MULTIPLE VIEWPOINT MELODIC PREDICTION WITHFIXED-CONTEXT NEURAL NETWORKS

Srikanth Cherla1,2, Tillman Weyde1,2 and Artur d’Avila Garcez21Music Informatics Research Group, Department of Computer Science, City University London

2 Machine Learning Group, Department of Computer Science, City University London{srikanth.cherla.1, t.e.weyde, a.garcez}@city.ac.uk

ABSTRACT

The multiple viewpoints representation is an event-basedrepresentation of symbolic music data which offers a meansfor the analysis and generation of notated music. Previ-ous work using this representation has predominantly re-lied on n-gram and variable order Markov models for mu-sic sequence modelling. Recently the efficacy of a classof distributed models, namely restricted Boltzmann ma-chines, was demonstrated for this purpose. In this paper,we demonstrate the use of two neural network models whichuse fixed-length sequences of various viewpoint types asinput to predict the pitch of the next note in the sequence.The predictive performance of each of these models is com-parable to that of models previously evaluated on the sametask. We then combine the predictions of individual mod-els using an entropy-weighted combination scheme to im-prove the overall prediction performance, and compare thiswith the predictions of a single equivalent model whichtakes as input all the viewpoint types of each of the indi-vidual models in the combination.

1. INTRODUCTION

We are interested in the computational modelling of melo-dies available in symbolic music data formats such as MIDIand KERN. For this purpose, we chose to work with a rep-resentation of symbolic music first proposed in [9] in rela-tion to multiple viewpoints for music prediction (which werefer to here as the “multiple viewpoints representation”).The multiple viewpoints representation is an event-basedrepresentation extracted from symbolic music data wherea given piece of music is decomposed into parallel streamsof features, known as viewpoint types. Each viewpoint typeis either a directly observable musical dimension such aspitch and note duration, or an abstract one derived fromthem such as inter-onset interval or pitch contour. In or-der to analyse musical structure using this representation,one can train a machine learning model on sequences of

c© Srikanth Cherla1,2, Tillman Weyde1,2 and Arturd’Avila Garcez2.Licensed under a Creative Commons Attribution 4.0 International Li-cense (CC BY 4.0). Attribution: Srikanth Cherla1,2, Tillman Weyde1,2

and Artur d’Avila Garcez2. “Multiple Viewpoint Melodic Prediction withFixed-Context Neural Networks”, 15th International Society for MusicInformation Retrieval Conference, 2014.

viewpoint types and apply it to tasks such as music gener-ation [6] and classification [3, 7]. This representation hasalso been the focus of more recent work related to musiccognition [14,17]. The novelty of this approach is in its ex-tension of previous work in language modelling to musicwith an information theoretic backing which facilitates anobjective evaluation of models for music prediction. Ap-proaches based on information theory have been of interestin musicology to understand structure and meaning in mu-sic in terms of its predictability [10, 11, 13].

In the original work on multiple viewpoints [9] and thatwhich followed [15, 21], Markov models were exclusivelyemployed for music modelling using this framework. Whilethis is a reasonable choice, Markov models are often facedwith a problem related to data sparsity known as the curseof dimensionality [2]. This refers to the exponential rise inthe number of model parameters to be estimated with thelength of the modelled sequences. Models which employdistributed architectures such as neural networks tend toavoid this problem, as they do not require enumerating allstate transition probabilities, but rather the weights of thenetwork encode only those dependencies necessary to min-imize prediction error. It was demonstrated more recentlyin [4] how a distributed model — the restricted Boltzmannmachine, is a suitable alternative in this context. It was alsosuggested in [8] that neural networks might be suitable al-ternatives to n-gram models for music modelling with mul-tiple viewpoints but no actual research in this direction hasensued.

