Dynamic cluster structure and predictive modelling of music creation style distributions

2.1. Data and analysed musical statistics

We used a dataset of Japanese popular songs (J-pop dataset) that comprised 1399 songs in the top sales charts provided by Oricon Inc.; the charts were mainly based on the number of phonograph record and CD sales. The vocal/main melodies of all songs that ranked within the top 20 in the yearly charts between 1950 and 2019 were transcribed and encoded in the MusicXML format (except for one song whose audio file was not accessible). Only notes in the main melody were analysed for instrumental music and songs with duets or a chorus, and notes with unclear pitches as in rap music were not included. For computational analysis, each melody was represented as a sequence of pairs of integers, one representing the pitches and the other representing the onset times of musical notes (rests were not used for the analysis). The pitch of a note was represented in units of semitones, and all songs were transposed to the C major or A minor key prior to the analysis. The onset time of a note was represented as an integer b ∈ {0, 1, …, 47} corresponding to its relative position in a bar (called the metrical position). For example, b = 0 indicates the downbeat position, and in 4/4 time, b = 12 indicates the second beat position. We fixed the temporal resolution such that a bar has 48 units, and notes with onset times that cannot be expressed in the resolution were excluded from the analysis.

For musical statistics, we used the pitch and rhythm bigram probabilities to represent the creation style of each song. These bigram probabilities are equivalent to the transition probabilities of Markov models and can describe different musical scales and rhythmic modes [30]. To obtain pitch bigrams, we used the extended pitch class representation, in which a transition from pitch p′ to pitch p is represented as a bigram (q′, q) (0 ≤ q′ ≤ 11 and 0 ≤ q ≤ 35) given by q′ ≡ p′ (mod 12) and q ≡ q′ + p − p′ (mod 36). Thereby, we obtained a representation independent of the pitch range and capable of discriminating pitch intervals between −17 and 17 semitones (an interval of 18 semitones and that of −18 semitones are identified). Thus, the pitch bigram probabilities had 12 × 36 elements. The rhythm bigram probabilities were calculated from the frequencies of the bigrams of metrical positions; they had 48 × 48 elements. Although we need both of the probabilities for pitches and metrical positions to construct a statistical model for melody generation, we treated them separately in the following analysis to make interpretation of the results easier.

After representing each song n with a tuple $({\mathit{\theta }}_n,t_n)$ of the probability vector ${\mathit{\theta }}_n$ of its pitch/rhythm statistics and its created time t_n, we applied a clustering method based on the discrete distribution mixture model. Specifically, we used the expectation–maximization (EM) algorithm [37] with random initialization to train K sets of discrete distributions representing the clusters. After convergence, each song was assigned to the cluster with the maximal likelihood. Created time information was not used in the clustering process.

2.2. Qualitative analysis of dynamics of creation style distributions

To obtain an intuitive overview of the creation style distribution dynamics, we represented the probability vectors of songs in a two-dimensional space using the t-distributed stochastic neighbour embedding (t-SNE) visualization method [38]. For this visualization, we first calculated the Jensen–Shannon (JS) divergence between each pair of probability vectors, which measures their ‘distance’. Then, the t-SNE method was used to map the probability vectors into the visualization space where the distances were retained as much as possible.

Before discussing the dynamics, we first discuss the cluster structure of the creation style distributions using the obtained visualization. From the results for the pitch statistics in figure 1a, we can find a small cluster on the left and another small cluster on the right, as well as a large cluster spread in the central region. Inside the large cluster, we can find some structures of density variations, but there are no clearly separated clusters. The clustering method, assuming five classes (K = 5), automatically identified the two small clusters and divided the large cluster into three smaller clusters (figure 1b). The average statistics for the clusters, represented as pitch-class probabilities, show that all five clusters can be associated with distinguishable musical scales. Cluster 1 can be associated with a minor pentatonic scale (known as Japanese in scale [39]), cluster 2 with a major pentatonic scale, cluster 3 with a minor hexatonic scale, cluster 4 with the minor diatonic scale, and cluster 5 with the major diatonic scale. These musical scales, except the hexatonic scale, match musical scales documented by musicologists studying Japanese popular music [40]. From a macroscopic viewpoint, clusters 3, 4 and 5 form a family of diatonic scales. These clusters in addition to the larger cluster that includes them form a hierarchical structure in the creation style distribution.

