Namunu | Automatic Structure Detection for Popular Music[END_REF].

As mentioned before, because of our limited chord dictionary, a mapping was performed between complex chords and their root triad. Chords of four notes often contain other triads than their root triad. For instance, a Cmin7 (C-Eb-G-Bb) contains C minor (C-Eb-G) chord and Eb Major (Eb-G-Bb) chord. Whereas the majority of songs in the evaluation dataset are composed of triad chords, some of them contain a lot of more complex chords, and it sometimes happens that the system recognizes other triad than the root triad of the complex chord analyzed, which makes also the recognition rate decrease.

Neighboring triad confusion: Most of the errors that occur correspond to harmonically close triad confusions: -parallel Major/ minor chords (EM being confused with em),relative chords (am being confused with CM), -dominant chords (CM being confused with GM) or -subdominant chords (CM being confused with FM). If the system does not recognize exactly a chord but makes such confusions, the result can still be useful for higher-level structural analysis such as key estimation, harmony progression or segmentation. Table 3 shows that if we consider close triads recognition as correct, the recognition rate of method 3 reaches up to 86%. It also becomes now the method with the smallest standard deviation, 9%. Limitation of chroma-based approach for inharmonic sounds: It is interesting to notice that we obtain much better results for the five first Beatles albums than for the last three (from the "Norwegian Wood (This Bird Has Flown)" on 1965's Rubber Soul). The reason for this, comes from the extended use of the Indian sitar instrument8 which produces highly inharmonic sounds. Since the chroma-based approach strongly relies on the presence of harmonic sounds it is not appropriate to use it for such music.

CONCLUSION AND FUTURE WORKS

In this paper we have proposed and compared several methods for the automatic estimation of chord progression of a music piece. All the methods are based on chroma representation of the audio signal and on modeling of the sequence of observation using a hidden Markov model. The methods have been compared on a large-scale evaluation. The best results were obtained with the modeling of the observation probabilities using a normalized correlation with a set of extended chord templates and a cognitive-based transition matrix. The templates are extended by considering the presence of higher harmonics of each pitch note of a chord. The transition matrix is derived from cognitive experiments on the perception of musical key. Current limitations of our system mainly come from the confusion between the various interpretation one can make about chords. A solution to that is to integrate extra (context) information such as musical key information. The integration of metrical information could also increase the robustness of the system.

  3.3, we study four approaches to define the transition matrix. Method A is based on music theory: the closeness of chords in the doubly-nested circle of fifths. Method B uses the results of cognitive experiments: the closeness of chords using Krumhansl's key profiles. Method C learns the transitions probabilities from the HMM training. We finally propose a new method, D, which learns the transitions from score transcriptions.

Fig. 2 .

 2 Fig. 2. Covariance matrices for a C Major and a C minor chord considering the presence of 4 harmonics (upper part, (a) and (b)) and proposed covariance matrices in [4] (bottom part, (c) and (d))

6 Fig. 1 .

 61 Fig. 1. Mean chroma vectors for the C Major (upper par of each figure) and C minor (lower part) chords using [from left to right]: Method 1 (trained using 7 CDs of the Beatles), Method 2 without harmonic contribution, Method 2 with 4 harmonics contribution, Method 3 with 6 harmonics contribution (in this case, the figures represent the chroma templates instead of the mean vectors).

  ure 4.

Fig. 3 .

 3 Fig. 3. Doubly-nested circle of fifths. From [4] Method B: Cognitive approach using correlation between key profiles

  #MEM FMF#MGMG#MAMA#MBM cmc#mdmd#mem fm f#mgmg#mama#

Fig. 4 .

 4 Fig. 4. State transition probability distribution between the 12 Major and minor chords using [from left to right]: method A, method B, and method D

A bit mask is a 12-dimensional vector corresponding to the 12 semitones of the pitch classes with 1 when the note belongs to the chord, 0 otherwise.

Smoothing of the semitone pitch spectrogram strengthens spectral envelope continuity, a physical property; while smoothing on the chromagram does not rely on any physical property. That is why the filtering is performed on the notes rather than on the chroma vectors.