In this paper, we first present two neural networks formodelling sequences of musical pitch. The first is a sim-ple feed-forward neural network [20], and the second isthe musical extension of the Neural Probabilistic LanguageModel [2] — a deeper feed-forward network with an addedweight-sharing layer between the input and hidden lay-ers. The latter was originally proposed for learning dis-tributed representations of words in language modelling.Both models predict a probability distribution over the pos-sible values of the next pitch given a fixed-length contextas input. Their predictive performance is comparable to orbetter than previously evaluated melody prediction mod-els in [4, 16]. The second network is further extended tomake use of additional viewpoint types extracted from thecontext, as inputs for the same task of predicting musi-cal pitch. We then combine the predictions of individualmodels with different viewpoint types as their respective

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inputs using an entropy-weighted combination scheme toimprove the overall prediction performance, and comparethis with the predictions of a single model which takes asinput all the viewpoint types of each of the individual mod-els in the combination.

We begin with an overview of the multiple viewpointsrepresentation in Section 2. This is followed by a descrip-tion of the two neural networks which are used with thisrepresentation, in Section 3. Section 4 presents an evalua-tion of the predictive performance of the two models alongwith a comparison to previous work. Finally, directions forfuture research are outlined in Section 5.

2. MULTIPLE VIEWPOINT SYSTEMS

In order to explain music prediction with multiple view-points, the analogy to natural language is used here. Instatistical language modelling, the goal is to build a modelthat can estimate the joint probability distribution of subse-quences of words occurring in a language L. A statisticallanguage model (SLM) can be represented by the condi-tional probability of the next word wT given all the previ-ous ones [w1, . . . , w(T−1)] (written here as w(T−1)

1 ), as

P (wT1 ) =T∏t=1

P (wt|w(t−1)1 ) . (1)

The most commonly used SLMs are n-gram models,which rely on the simplifying assumption that the proba-bility of a word in a sequence depends only on the imme-diately preceding (n− 1) words [12]. This is known as theMarkov assumption, and reduces (1) to

P (wT1 ) =T∏t=1

P (wt|w(t−1)(t−n+1)) . (2)

Following this approach, musical styles can be inter-preted as vast and complex languages [9]. In predictingmusic, one is interested in learning the joint distributionof musical event sequences sT1 in a musical language S.Much in the same way as an SLM, a system for music pre-diction models the conditional distribution p(st|s(t−1)1 ), orunder the Markov assumption p(st|s(t−1)(t−n+1)). For eachprediction, context information is obtained from the eventss(t−1)(t−n+1) immediately preceding st. Musical events have a

rich internal structure and can be expressed in terms of di-rectly observable or derived musical features such as pitch,note duration, inter-onset interval, or a combination of twoor more such features. The framework of multiple view-point systems for music prediction [9] was proposed in or-der to efficiently handle this rich internal structure of musicby exploiting information contained in these different mu-sical feature sequences, while at the same time limiting thedimensionality of the models using these features. In theinterest of brevity, we limit ourselves to an informal discus-sion of multiple viewpoint systems for monophonic musicprediction and refer the reader to [9] for a more detailedexplanation.

A musical event s refers to the occurrence of a note in amelody. A viewpoint type (or simply type) τ refers to anyof a set of musical features that describe an event. The do-main of a type, denoted by [τ ] is the set of possible valuesof that type. A basic type is a directly observable or givenfeature such as pitch, note duration, key-signature or time-signature. A derived type can be derived from any of thebasic types or other derived types. Two or more types canbe “linked” by taking the Cartesian product of their respec-tive domains, thus creating a linked viewpoint type. A mul-tiple viewpoints system (MVS) is a set of models, each ofwhich is trained on subsequences of one type, whose indi-vidual predictions are combined in some way to influencethe prediction of the next event in a given event sequence.Given a context s(t−1)(t−n+1) and an event st, each viewpoint τ

in an MVS must compute the probability pτ (st|s(t−1)(t−n+1)).In order to input the viewpoint type sequences to the

neural network models, we first convert each input typevalue into a binary one-hot encoding. When a contextevent is missing or undefined, each element of the vectoris initialized to 1/|S|. When there is more than one inputtype, one-hot vectors corresponding to all the input typesfor a musical event are concatenated to obtain an input vec-tor for that event. As we are dealing with models of fixedcontext-length l, the final input feature vector input to themodel is a concatenation of l such vectors. In doing so, weare effectively bypassing the need to compute a Cartesianproduct to link viewpoint types before using them as inputto a single model which has been the practice when usingn-gram and variable order Markov models.