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Similar observations can be drawn from the results for the rhythm statistics. From figure 2a, we find three clearly separated small clusters on the right and a large cluster with visible internal structures on the left. Assuming five classes, the clustering method identified three clusters inside the large cluster and assigned two clusters to the separated clusters on the right (figure 2b). Again, we can musically interpret the five clusters using their average statistics, represented as metrical position probabilities. Cluster 1 can be associated with a dotted rhythm, cluster 2 with a four-beat rhythm, cluster 3 with ternary metres, cluster 4 with an eighth-note rhythm and cluster 5 with a 16th-note rhythm. This creation style distribution also has a hierarchical structure with a large cluster containing smaller clusters that overlap with each other.

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These results provide some implications for the applicability of representing creation style distributions as a mixture of clusters. On the one hand, representation using clusters can capture the internal structure of a creation style distribution more precisely than representing it as a simple (e.g. unimodal) distribution, and to some extent, it can reproduce classifications of musical patterns studied by musicologists. On the other hand, the clusters often overlap significantly and possibly form a hierarchical structure, which invalidates the notion of definitive clusters. Despite these limitations, cluster representation is still effective for approximating the complex structure of creation style distributions using simple and tractable mathematical models and interpreting the results.

We now discuss the dynamics of creation style distributions. From figures 1a and 2a, we find drastic changes in the creation style distributions for both statistics. For the pitch statistics, most songs were located in the leftmost or rightmost regions in the 1950s and 1960s, and in the central region in the 1990s and later. This overall change is clearly visible in the evolution of cluster frequencies, which shows the dominance of clusters 1 and 2 in the early years and the dominance of clusters 4 and 5 in later years (figure 1c). Similarly, for rhythm statistics, most songs were located in the upper-right region in the panels for the 1950s and 1960s, and in the lower-left region in the panels for the 1980s and later. The evolution of cluster frequencies in figure 2c also reflects this overall transition. Therefore, the major transitions in the creation style distributions of Japanese popular songs can be captured by the inter-cluster dynamics, similar to the observations made in the evolution of US popular music [12] and Western classical music [14] (see also §4).

We also observed some dynamics within the clusters. First, the distribution of the pitch statistics in cluster 5 reduced its variance from the 1990s to the 2010s. This contraction of the cluster is ascribed to the movement towards market concentration during the period: 103 distinct artists appeared on the chart in the 1990s, whereas only 76 (resp. 32) distinct artists appeared in the 2000s (resp. the 2010s). Second, the distribution of pitch statistics in cluster 3 exhibits a significant shift of a concentrated region from the 1950s to the 1980s. This reflects the movement of musical preferences from the minor pentatonic scale to the minor diatonic scale in songs that appear on the chart. A similar shift of a concentrated region can be observed in cluster 4 of the rhythm statistics between the 1960s and the 1980s. This shift represents the movement towards widely using syncopated rhythms (see electronic supplementary material).

These observations show that the creation style distributions exhibit dynamical modes that cannot be purely represented as changes in the frequencies of clusters having stationary distributions. To capture the collective nature of a contraction or shift of a cluster, we should consider the intra-cluster dynamics, or equivalently, represent the cluster by its smaller components and consider their coherent frequency dynamics that also depend on their content features. Therefore, the present analysis indicates that both inter- and intra-cluster dynamics are relevant for an effective modelling of the evolution of music creation style distributions.

2.3. Quantitative analysis of intra-cluster dynamics using Dirichlet distribution

While the qualitative analysis in the previous section was intuitive, the two-dimensional visualization should be seen as an approximate picture of the distribution of creation styles, which were in fact represented as high-dimensional probability vectors. To quantitatively analyse intra-cluster dynamics in the creation style distribution, we need to mathematically formulate clusters and describe their attributes, such as the size and centre position, in the space of probability vectors. Note that a standard statistical formulation uses the Gaussian distribution to represent a cluster in the Euclidean space [37]. The Gaussian distribution has parameters that can represent the basic properties of a cluster (the mean vector represents the cluster’s centre while the covariance matrix represents the cluster’s size and direction) and is well-defined in the Euclidean space. By extending this idea to the case of our concern, we can use the Dirichlet distribution, which is commonly used in Bayesian statistics to represent a cluster in the space of probability vectors [37]. It is also known that the Dirichlet distribution can approximately reproduce the distribution of the Shannon entropies for rhythm statistics in popular music melodies [41].