For instance, if 90% of the training set is in C Major we are more likely to observe a II/V/I transition in C Major, i.e. dM/GM/CM, than a II/V/I transition in F#M, i.e. g#m/D#M/F#M.

www.elec.qmul.ac.uk/digitalmusic/

It should be noted however that we did not recover the high results given in[START_REF] Bello | A robust mid-level representation for harmonic content in music signal[END_REF] with Method 1 and nbh = 1 which is very close to the one presented in[START_REF] Bello | A robust mid-level representation for harmonic content in music signal[END_REF] and tested on the same dataset.

Sitar instrument is a stringed instrument that uses sympathetic strings along with regular strings. This produces very lush sound with inharmonic components.

ACKNOWLEDGEMENTS

Part of this work was conducted in the context of the French RIAM project "Ecoute".


Hélène Papadopoulos

Geoffroy Peeters
email: peeters@ircam.fr

SIMULTANEOUS ESTIMATION OF CHORD PROGRESSION AND DOWNBEATS FROM AN AUDIO FILE

Keywords: HMM, Chroma, Chord, Downbeat, Metrical structure

Harmony and metrical structure are some of the most important attributes of Western tonal music. In this paper, we present a new method for simultaneously estimating the chord progression and the downbeats from an audio file. For this, we propose a specific topology of hidden Markov models that allows us to model chords dependency on metrical structure. The model is evaluated on a dataset of 66 popular music songs from the Beatles and shows improvement over the state of the art.

INTRODUCTION

Musical signals are highly structured in terms of harmony and rhythm. Thus, these two components are essential in the understanding of music. Harmonic analysis and rhythm analysis find many applications within the context of music information retrieval such as music classification, structural audio segmentation or in general all applications based on music content analysis. In Western tonal music, the chord progression determines the harmonic structure of a piece of music. It is strongly related to the metrical structure of the piece [START_REF] Goto | An audio-based real-time beat tracking system for music with or without drum sounds[END_REF]. The meter is "the sense of strong and weak beats that arises from the interaction among hierarchical level of sequences having nested periodic components" [START_REF] Eck | A graphical model for chord progressions embedded in a psychoacoustic space[END_REF]. In a piece of music, each chord is locally related to the surrounding chords according to the harmonic progression of the piece (local dependency). Furthermore, the position that a chord occupies in a measure or more generally in the global metrical structure has to be taken into account (global dependency). For example, chords will change more often on strong beats than on other positions. This musical characteristic has already been explored in previous works ( [START_REF] Maddage | Automatic Structure Detection for Popular Music[END_REF], [START_REF] Shenoy | Key, chord and rhythm tracking of popular music recordings[END_REF] [2], [START_REF] Raphael | Harmonic analysis with probabilistic graphical models[END_REF], [START_REF] Thornburg | Detection and Modeling of Transient Audio Signals with Prior Information[END_REF]).

Metrical level is a hierarchical structure. The most salient metrical level, called the tactus or beat level, is a moderate metrical level which corresponds to the foot tapping rate. Here, we will also consider another common metrical level called tatum. The tatum level corresponds to the "shortest durational values in music that are still more than accidentally encountered " [START_REF] Klapuri | Analysis of the meter of acoustical musical signals[END_REF]. Musical signals are divided into units of equal time value called measures or bars. The relationship between measures and tactus/tatum is defined by the meter which is usually indicated by a time signature. One important problem related to metrical analysis is finding the position of the downbeat or the first beat of each measure.

In the last few years, there has been an increasing interest in modeling higher-level information with low-level signal features in Thanks to ANR Projet Ecoute and AII Project Quaero for funding. the context of music analysis. Two paths have been explored. On the one hand, hierarchical frameworks based on rule-based approach have been proposed (see for instance [START_REF] Maddage | Automatic Structure Detection for Popular Music[END_REF], [START_REF] Shenoy | Key, chord and rhythm tracking of popular music recordings[END_REF]). On the other hand, statistical framework including graphical models and Bayesian approach have been proposed (see [START_REF] Eck | A graphical model for chord progressions embedded in a psychoacoustic space[END_REF], [START_REF] Cemgil | Bayesian Music Transcription[END_REF], [START_REF] Temperley | Bayesian models of musical structure and cognition[END_REF], [START_REF] Smith | Bayesian identification of closely-spaced chords from single-frame stft peaks[END_REF], [START_REF] Thornburg | Detection and Modeling of Transient Audio Signals with Prior Information[END_REF]). Statistical approaches are more flexible than rule-based approaches and offer large opportunities to explore the interaction between low-level features with high-level music information. Our purpose is to show how the metrical information and the harmonic information of a piece of music interact and how this can be used into a mutually informing manner to improve both the estimation of the chord progression and the downbeat positions. For this, we propose a specific topology of HMM that allows us to extract simultaneously the chord progression and the downbeats from an audio file. Our approach is somehow related in spirit to Bayesian modeling. Indeed, we intend to model global dependencies within the chords. Although HMM usually concentrate on local dependencies, it is not the case here.

