Abstract
If a cultural feature is transmitted over generations and exposed to stochastic selection when spreading in a population, its evolution may be governed by statistical laws and be partly predictable, as in the case of genetic evolution. Music exhibits steady changes of styles over time, with new characteristics developing from traditions. Recent studies have found trends in the evolution of music styles, but little is known about their relations to the evolution theory. Here we analyze Western classical music data and find statistical evolutionary laws. For example, distributions of the frequencies of some rare musical events (e.g. dissonant intervals) exhibit steady increase in the mean and standard deviation as well as constancy of their ratio. We then study an evolutionary model where creators learn their datageneration models from past data and generate new data that will be socially selected by evaluators according to the content dissimilarity (novelty) and style conformity (typicality) with respect to the past data. The model reproduces the observed statistical laws and can make nontrivial predictions for the evolution of independent musical features. In addition, the same model with different parameterization can predict the evolution of Japanese enka music, which is developed in a different society and has a qualitatively different tendency of evolution. Our results suggest that the evolution of musical styles can partly be explained and predicted by the evolutionary model incorporating statistical learning, which can be important for other cultures and future music technologies.
Introduction
A prominent feature of humans is that they learn and transmit cultural traits over generations^{1}. Although many cultural traits (e.g. style of music/language/fine art, fashion, unscientific beliefs, etc.) seem to make little direct contribution to an individual’s biological fitness, some of them (e.g. music and fashion) have evolved into highly complex forms and have rather large influence on human behaviour. To understand human’s behaviour, it is important to uncover some possible laws in cultural evolution and seek for a theory that can explain them^{2,3}. Moreover, a theory that can quantitatively predict parts of cultural evolution can serve as a base for the development of new technologies to predict cultural trend and enhance cultural production.
In this study, we consider the evolution of musical styles, which has gathered growing attention^{4,5,6,7,8,9,10}. It has been observed in a recent paper^{11} that some features of music, e.g. the frequency of tritones, have steadily increased during the history of Western classical music. (The tritone is a pitch interval consisting of six semitones. It is regarded as a “dissonant” interval in traditional music theories^{12}.) Although these clear trends imply some driving force for the evolution of the music style, their theoretical origins and how much they can be predicted are not understood. Moreover, while previous studies^{4,5,6,11} have focused on the mean of musical features, other statistics including the standard deviation and distribution form are important for studying their dynamics in relation to evolutionary models^{13}.
Data Analysis
We first analyze the time evolution of statistics of musical features of Western classical music data. In particular, we study two major aspects of tonal content of music, dissonance and tonality, on which musicologists have focused^{14,15}. More specifically, we focus on the frequencies of two features that represent these aspects and can be computationally analyzed without interpretation of data by human. One of them is tritone, which is a representative interval historically considered dissonant^{14} and has been studied also in a previous study^{11} (see Refs. ^{16,17} for psychological studies on consonance and dissonance of musical sound). The other one is nondiatonic motion, which is defined as bigrams of pitchclass intervals that cannot be realized on a diatonic scale. (The Cmajor scale or “the scale of white keys” (CDEFGAB) is one instance of diatonic scales. In general, a diatonic scale can be transposed to the Cmajor scale by a global pitch shift.) The nondiatonic motion is an indicator of chromatic motions and modulations (key changes) to a distant key^{18} (see Methods for a detailed definition).
Figure 1(a) shows that the mean and standard deviation of the frequency (probability) of tritones steadily increased during the years 1500–1900 while their ratio stayed approximately constant over that time (see Methods for details of the analysis). Here and in what follows, zerofrequency data points are excluded from the analysis in order to obtain statistically reliable results. Figure 1(b) shows actual distributions of the frequency of tritones obtained for each century. We find that the distribution can be approximately fitted by a beta distribution (its function form is given in Eq. (1)), which is a mathematically simple distribution defined over the range of real numbers between 0 and 1.
These statistical tendencies can also be found for the frequency of nondiatonic motions. Statistical data of their frequency corresponding to Fig. 1 are given in Supplemental Material. As a general set of musical features including these two, we can consider bigram probabilities of pitchclass intervals^{18}, which have 121 elements (see Methods). Figure 1(c) shows how the means of these 121 features evolved over the centuries. We see that lowprobability features exhibit exponentiallike growth. Still, the number of observations is small and statistics of such rare events may not be so reliable. (The distributions of very rare events typically have a peak at zero and the standard deviation is larger than the mean. This limitation of observation is caused by the fact that a musical piece usually consists of 10^{2}–10^{3} musical notes.)
To summarize our data analysis results, we have found the following statistical evolutionary laws in lowprobability features of Western classical music data:

1.
Betalike distribution of frequency features

2.
Steady increase of the mean and standard deviation

3.
Nearly constant ratio of the mean and standard deviation, which is slightly less than unity

4.
(Possibly) exponentiallike growth of the mean
These findings reveal that the evolution of styles in the classical music data exhibits much more regularities than previously found^{11}. The last two laws indicate that the dynamics is scale invariant, that is, the dynamics at one value of the features looks similar to that at a different value of the features. Since these statistical laws are found in the music data of various composers in four consecutive centuries, they may be caused by general mechanisms of transmission and selection of cultural style rather than by the circumstances of individual composers or social communities of individual time periods.