Each model in an MVS relies on a different source of in-formation (its respective input types) to make a predictionabout the target viewpoint type. The accuracy of the pre-diction depends on how informative these input types areof the target type. It is possible to combine the informationprovided by different input types for possibly better pre-dictive performance. Here, we consider two ways of doingthis - implicitly in a single model which is trained using aset of input types, and explicitly by combining the prob-ability distributions of multiple models, each of which istrained separately on a mutually exclusive subset of theseinput types. While the former is only a special case of whathas been described so far, we provide an explanation of thelatter below in Section 2.1.

2.1 Combining Multiple Models

It was demonstrated in [9, 15] that an entropy-weightedcombination of the predictions of two or more n-gram orvariable order Markov models typically results in ensem-bles with better predictive performance than any of the in-dividual models. As it is the predicted distributions whichare combined, this approach is independent of the types ofmodels involved. Here, we briefly describe two approachesfor creating such ensembles. Let M be a set of models andpm(s) be the probability assigned to symbol s ∈ [τtgt] bymodel m, where [τtgt] is the domain of the target type.The first approach involves taking a weighted arithmeticmean of their respective predictions. This is the mixture-

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of-experts combination, and is defined as

p(s) =

∑m∈M wmpm(s)∑

m∈M wm

where each of the weights wm depends on the entropy ofthe distribution generated by the corresponding model min the combination such that greater entropy (and henceuncertainty) is associated with a lower weight [5]. Theweights are given by the expression wm = Hrel(pm)−b,where the relative entropy Hrel(pm) is

Hrel(pm) =

{H(pm)/Hmax(pm), if Hmax([τtgt]) > 0

1, otherwise

The best value of the bias b is determined through cross-validation. The quantities H and Hmax are respectivelythe entropy of the prediction and the maximum entropy ofpredictions over the symbol space [τtgt], and are defined as

H(p) = −∑

s∈[τtgt]

p(s) log2 p(s) . (3)

Hmax(p) = log2 |S|.

where p(s ∈ [τtgt]) = p(χ = s) is the probability massfunction of a random variable χ distributed over the dis-crete alphabet [τtgt] such that the individual probabilitiesare independent and sum to 1.

The second combination method — product-of-experts,is computed similarly as the weighted geometric mean ofthe probability distributions. This is given by

p(s) =1

R

( ∏m∈M

pm(s)wm

) 1∑m∈M wm

whereR is a normalisation constant which ensures that theresulting distribution over S sums to unity. The weightswm in this case are obtained in the same manner as forthe mixture-of-experts case. It was observed in a previousapplication of these two combination methods to melodymodelling [15], that product-of-experts resulted in a greaterimprovement in predictive performance.

3. FIXED-CONTEXT NEURAL NETWORKS

In this section, we provide a brief overview of the twofixed-context neural network models which we employedfor the task of predicting the pitch of the next note in amelody, given a viewpoint type context which leads up toit. These are (1) a feed-forward neural network, and (2)a neural probabilistic melody model. The key differencebetween the two is the presence of an additional weight-sharing layer in the latter which transforms the binary rep-resentation of the viewpoint types into lower-dimensionalreal-valued vectors before passing these on as inputs to afeed-forward network (much like the former).