The paper is organized as follows. In section 2.1, we present the extraction of a set of meter-related feature vectors that represent the audio signal. We introduce a probabilistic model for simultaneous chord progression and downbeat positions estimation in section 2.2. In section 3, the proposed model is evaluated on a set of hand-labeled songs of the Beatles.

MODEL

In order to extract the chord progression and the downbeats from the audio signal, one first needs to extract a set of meter-related feature vectors that describe the signal. Pitch Class Profiles [START_REF] Fujishima | Real-time chord recognition of musical sound: A system using common lisp music[END_REF] or chromabased representation [START_REF] Wakefield | Mathematical representation of joint time-chroma distribution[END_REF] have become common features to automatically estimate chords or musical key from audio recordings ( [START_REF] Sheh | Chord segmentation and recognition using em-trained hmm[END_REF], [START_REF] Bello | A robust mid-level representation for harmonic content in music signal[END_REF], [START_REF] Harte | Automatic chord identification using a quantised chromagram[END_REF], [START_REF] Peeters | Chroma-based estimation of tonality from audio-signal analysis[END_REF], [START_REF] Gomez | Tonal description of polyphonic audio for music content processings[END_REF]). PCP/chroma vectors represent the intensity of the twelve semitones of the pitch classes. The chord progression is represented using a hidden Markov model that takes into account global dependencies on meter. The tactus/tatum positions have been extracted using the method proposed in [START_REF] Peeters | Template-based estimation of time-varying tempo[END_REF]. In our evaluation, we have only considered songs with 100% tactus recognition rate.

Our model is general and could be applied to songs with any kind of time-signature (3/4, 4/4, . . . ). However, because of dataset availability, we have concentrated our evaluation on the case of popular music and limited our experiments to songs built on four-beat meters (most common case in popular music). We will assume that the time signature is known (4/4) and constant. We will also assume that chord changes can only occur on beats or after beats. Theses hypothesis correspond to the characteristics of a wide part of popular music. For instance, if we consider the first eight CDs of the Beatles (110 songs), only 3 songs do not fit the assumptions we made.

Features extraction

We work directly on the audio signal. In our analysis, the signal is down-sampled to 11025Hz, converted to mono and converted to the frequency domain by a DFT using a Blackman window of length 0.48s with 25% overlap. Because of frequency resolution limits (the frequency distance between adjacent semitone pitches becomes small in low frequencies), we only consider frequencies above 60Hz. The upper limit is set to 1kHz because the fundamentals and harmonics of the notes in popular music are usually stronger than the non-harmonic components up to 1kHz [START_REF] Maddage | Automatic Structure Detection for Popular Music[END_REF]. This choice is also supported by the fact that the mapping operated between the energy of the harmonics and the chroma vectors is only valid for the lowest harmonics, hence the lowest part of the spectrum. The tuning of the track is estimated using the method proposed in [START_REF] Peeters | Chroma-based estimation of tonality from audio-signal analysis[END_REF]. The signal is then re-sampled so that the rest of the system can be based on a tuning of the standard A4 = 440Hz. The temporal sequence of chroma vectors over time is known as chromagram. It is computed using the method proposed in [START_REF] Peeters | Chroma-based estimation of tonality from audio-signal analysis[END_REF]. First, the values of the DFT are mapped to a semitone pitch spectrum using the mapping function:

n(f k ) = 12 log 2 ( f k 440 ) + 69, n ∈ ℜ +
, where f k are the frequencies of the Fourier transform and n correspond to the semitone pitch scale values. Then, the semitone pitch spectrum is smoothed over time using a median filtering. This provides a reduction of transients and noise in the signal. Finally, after this smoothing, the semitone pitches n are mapped to the the semitone pitch classes c using the mapping function: c(n) = mod (n, 12). We obtain a sequence of 12-dimensional vectors that are suitable feature vectors for our analysis. Tactus/tatum-related chroma vectors: Since we want to study the relationship between chords and metrical structure, we need to deal with observation features that are related to the meter. The frame by frame analysis does not fit our needs: we need to proceed to a beat related analysis. To this end, the chromagram is averaged so that we obtain one feature per tactus/tatum 1 . In our study, we have considered two cases. The chromagram has been averaged with respect to the beats or quarter notes (tactus) in the first case, and with respect to the eighth notes (tatum) in the second case. We will further discuss the relevance of both approaches.

Chord progression and downbeat estimation from the chroma vectors using a "double state" HMM

Overview of the model

We consider an ergodic I * K-states HMM where each state s ik is defined as an occurrence of a chord ci, i ∈ [1 : I] at a "position in the measure" (position of a beat or tatum inside a measure)

pim k , k ∈ [1; K]: s ik = [ci, pim k ].
In our experiments, our chord lexicon is composed of I = 24 Major and minor triads (C Major, . . . , B Major, C minor, . . . , B minor). We assume that chord changes can only occur on beats or on after beats. The positions in the measure where chord changes occur will be referred to "position in the measure" and denoted by pim. For a song built on four-beats meter, K = 4 if we consider the tactus-level (k ∈ [1; 4] for a 4/4 measure) and K = 8 if we consider the tatum-level (k ∈ [1; 8] for a 4/4 measure). If there are K possible pim in a measure, the total number of states is thus I chords by K pim i.e. I * K states. Each state in the model generates with some probability an observation vector O(tm) at time tm. This is defined by the observation probabilities. Given the observations, we estimate the most likely chord sequence over 1 The tactus/tatum positions are considered as inputs of our system. time and the downbeat positions in a maximum likelihood sense. In what follows, we denote by π and T the initial state distribution and state transition probability distribution.

Initial state distribution π

The prior probability π ik for each state is the prior probability to observe a specific chord i occurring on pim k. Since we do not know a priori which chord the piece begins with and which pim the piece starts with, we initialize π at 1 I * K for each of the I * K states.

Observation probabilities P (O(tm)|s ik )

The observation probabilities are computed in the following way. Let P (O(tm)|s ik ) denote the probability that observation O(tm) has been emitted at time instant tm given that the model is in state s ik . Let P (O(tm)|ci) denote the one that it has been emitted by chord ci and P (O(tm)|pim k ) the one that it has been emitted given that the chord is occurring on pim k. As said before, we rely upon the assumption that chord changes are more likely to occur at the beginning of measures than at other pim. We now assume independence between chord type (CM, C#M, . . . , cm, . . . , bm) and pim.

For instance, we consider that in any given song, even if we favor chord changes on pim = 1, we do not favor any chord type: a D major chord is as likely to occur at the beginning of a measure as a C major chord. The observation probabilities are computed as:

P (O(tm)|s ik ) = P (O(tm)|ci)P (O(tm)|pim k ) (1)
Observation chord symbol probability distribution: The observation chord symbol probabilities P (ci|O(tm)) are obtained by computing the correlation between the observation vectors (the chroma vectors) and a set of chord templates which are the theoretical chroma vectors corresponding to the I = 24 major and minor triads. For more details, see [START_REF] Papadopoulos | Large-scale study of chord estimation algorithms based on chroma representation and hmm[END_REF].

Observation pim probability distribution: The pim probability distribution P (pim k |O(tm)) is considered here as uniform ( 1 K for each pim in the measure). Note that this probability distribution could be derived from information given by the signal. Future works will concentrate on that.

State transition probability distribution T

The main reason why the problem is modeled using a Markov model is that in music pieces, the transitions between chords result from musical rules. Using a Markov model, we can model these rules in the state transition matrix T . According to [START_REF] Goto | An audio-based real-time beat tracking system for music with or without drum sounds[END_REF], chords are more likely to change at the beginning of a measure than at other pim. Starting from this statement, we detect the downbeats by giving lower self-transition probabilities in the state transition matrix for chords occurring on the K th beat.