Theoretical Model
Let us now discuss a possible evolutionary model that may explain the origin of the observed statistical laws. Following the general framework of Darwinian evolution, we construct a theoretical model based on information transmission and stochastic selection. A feature of music culture is that creation styles are learned and transmitted via data (e.g. musical scores and audio signals), and recent studies have suggested the importance of statistical learning for music composition (e.g.^{19,20,21}) and for lister’s understanding (e.g.^{22}). As is commonly done in the field of music informatics (e.g.^{20,21}), we represent creators (composers) with statistical models for data generation and try to capture the evolution of music styles through dynamic changes of creators’ models. As a driving force for time evolution, we consider social selection by contemporary evaluators (listeners). Specifically, we study a dynamical system of creators that statistically learn their datageneration models from past data and then generate new data, and of evaluators that determine the fitness of the generated data. Since evaluators should also learn their dataevaluation models from existing data, it is legitimate to consider a dynamic change in the fitness depending on other agents/data, as in evolutionary game theory^{23}. Similar models of iterated learning have been studied in the context of language evolution^{24,25,26}.
A dynamical system we call a statistical creatorevaluator (SCE) model is formulated as follows. Each creator at generation t generates a dataset X_{t} of musical pieces according to a distribution (datageneration model) \({\varphi }_{t}\), and the generated data are evaluated with the fitness defined below. Following this evaluation, the creator’s model of the next generation \({\varphi }_{t+1}\) is determined by statistical learning. With this procedure, the creator’s datageneration model evolves over generations. The creator’s model \({\varphi }_{t}(\theta )\) is defined over a probability parameter θ ∈ (0, 1) (e.g. frequency of some musical events). An evaluator is similarly modelled by a distribution ψ_{t}(θ). We assume that \({\varphi }_{t}\) is described as a beta distribution (a_{t}, b_{t} > 0):
The beta distribution is introduced here because it is a simple distribution function for the probability variable whose value is restricted between 0 and 1, where the emergence of a new feature is represented by a shift of a peak at 0 towards 1, and because it approximates the data well as shown in Fig. 1(b). For a_{t}, b_{t} > 1, a beta distribution satisfies boundary conditions \({\varphi }_{t}(0)={\varphi }_{t}(1)=0\) and the parameters a_{t} and b_{t} specify the power of θ at the boundaries. The datageneration process is described in two steps: (i) a value of θ is drawn from \({\varphi }_{t}(\theta )\) for each data unit (e.g. musical piece), here called a (data) chunk, and (ii) data samples (e.g. musical notes) in that chunk are sampled with the chosen θ. It is assumed that the data selection is carried out in the space of θ so that the model is described in terms of the space of θ without referring to data samples, so we treat θ as a directly observable quantity.
The dataselection (evaluation) process is described as follows (see Fig. 2). At each generation t, a dataset X_{t} of chunks is generated by \({\varphi }_{t}(\theta )\). Data chunks in X_{t} are then evaluated by the evaluators assigning weights \({e}^{\beta {R}_{t}(\theta )}\), where R_{t}(θ) is a functional of ψ_{t} and X_{t} called an evaluation function and β is a selection coefficient. The nextgeneration creator’s model \({\varphi }_{t+1}(\theta )\) and evaluator’s model ψ_{t+1}(θ) are learned from the dataset of chunks denoted by Y_{t} (i.e. Y_{t} is used as training data). The data Y_{t} consists of data chunks selected randomly from X_{t} with probabilities proportional to \({e}^{\beta {R}_{t}(\theta )}\). It is assumed that \({\varphi }_{t}\) and \({\varphi }_{t+1}\) have the same distribution form (1) and the parameters of \({\varphi }_{t+1}\) are chosen to approximate Y_{t} as much as possible (the learning scheme is specified later), even though the data Y_{t} are distributed differently from \({\varphi }_{t+1}\) in general. In the limit of infinite data size, X_{t} is distributed as \({\varphi }_{t}(\theta )\) and Y_{t} is distributed in proportion to \({\varphi }_{t}(\theta ){e}^{\beta {R}_{t}(\theta )}\). Here, we consider a simple case where ψ_{t}(θ) is learned in the same way as \({\varphi }_{t}(\theta )\) so that these distributions are in fact identical. The dynamics is summarized as
where the arrow means that the distribution on the lefthand side is learned from the data on the righthand side. Although we mainly focus on the case where \({\varphi }_{t}\) is given as in Eq. (1), \({\varphi }_{t}\) in the SCE model in Eq. (2) can be described with other distributions in general.