. . . y

. . . h

. . . . . . . . . x

W(0)

W(1)

(a) Feed-forward Neural Network. . . y

. . . h

. . . . . . . . . v

. . . . . . . . . x

W(c) W(c)

W(0)

W(1)

(b) Neural Probabilistic Melody Model

Figure 1: The two models employed for multiple view-point melodic prediction in this paper (biases ignored inthe illustration). A concatenation of the fixed-length inputtype context is presented to each model in its visible layerand the predictions are made in the output layer.

3.1 Feed-forward Neural Network

In its simplest form, a feed-forward neural network (Fig-ure 1) consists of an input layer x ∈ Rn, a hidden layerh ∈ Rm and an output layer y ∈ Rl. The input layeris connected to the hidden layer by a weight-matrix W (0)

and likewise, the hidden layer to the output layer by a ma-trix W (1). Each unit in the hidden layer typically appliesa non-linear function to the input it receives from the layerbelow it. Similarly, each unit of the output layer applies afunction to the input it receives from the hidden layer im-mediately preceding it. In a network with a single hiddenlayer, this happens according to the following equations

u(0) = b(0) +W (0)x (4)

h = f (0)(u) (5)

u(1) = b(1) +W (1)h (6)

y = f (1)(v) (7)

where b(0) and b(1) are the hidden and output layer biases,f (0) and f (1) are functions applied to the input receivedby each node in the hidden and output layers respectively.Thus, for a given input x, the output y is calculated as

y = f (1)(b(1) +W (1) · f (0)(b(0) +W (0)x)) (8)

In the present case, f (0) is the logistic sigmoid func-tion and f (1) is the softmax function. The network can

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be trained in a supervised manner using the backpropaga-tion algorithm [20]. This algorithm applies the chain ruleof differentiation to propagate the error between the targetoutput and the output of the model backwards into the net-work, and use these derivatives to appropriately update themodel parameters (the network weights and biases).

3.2 Neural Probabilistic Melody Model

Next we consider the neural probabilistic melody model(NPMM), which was originally introduced in [2] as a lan-guage model for word sequences. It consists of a feed-forward network such as the one described in Section 3.1,with an additional embedding layer below it (Figure 1).This model takes as input a concatenation of binary view-point type vectors (cf. Section 3) which represent a fixed-length context. The first layer of the network maps eachof these sparse binary vectors to lower-dimensional densereal-valued vectors which make up the input layer of whatis essentially a feed-forward network above it. This map-ping is determined by a shared weight matrix W (c) whichis learned from data, and is given by

v =W (c)x. (9)

The hidden layer in the case of the NPMM consists ofhyperbolic-tangent activation units. The output layer con-tains softmax units. The model is trained with backprop-agation using gradient descent as in the case of a standardfeed-forward neural network.

4. EVALUATION

The first goal of this paper is to demonstrate the suitabil-ity of fixed-context neural networks for multiple viewpointmelodic prediction. To this end, we compare the two mod-els described in Section 3 with variable-order Markov mod-els (VOMMs) and restricted Boltzmann machines (RBMs).It was observed that the predictive performance of each ofthe neural network models is either comparable to or bet-ter than that of the best VOMMs of both bounded and un-bounded order [16], while slightly worse than the RBMof [4] (Figure 2). Second, we wish to compare the predic-tions of a single neural network which uses multiple inputtypes with that of an ensemble of networks with smallerinput dimensions, each of which uses a subset of the inputtypes of the former, and combined with the entropy-basedweighting scheme described in 2.1. We found that, whilethe addition of viewpoint types does improve the predic-tive performance in both cases, that of the single networkis slightly worse than the ensemble (Figure 3). Moreover,the extent of this improvement diminishes with an increasein context length.