For D-dimensional probability vectors $\mathit{\theta }={\left({\theta }_i\right)}_{i=1}^D$ $(\sum _i{\theta }_i=1)$, the Dirichlet distribution is defined as

where the parameter α > 0 represents the concentration and the probability vector $\mathit{\mu }={({\mu }_i)}_{i=1}^D$ is the mean (or base) distribution. A Dirichlet distribution is a probability distribution of probability vectors, and the component-wise mean and variance are given as E(θ_i) = μ_i and V(θ_i) = μ_i(1 − μ_i)/(α + 1), respectively. Thus, the mean $\mathit{\mu }$ can be interpreted as the centre of the cluster, and the concentration α is a parameter related to the cluster’s size (the variance is inversely proportional to α + 1). Given a set of probability vectors assigned to a cluster, we can estimate these parameters for the cluster using the maximum likelihood estimation method [42].

Figure 3 shows the evolution of the estimated parameters of the Dirichlet distributions for some clusters computed after the data samples were divided into time slices. To quantify the temporal differences in the mean distributions, we used the symmetric Kullback–Leibler (SKL) divergence, which is a distance measure for general probability distributions. Figure 3a illustrates the evolution of cluster 5 of the pitch statistics; the contraction of the cluster from the 1990s to the 2010s is represented as an increase in the concentration parameter. Note that the SKL divergences of the mean distributions between any pair of time slices remained far below unity during this time period, indicating that the shift of the cluster centre was relatively small. Figure 3b illustrates the evolution of cluster 3 of the pitch statistics; the shift of the cluster centre from the 1950s to the 1980s is reflected in the relatively large off-diagonal elements in the SKL divergence matrix during this period. Similarly, in figure 3c, which illustrates the evolution of cluster 4 of the rhythm statistics, the off-diagonal elements in the SKL divergence matrix for the period between the 1960s and the 1980s take considerably larger values than those for the period between the 1990s and the 2010s, quantitatively demonstrating that a shift of the cluster centre occurred in the early years.

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These results show that the Dirichlet distribution can be used to quantitatively analyse the intra-cluster dynamics of the creation styles. In particular, a contraction (and possibly expansion) and shift of the centre of a cluster can be described as evolutions of the concentration and mean distribution, respectively. Moreover, as shown in the next section, together with a dynamical model for the parameter evolution, the Dirichlet distribution can be used as a statistical model for predicting the future creation style distribution. This formulation also allows us to evaluate the predictive accuracy by measuring the likelihood, following the general statistical framework.

3.1. Dynamic Dirichlet mixture model

In §2, we found that both inter- and intra-cluster dynamics are important aspects of the evolution of music creation style distributions. Here, we construct a model incorporating these dynamics to predict future creation style distributions and examine the relevance of intra-cluster dynamics for predictive modelling. As explained in §2.3, the clusters of creation styles in the space of probability vectors can be described by Dirichlet distributions. Thus, the distribution of creation styles composed of multiple clusters can be described by a Dirichlet mixture model (DMM) that is defined as

where $\mathit{\theta }$ represents a probability vector as in equation (2.1), k indexes clusters and K denotes the number of clusters. The quantity π_k (called the mixture probability) represents the relative frequency of cluster k (we have $\sum _k{\pi }_k=1$).

In equation (3.1), the dynamics of the creation style distribution are decomposed into the time evolutions of the parameters on the right-hand side, which is the basic assumption of the present model. The inter-cluster dynamics are incorporated in the time-dependent mixture probabilities π_k(t), representing the changes in the cluster frequencies. The intra-cluster dynamics are incorporated in the time-dependent means ${\mathit{\mu }}_k(t)$, representing the shifts of the cluster centres, and the time-dependent concentration parameters α_k(t), representing the contractions and expansions of the clusters. Hereafter, the model in equation (3.1) is called the DDMM.