The I * K-states transition matrix T used in our HMM takes into account both the chord transitions and their respective positions in the measure. It is derived from a I-states chord transition matrix Tc based on music-theoretical knowledge about key-relationships. We refer the reader to [START_REF] Papadopoulos | Large-scale study of chord estimation algorithms based on chroma representation and hmm[END_REF] for more details. We note Tc(i, i ′ ) the transition probability between chord i and chord i ′ . This matrix is represented in Figure 1 [left].

We also define a pim transition matrix Tpim which represents the probability to transit from pim k to pim k ′ . Since we do not allow our present system to jump over a pim (i.e. skip over or add one or several beats), only the values Tpim(k, k ′ ) for k ′ = k + 1[K] are non-zero. All non-zero values are set to the same value. This matrix is represented in Figure 1 [right,top].

We need here to distinguish between two cases: the first case concerns transitions between two different chords (i ′ = i), the second case concerns self-transitions (i ′ = i) and corresponds to the diagonal blocks of T . Since we want to favor chord changes on downbeats, i.e. disfavor self-transition between the last pim of a measure and the first pim of the next measure, we need to define a specific transition matrix for the self-transition case (i ′ = i). This specific matrix is denoted by T ′ pim . This matrix is represented in Figure 1 [right, bottom]. As one can see T ′ pim differs from Tpim only in the value T ′ pim (K, 1) which is lower than Tpim(K, 1). The consequence of this lower value is that T ′ pim disfavors transition between identical chords (self-transition) at measure boundaries. In our experiments (case 4/4 time-signature), we have attributed empirical values to T ′ pim (k, k ′ ), k, k ′ ∈ [1; 4] with respect to the fact that we want to favor chord changes on downbeats 2 . Note that these values could be learned from the dataset by counting the proportion of chord changes on each measure position in the dataset.

From Tc, Tpim and T ′ pim , we construct the global transition matrix T normalized so that the sum of each row is equal to 1 (Figure 1 [middle]). Each block B ii ′ (k, k ′ ) of this matrix represents the transition from chord i at pim k to chord i ′ at pim k ′ :

 B ii ′ (k, k ′ ) = Tc(i, i ′ ) • Tpim(k, k ′ ) if i = i ′ , = Tc(i, i ′ ) • T ′ pim (k, k ′ ) if i = i ′

Chord progression and downbeats detection

The optimal succession of states [ci, pim k ] over time is found using the Viterbi decoding algorithm [START_REF] Gold | Speech and audio Signal Processing: Processing and Perception of Speech and Music[END_REF] which gives us the most likely path trough the HMM states given our sequence of observations. It gives us simultaneously the best sequence of chords over time and the downbeat positions.

EVALUATION AND RESULTS

Test set

The proposed model has been tested on a set of 66 hand-labeled songs of the Beatles 3 . All the songs are built on four-beat meter with constant time signature. The chord annotations were kindly provided by C. Harte from QMUL. Note that since our chord lexicon only represents major and minor triads, we have performed a mapping of complex chords in the annotation (such as major and minor 6 ths , 7 ths , 9 ths , augmented and diminished chords) to their root triads. The tactus were obtained using the method proposed in [START_REF] Peeters | Template-based estimation of time-varying tempo[END_REF]. The ground-truth downbeats have been annotated by hand by the authors. All the recordings are polyphonic, multi-instrumental songs containing drums and vocal parts.