4.1 Dataset

Evaluation was carried out on a corpus of monophonicMIDI melodies that cover a range of musical styles. Itconsists of 4 datasets - Bach chorale melodies, and folkmelodies from Canada, China and Germany, with a total

0 2 4 6 82.6

2.7

2.8

2.9

Context length

Cro

ssE

ntro

py

V OMM(b)

V OMM(u)

RBM

FNN

NPMM

Figure 2: Comparison between the predictive perfor-mances of the best bounded and unbounded variable-orderMarkov models (VOMM(b) and VOMM(u) respectively),the best restricted Boltzmann machine (RBM), the feed-forward neural network (FNN) and the neural probabilisticmelody model (NPMM) averaged over the datasets.

of 37, 229 musical events. These were also used to eval-uate RBMs and variable order Markov models for musicprediction in [4, 16]. To facilitate a direct comparison, themelodies are not transposed to a default key.

4.2 Evaluation Measure

In order to evaluate the proposed prediction models, weturn to a previous study of Markov models for music pre-diction in [16]. There, cross entropy was used to measurethe information content of the models. This is a quantityrelated to entropy (3). The value of entropy, with referenceto a prediction model, is a measure of the uncertainty of itspredictions. A higher value reflects greater uncertainty. Inpractice, one rarely knows the true probability distributionof the stochastic process and uses a model to approximatethe probabilities in (3). An estimate of the goodness ofthis approximation can be measured using cross entropy(Hc) which represents the divergence between the entropycalculated from the estimated probabilities and the sourcemodel. This quantity can be computed over all the subse-quences of length n in the test data Dtest, as

Hc(pmod,Dtest) =−∑sn1∈Dtest

log2 pmod(sn|s(n−1)1 )

|Dtest|(10)

where pmod is the probability assigned by the model to thelast pitch in the subsequence given its preceding context.Cross-entropy approaches the true entropy as the numberof test samples (|Dtest|) increases.

4.3 Model Selection

Different neural network configurations were evaluated bya grid search over the learning rate η = {0.05, 0.1}, thenumber of hidden units nhid = {25, 50, 100, 200, 400},number of embedding units nemb = {10, 20} (only forthe NPMM), and weight decay wdecay = {0.0000, 0.0001,0.0005}. Each model was trained using mini-batch gradi-ent descent up to a maximum of 1000 epochs with a batchsize of 100 samples. Early-stopping [19] and weight-decay

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were also incorporated to counter overfitting. The momen-tum parameter µ, was set to 0.5 during the first five epochsand then increased to 0.9 for the rest of the training. Eachmodel was evaluated with 10-fold cross-validation, withfolds identical to those used in [4, 16] for the sake of com-parison.

4.4 Model Comparison

We carried out a comparison between the predictive perfor-mance of the two neural network models presented here,and models previously evaluated on the same datasets [4,16]. It is to be noted that, since neither of our models isupdated online during prediction, the comparison with thevariable order Markov models of [16] is limited to theirbest performing Long-Term Models. These are of orderbound 2 and unbounded order (labelled there as C*I). It isevident from Figure 2 that both the neural network modelsare able to take advantage of information in longer contextsthan the bounded order n-gram models. This is also a fea-ture of the RBM, whose best case of context-length 5 out-performs the rest of the models in the plot. The slight de-terioration in the performance of the feed-forward networkfor longer contexts is possibly due to poor optimizationof its parameters. This is considering the fact that weight-decay and early-stopping were implemented in the trainingalgorithm to prevent overfitting. While it was not possibleto incorporate further steps for better parameter optimiza-tion in this paper, the results are still illustrative of the net-works’ suitability at the given task and the improvement inperformance with context consistent with each other andwith that of the RBMs. Possible optimizations have beenleft as future work, and will be discussed in Section 5.