To use the DDMM for predicting future creation style distributions, given data up to the present, two problems should be addressed. The first is the parameter estimation problem, that is, estimating the parameters π_k(t), α_k(t) and ${\mathit{\mu }}_k(t)$ at each time point up to the present that optimally fit the given data. The second is the parameter prediction problem, that is, predicting future values of the parameters given the parameter values up to the present. We discuss the first problem in §3.2 and the second problem in §3.3. As a problem set-up, we consider the situation in which the data samples (musical statistics) are assigned to one of the clusters, as in §2.

3.2. Parameter estimation problem

A simple solution to the parameter estimation problem is to learn the DDMM parameters for each year (or other units of time slice), t, using the data samples created in that year. The mixture probabilities π_k(t) and mean distributions ${\mathit{\mu }}_k(t)$ can be obtained by taking the sample averages, and the concentration α_k(t) can also be determined using the maximum likelihood estimation method [42]. However, this simple method suffers from the data sparseness problem when the available data are not sufficient and the number of analysed statistics is large, as is the case for the present analysis.

To address this problem, one approach is to use all data samples created by time t with certain weighting factors to learn the parameters. Using the standard exponential decay factor, this weighted average method can be formulated as follows. The mixture probabilities and mean distributions can be obtained as

andwhere ${\pi }_k^S(s)$ and ${\mathit{\mu }}_k^S(s)$ are the values obtained using the aforementioned simple method, e^(s−t)/τ is the decay factor, τ is the time constant and $C(t)=\sum _{s=-\mathrm{\infty }}^t\hspace{0.17em}{\mathrm{e}}^{(s-t)/\tau }$ is a normalization factor. For the concentration parameter, we can use the log-likelihood obtained from a single-time log-likelihood $L_k^S(\alpha ;s)$ calculated from equation (2.1) for cluster k, asand then estimate its value as ${\alpha }_k^{\mathrm{WA}}(t)={\mathrm{argmax}}_{\alpha }{L}_k^{\mathrm{WA}}(\alpha ;t)$.

A suitable value for the time constant τ can be determined from the data by optimizing the likelihood of the estimated DMM. Figure 4 shows how the likelihood depends on the value of τ. To obtain this result, we set the referential year (tentative ‘present year’ for the analysis) t₀ to each value in the range [1960, 2018]. Then, we estimated the DMM parameters and calculated the likelihood of the data created in year t₀ + 1 (assuming that the creation style distribution would not change significantly in 1 year). The likelihood was optimal at τ = 10 for the pitch statistics and τ = 8 for the rhythm statistics. We used these values in the following analysis. For comparison, the result of estimating the DMM parameters using data samples in a time window of various widths (using a 1-year width is the same as the simple method) is also shown in figure 4, demonstrating the superiority of the weighted average method.

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3.3. Parameter prediction problem

To solve the parameter prediction problem, we construct a model for the inter- and intra-cluster dynamics within the framework of cultural evolution theory [1]. The model we construct here is a simplified description of the generative process for songs produced by creators who learn their creation styles from past creators. As shown later in this section, the outcome of this model is a statistical tool used for data analysis and prediction, rather than a tool for theoretically understanding the evolutionary process. However, we explain how the model attempts to capture the elements of the underlying process, to clarify the simplifications and abstractions made during the model construction and to make this paper more useful for those who wish to develop more refined models or apply the method to other cultural domains or in other situations.

Each creator in our model has a set of cultural traits $(k,\mathit{\theta })$, where k represents the cluster to which it belongs and $\mathit{\theta }$ represents the musical statistics describing its creation style. With its statistics $\mathit{\theta }$ a creator produces songs, which are presented to the imaginary society of the model and cause the creator’s evaluation. Although a creator can produce multiple songs in reality, in our model a creator produces only one song and the song’s statistics are identified with the creators statistics, since we do not analyse artist labels for the songs of the datasets. Therefore, creators and songs are effectively the same thing in our model, but we consider creators (not songs) as replicating organisms in the sense of evolution theory.