Overall results

The results are indicated in Table 1. Let S denote the total number of songs in the dataset and let consider a song s divided into N frames. Each signal frame (tactus-frame or tatum-frame) of the ground truth has been mapped to a chord of our lexicon. Let Ĉn, n ∈ [1; N ] denote the theoretical chord corresponding to frame n and let Cn 3 The list of the tracks can be found at the following URL: http://recherche.ircam.fr/equipes/analyse-synthese/papadopoulos/. denote the estimated chord at that frame. We compute the correct chord recognition rate for song s as:

2 T ′ pim (1, 2) = T ′ pim (2, 3) = T ′ pim (3, 4) = 1.1 , T ′ pim (4, 1) = 0.85, T ′ pim (k, k ′ ) = 0 otherwise.
µs = 1 N P n∈[1;N] (Cn = Ĉn) (2)
The results we give in Table 1 correspond to the mean and standard deviation of correctly identified frames per song:

µ = 1 S P s∈[1;S] µs and σ = 1 S q P s∈[1;S] (µ -µs) 2 (3)
• NM (No Meter)/WM (With Meter) columns correspond to the exact recognition rate on all the frames without/when taking into account chords dependency on the metrical structure in the model. TAC corresponds to a tactus-frame analysis, TAT to a tatum-frame analysis.

• DU (Downbeat Unknown) row corresponds to the case where downbeats are estimated simultaneously with the chords. DK (Downbeat Known) row corresponds to the case where downbeats are given by manual annotation. In this case, we only intend to evaluate the influence of the knowledge of downbeats on chord recognition.

• D corresponds to the percentage of songs where the downbeats have been correctly estimated. Note that the dataset contains only pieces without skipped or added beats. For a given song, the estimated downbeats are then either all correct or all incorrect.

Analysis of the results

Downbeat estimation:

The percentage of downbeats correctly estimated is encouraging. It achieves 92% (79% ) of correct estimation in the case of tactus-frame (tatum-frame) analysis. Note that without chord information, the downbeats estimation would be 25% for tactus-frame analysis and 12.5% for tatum-frame analysis. Importance of the knowledge of the downbeat positions: In Table 1 [left], we can see that the chord estimation task benefits from the knowledge of the downbeat positions either given manually (DK) or estimated through the model (DU). Taking into account the pim of the chords in the measures allows to improve the chord recognition task by 4.9% relative improvement in the case of tactus-frame analysis and 3.5% relative improvement in the case of tatum-frame analysis. It is important to note that when we perform simultaneous estimation of chord progression and downbeats, the global rate for chord recognition is better than when we do not take into account the influence of the metrical structure, even if the downbeats are not correctly estimated in all the pieces. Tactus-frame/tatum-frame analysis: Table 1 indicates that the tatum-frame analysis performs better in general than the tactusframe analysis. Some chords in our dataset do not change exactly on the beats (voice effects, after beats). The tatum-frame analysis allows to take into account chord changes on more positions than the tactus-frame analysis and thus gives better results. Chord changes and boundaries errors: The example in Figure 2 clearly shows how the chord progression estimation task can benefit from modeling chords dependency on the metrical structure. This piece is in C major key and it transits between C major and G major chords about every two measures (truth line). Without taking into account global dependencies (NM line), chord transitions are badly detected and the estimated chord progression remains almost all the time on G major chord instead of transiting between G major and C major. The knowledge of downbeat positions (WM line) allows to better detect transitions. Furthermore, using the chords dependency on the metrical structure also allows to improve the exact location of chord changes (boudaries). Without taking into account this dependency, chord changes are often detected a beat before or after their theoretical positions.

CONCLUSION AND FUTURE WORK

In this paper, we have proposed a method that allows to estimate simultaneously the chord progression and the downbeats from an audio file. This method has been evaluated on a large set of handlabeled files and gives very encouraging results. From this evaluation, we can state that the chord progression of a piece of music benefits from the knowledge of downbeat positions and conversely that the downbeats of popular music songs can be estimated using harmonic information. An interesting result from our evaluation is that tatum-related analysis is better than tactus-related analysis for the estimation of chord progression. Future works will consist in evaluating the performances of our model on songs with other time signatures. Including a time-signature estimation algorithm in our system will allow us to deal with pieces with meter changes.

Fig. 1 .

 1 Fig. 1. Chord transition matrix for a singles-state HMM [left], transition matrices for Major to Major chords in the case of double-states HMM, without taking into account the pim of the chord in the measure [middle left] and taking into account the pim of the chord [middle right], pim transition matrices [right].

Fig. 2 .

 2 Fig. 2. Chord progression for the song Love Me Do not considering a)/considering b) chords dependency on meter, ground truth c).

Table 1 .

 1 Chord estimation rate and downbeat estimation rate.