4.5 Model Combination

In order to evaluate the combination of viewpoint types, weselected one type which is related to the “what” in music— scale-degree (intfref ), and another which is related tothe “when” — inter-onset interval (ioi), from the severalpossible choices that exist. Furthermore, this experimentwas performed using the NPMM and only on the Chinesefolk melody dataset for the purpose of illustration, withthe assumption that a similar trend would be observed withthe other model and datasets. As our target viewpoint typei.e. the one being predicted, is musical pitch (seqpitch),the first model has the input types seqpitch and intfrefand the second one seqpitch and ioi. The additional view-points are incorporated as explained in Section 2. The pre-dictions of these two models are combined explicitly usingthe mixture- and product-of-experts schemes. On the otherhand, the implicit combination of these two is a singlemodel whose input types are seqpitch, intfref and ioi.Figure 3 compares the predictions of the pitch-only versionof the NPMM and the three models using the additional in-put types. It can be seen that each of these three models hasa better predictive performance than its pitch-only coun-terpart, thus confirming the relevance of the added view-point types to musical pitch prediction. Both the mixture-and product-of-experts combination schemes (seqpitch ×

0 2 4 6 8

2.8

3

Context length

Cro

ssE

ntro

py

seqpitch

seqpitch × intfref × ioi

seqpitch × (intfref +m ioi)

seqpitch × (intfref +p ioi)

Figure 3: Comparison between the predictive perfor-mances, on the Chinese folk melody dataset, of the pitch-only NPMM, its extension which uses the intfref and ioitypes as additional input, and ensembles each of whichcombines two models of input types (a) seqpitch and intfref(b) seqpitch and ioi using the mixture (+m) and product(+p) combination schemes.

(intfref +m ioi) and seqpitch × (intfref +p ioi) re-spectively in the plot) result in very similar predictive per-formance, with the latter working only slightly better forshorter context-lengths of 1, 2 and 3. Moreover, both theseexplicit combinations of viewpoint types perform betterthan the single implicit combination of types (seqpitch ×intfref × ioi in the plot). One will, however, noticethat the cross entropy of the predictions slightly worsens atlonger context-lengths, and that the discrepancy betweenthe implicit and explicit combinations gradually increasesin these cases. As mentioned earlier, we attribute this tothe optimization of the network parameters, which is to bedealt with in future work.

5. CONCLUSIONS & FUTURE WORK

The two neural network models for melodic prediction pre-sented here have been found to have a predictive perfor-mance comparable to or better than previously evaluatedVOMMs, but slightly worse than that of RBMs. Predic-tive performance can be further improved by the additionof viewpoint types to the same model, or by combiningmultiple models using an entropy-weighted combinationscheme. In our experiments, the latter tended to be better.

One open issue that remains is the parameter optimiza-tion in the two networks presented here. It was observedthat, particularly when the input layer of a network is largeand the dataset relatively small, the predictive performancedoes not improve as expected with context-length and theaddition of viewpoint types. We note here that the re-sults presented have been generated with models imple-mented in-house 1 for use with the Python machine learn-ing library scikit-learn [18], and were thus limited in thevarious initialization and optimization strategies used intheir learning algorithms. We also suspect this to be thereason for the limited success of the NPMM which ex-hibited relatively more promising results in its language

1 Code available upon request.

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modelling application in [2]. Many more measures to im-prove generalization and overall prediction accuracy (suchas dropout, different weights initialization strategies andlayer-wise pre-training) have been suggested in [1]. Incor-porating these measures (or using an existing neural net-work library which does) can further improve the results.

Apart from this, there are three other aspects which areof immediate interest to us. The first is the incorporationof a short-term element in the prediction model which up-dates its parameters as data is presented to it, and has beenshown to result in improved prediction performance andhuman-like predictions [15]. Secondly, while the num-ber of parameters of the fixed-context models presentedhere increases linearly with the context-length (assuminga fixed number of hidden units), we are at present experi-menting with recurrent networks where this problem doesnot arise due to their recurrent connections. And finally,the extension of the said models to polyphonic multipleviewpoints representations is also an open issue at the mo-ment which we hope to address in the future.

6. ACKNOWLEDGEMENTS

Srikanth Cherla is supported by a PhD studentship fromCity University London. The authors would like to thankMarcus Pearce for his valuable advice, and the anonymousreviewers for their feedback on the submission.

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