At each year (the generation time of our model) a population of new creators is introduced, and each new creator inherits its cultural traits from a creator in the previous generation. The inheritance process is described in two steps. First, a new creator n selects a cluster k_n with a probability proportional to the cluster’s relative frequency weighted by the fitness $w_{k_n}$ of the cluster. This fitness represents the inclusive effect of selective pressure resulting from creators’ preferences, the public’s evaluation, and other possible selection biases (such as the effect of advertisement and reputation [43]) on the cluster. The fitness drives the inter-cluster dynamics. Next, the new creator selects a cultural parent among the creators of the previous generation belonging to cluster k_n and inherits the parent’s statistical traits. The cultural parent $\tilde{n}$ is selected with a probability proportional to the fitness $w({\mathit{\theta }}_{\tilde{n}},k_n)$ of its statistical traits ${\mathit{\theta }}_{\tilde{n}}$. This fitness represents the inclusive effect of creators’ preferences etc. on the musical elements and is defined for each selected cluster k_n. The fitness drives the intra-cluster dynamics.

As in the models in evolutionary biology [7,8], iterations of these processes gradually change the distribution $P(k,\mathit{\theta },t)$ of the clusters and the statistical traits within the population of creators at time t. For mathematical simplicity, we consider an infinite-size population model, in which case the distribution is a continuous function of $\mathit{\theta }$. We also assume that the distribution has the form $P(k,\mathit{\theta },t)={\pi }_k(t)\hspace{0.17em}\mathrm{Dir}(\mathit{\theta };{\alpha }_k(t),{\mathit{\mu }}_k(t))$, as given in equation (3.1), which can be realized by a suitable choice of initial condition and form of the fitness function $w(\mathit{\theta },k)$.

To use this evolutionary model for the prediction problem, it is necessary to estimate fitness from past data. Since such evolutionary processes are usually stochastic, while we can only observe a finite (and often small) number of data samples, statistical noise should be considered for the fitness estimation. In addition, as musical preferences themselves are also likely to change over time [15], a model that allows extensions to incorporate the evolution of fitness is needed. In the following paragraphs, we formulate a fitness-based model within a statistical framework that allows a robust estimation of fitness and extensions to dynamically changing fitness.

We first present a model for predicting the evolution of mixture probabilities π_k(t). The basic equation of fitness-based evolutionary dynamics [7] can be represented as

orwhere w_k denotes the fitness (reproduction rate) for cluster k, and the average fitness $\overline{w}$ is defined as $\overline{w}=\sum _{\ell }{w}_{\ell }{\pi }_{\ell }(t)$. The second equation, which is a simple arrangement of the first equation, can be viewed as the equation of motion for the log probabilities, where the logarithm of the relative fitness $w_k/\overline{w}$ corresponds to the velocity.

There are two points to be discussed before moving forward. First, although the relative fitness depends on all π_ℓ(t) through $\overline{w}$, the effect of the other π_ℓ(t) on the dynamics of π_k(t) is small if the speed of evolution is slow. In this approximation of slow evolution, which we adopt in this study, the dynamics of π_k(t) are decoupled from each other and can be treated independently (until they are finally normalized after solving the dynamics). Second, the above equations describe a model for a large population and do not include the statistical noise observed in a finite set of data samples. If we write x(t) = ln π_k(t) and $v(t)=\ln (w_k/\overline{w})$, and represent the noise added to the observed x(t + 1) by ε(t), then the modified model is given as

Specifically, ε(t) is described by a Gaussian noise that satisfies 〈ε(t)〉 = 0 and 〈ε(t)ε(s)〉 = σ²δ_ts. In the approximation of a constant fitness, v(t) is independent of time.

We can extend the model to include the dynamics of the fitness by treating v(t) as a time-dependent variable and adding a dynamical equation. We obtain a natural model for v(t) by introducing a variable η(t) representing the temporal variation of v(t) as

where η(t) is also described by Gaussian noise. In general, we can also introduce variables representing higher-order time derivatives and construct a general model (see electronic supplementary material). Equations (3.7) and (3.8) (with possible additional equations) are known as the state-space model in statistics [37]. An advantage of the present formulation is that we can apply the well-developed statistical theory of state-space models to estimate the fitness (or velocity v(t)) from the data and predict the future values of x(t). The details of this method are presented in electronic supplementary material.

We can derive the state-space models for predicting the evolutions of the mean distributions μ_ki(t) and concentrations α_k(t) in a similar fashion (see electronic supplementary material). In the case of mean distributions, we can interpret the fitness as representing the preferences of the creators and the public with respect to constituent musical elements indexed by i. For concentrations, we could not find such an intuitive interpretaton of the corresponding fitness. Nevertheless, if the evolution of the concentrations results from dynamics where their velocities vary smoothly over time, a state-space model can serve as an approximating model for the dynamics. Therefore, we apply such a model in the following analysis.

In a preliminary analysis, we found that using predictions that are more conservative than those directly obtained by the model often leads to higher accuracy. More precisely, after the velocity is predicted by the model, it is reduced by a factor called the fitness reduction factor before it is used for making predictions. In our analysis, the reduction factor was optimized for each set of parameters (mixture probabilities, concentrations and mean distributions) in the range [0.01, 1]. See electronic supplementary material for details.

3.4. Evaluation of predictive accuracies

To evaluate the predictive model for creation style distributions, we conducted two computational experiments. In the first experiment, we evaluated the predictive accuracies of the DDMM parameters by component and compared them with other possible methods to examine the effectiveness of the fitness-based evolutionary model. In the second experiment, we evaluated the overall predictive accuracy in terms of the likelihood of predicted distributions to examine the relevance of the predictive models for the component parameters and the dependence on prediction times.

In the first experiment (i.e. DDMM parameter prediction), we estimated the DDMM parameters using the music data created before and in a referential year t₀ by the method presented in §3.2. We then predicted the model parameters for later years using the method presented in §3.3. This prediction method will be referred to as the state-space evolutionary model (SSEM). For comparison, we refer to a method that simply uses as predictions the model parameters estimated for the referential year as the ‘static model’. We also compared the predictions made by the Gaussian process (GP) model [44], which is a general-purpose regression method, and in particular, a generalization of the linear regression model. We used a multi-kernel GP composed of a linear kernel, Gaussian kernel, Matérn kernel with degree ν = 5/2, and a bias term. When applying the SSEM and GP to predict the values of the mean distributions, we used these methods only for components with an average probability in the past data above a threshold of 0.01 and used the static model for the other rare components. This is because the probabilities of rare components are susceptible to large statistical noise and cannot be reliably estimated. The weights and kernel parameters of the GP model were optimized to predict each DDMM parameter (the GPy library [45] was used in the analysis). To evaluate the predictive ability, we computed the errors for each set of DDMM parameters. We used the SKL divergence to measure the prediction errors for the mixture probabilities and mean distributions, and used the log-squared error for the concentration parameters.

The results are summarized in figure 5, where the aforementioned three methods are compared for both pitch and rhythm statistics for the range t₀ ∈ [1969, 2009]. For the mixture probabilities, the SSEM had significantly smaller errors than the static model in all cases. The GP model had larger errors than the static model, except for the pitch statistics and the number of clusters K = 5. For the concentrations, the SSEM had smaller errors than the static model, except for the case of rhythm statistics and K = 5, where the errors for these models were similar. The GP model had a similar or larger error than the static model for all cases. For the mean distribution parameters, the SSEM and the static model had similar errors in the case of pitch statistics, while the former model had slightly smaller errors in the case of rhythm statistics. The GP model had the largest errors in all cases.

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Overall, the GP model often had larger prediction errors than the static model, indicating that the prediction of the DDMM parameters is a non-trivial problem. In most cases, the SSEM can predict the parameters more accurately than the static model. In particular, the reduction in error in the mixture probability predictions was considerably large, suggesting the effectiveness of describing the dynamics by a fitness-based evolutionary process with smoothly varying fitness values for the clusters. However, the error reductions in the mean distribution parameters were relatively small, implying a possibility that the parameters follow more complicated dynamics.

Examples of predicted and actual evolutions of the DDMM parameters of the pitch and rhythm statistics, when the reference year was set to t₀ = 1999 and the number of clusters was K = 5, are shown in figure 6. For the mixture probabilities and concentrations, the SSEM linearly extrapolates recent trends and in most cases succeeds in predicting the directions of evolution. For the mean distributions, the predictions are conservative and only slightly differ from constant values. Significant errors are observed especially for musical elements with small probabilities.

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The future values of the DDMM parameters predicted by the SSEM are shown in figure 7 for the case with K = 5 clusters. To make it easier to examine trends, the fitness reduction factor was set to unity for the mean distribution parameters to obtain these results. For both pitch and rhythm statistics, it is predicted that the fifth cluster will continue to rise and become dominant. For rhythm statistics, the second cluster is also predicted to rise in the near future. The concentrations for all rising clusters are predicted to increase, indicating a decrease in the variabilities within individual clusters.

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Focusing on the fifth cluster of the pitch statistics, some changes in the probabilities of the most frequent bigrams are predicted. In particular, the probabilities of stepwise motions, such as (C,D), (F,E) and (B,C), are predicted to increase, whereas those of same-tone motions, such as (C,C), (E,E) and (D,D), are predicted to decrease. For the fifth cluster of the rhythm statistics, the probabilities of currently the most frequent bigrams are predicted to eventually decrease, indicating a general uniformization of bigram probabilities. Notable exceptions are bigrams with a 16th-note length, such as (6,9), (30,33) and (33,36), for which relatively rapid increases in the probabilities are predicted. The observed trends in the to-be-dominant clusters of the pitch and rhythm statistics are consistent in reducing monotonic motions (same-tone or eighth note rhythm). To illustrate the effect of these parameter changes, some melodies generated by the mean distributions observed in 2019 and those predicted in 2040 are given in electronic supplementary material. Perceptually, however, it is difficult to clearly distinguish the styles of these melodies.

In the second experiment (i.e. prediction of the creation style distributions), we measured the likelihood of the probability vectors in a range of 20 years after a referential year t₀, using the parameters predicted by the SSEM. For predicting the creation style distribution at year t_p after t₀, we used the parameters predicted for year t_p + Δ_p, where Δ_p is a ‘look-ahead’ parameter that compensates for the effect of the weighted average smoothing. The value of Δ_p was optimized to maximize the likelihood, resulting in values Δ_p = 8 and 15 for the pitch and rhythm statistics, respectively. For the same purpose, we also used increased values for concentrations because the creation style distribution of an individual year was expected to have a larger concentration (smaller variance) than the smoothed one. For the analysis, an increasing factor of 5 was used for all data. For comparison, we measured the likelihood of the same data using the static model explained above. We evaluated cases K = 5 and 10 for the number of clusters, and used the static model with a single cluster (K = 1) as the baseline.

Figure 8 shows the dependence of the log-likelihood gains (compared with the K = 1 static model) on the predictive intervals (which is defined as the year of prediction—the referential year), averaged over all referential years in the range t₀ ∈ [1969, 2009]. First, in all cases, the SSEM had a larger likelihood than the static model, indicating that the DDMM parameter prediction made by the SSEM can assist in predicting the distribution of the creation styles up to a prediction interval of at least 20 years. Second, for both the pitch and rhythm statistics, the likelihood gains of the models with K = 10 tended to decrease for longer prediction intervals and fell beneath the gains of the model with K = 1 (pitch statistics) or K = 5 (rhythm statistics) for predictive intervals of nearly 20 years. This result indicates the difficulty of predicting the distribution increases for large numbers of clusters. In addition to the possible effects of overfitting and data sparseness, the increase in the number of parameters that are difficult to predict might have led to the deterioration of the likelihood gains.

Finally, to examine the contributions of individual elements of the DDMM parameters to the likelihood gains, we compared the gains obtained by using the predictions by the SSEM for partial sets of the parameters and using the static model’s predictions elsewhere. The results in figure 9 show that the SSEM’s predictions of the mixture probabilities and concentrations contribute to the likelihood gains to some extent, and their contributions are often comparative. This indicates that the predictions of both inter- and intra-cluster dynamics have positive effects on likelihood gains. However, the SSEM’s predictions of the mean distribution parameters have small and sometimes even negative effects. Again, this is mainly due to the difficulty of predicting these parameters.

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