Since the fundamental process of evaluating musical data is unknown, we attempt to derive a reasonable form of the evaluation function R_{t} based on a theoretical argument. Rather than introducing biases depending on particular features of music content, we here focus on two viewpoints for evaluation considered most fundamental and general^{2}: the content dissimilarity (novelty) and style conformity (typicality) with respect to the past data. Naively, novelty is important because newly generated data chunks whose content is very similar to that of an existing data chunk do not increase the experience of the evaluators and thus are not favoured. The fast updates of popular music album charts suggest the possibility of this bias^{27}. Typicality is also important because a data chunk that deviates significantly from the style of the past data cannot be understood easily by the evaluators and are not approved. Some critics’ denials of innovative musical works such as Berlioz’s Symphonie Fantastique^{28} and Stravinsky’s Rite of Spring^{15} at their premieres suggest the relevance of this bias.
We propose to mathematically formulate these two metrics in terms of information measures. Novelty can be formulated by considering the effective amount of information obtained from the evaluator’s perspective. For each value of θ, novelty can be measured with the amount of similar data chunks in X_{t} = ϕ_{t}, which is proportional to \({\varphi }_{t}(\theta )\) in the limit of infinitesimal precision of discriminating musical features (see Methods for a detailed derivation). Typicality can be formulated by considering the difficulty of understanding, or memorizing, a data chunk according to the evaluator’s model ψ_{t}. Thus, in informationtheoretical terms, typicality can be described as the number of bits needed to encode the information contained in a data chunk θ using the model ψ_{t}, which is proportional to −lnψ_{t}(θ)^{29}.
In other words, to gather the information contained in a data chunk θ, the evaluator must first spend cost proportional to \({\varphi }_{t}(\theta )\) to obtain that data chunk (together with unavoidable similar data chunks) and then spend cost proportional to −lnψ_{t}(θ) to memorize the contained information. In this way, the evaluation function constructed as a sum of the novelty and typicality defined here can be interpreted as the effective amount of cost necessary for the evaluator to gather information. In this sense, the novelty and typicality biases may have relation to biological fitness, as the ability to gather information about the environment is essential for surviving.
By using the analogy of the above selection probability with a Boltzmann distribution in statistical physics, where β and R_{t} correspond to the inverse temperature and negative energy (cost), the form of R_{t} is given as
where β_{T} and β_{N} are constant factors, the first and second terms respectively represent the typicality and novelty of chunk θ, and we have used the relation ψ_{t} = ϕ_{t}. Substituting Eq. (3) into Eq. (2), we have
The signs of the two terms in Eq. (3) are chosen so that when β_{T} and β_{N} are positive, evaluators favour both typical and novel data chunks. Theoretically, these parameters can take negative values in general.
To complete a mathematical formulation, we specify the learning process. The creator learns the data distribution \({\varphi }_{t+1}(\theta )\) from \({\tilde{\varphi }}_{t}(\theta )\) so that \({\varphi }_{t+1}(\theta )\) is assimilated by the beta distribution by optimizing its parameters. To be specific, noting that the pair (a_{t}, b_{t}) has onetoone correspondence with the pair of mean and standard deviation (μ_{t}, σ_{t}) (see Methods), we use the moment matching method to learn \({\varphi }_{t+1}\) from \({\tilde{\varphi }}_{t}\). That is, we choose the parameters of \({\varphi }_{t+1}\) so that its mean μ_{t+1} and standard deviation σ_{t+1} exactly match those of \({\tilde{\varphi }}_{t}\). If we take the statistics μ_{t} and σ_{t} as state variables, the update equation (2) is described as a twodimensional map (μ_{t}, σ_{t}) → (μ_{t+1}, σ_{t+1}).
Let us analyze the model. See Methods for mathematical details. Qualitatively, positive β_{T} and negative β_{N} put higher weights on more probable θ, causing \({\varphi }_{t+1}\) to be sparser than ϕ_{t}. Conversely, negative β_{T} and positive β_{N} make \({\varphi }_{t+1}\) less sparse. The case β_{T} < −1 puts infinite weights on zeroprobability θ and is thus illdefined. In the following, we focus on the case β_{T}, β_{N} ≥ 0, μ_{t} < 1/2, and σ_{t} < μ_{t}, and in particular the regime where μ_{t} is small, to analyze the dynamical system quantitatively.
For small β_{T} and β_{N}, which are of our interest, the discretetime dynamics of the system is relatively smooth and vectors in Fig. 3(a–c) show how an update changes μ and σ at each point. When β_{N} = 0 (i.e. only typicality is evaluated), both the mean and standard deviation decrease over time (Fig. 3(a)). More specifically, the mean will converge to the mode (peak position) whereas the standard deviation will converge to 0 for t → ∞. This is shown analytically in Methods.
When β_{N} > 0 and β_{T} = 0 (i.e. only novelty is evaluated), both the mean and standard deviation increase over time and the orbits converge to a fixed point with μ_{t=∞} = 1/2 (Fig. 3(b)). The reason the mean increases can be understood intuitively from the shape of the distribution. When β_{N} is not too small, the weighted data \({\tilde{\varphi }}_{t}\) has two peaks around the mean of ϕ_{t} and the left one is narrower due to the boundary at zero (as in Fig. 2) so that the distribution \({\varphi }_{t+1}\) is pushed to the right.
A notable feature of this case is the presence of a “slow manifold”. The dynamics quickly fall onto the manifold (i.e. subspace of the parameter space) with σ_{t}/μ_{t} ≈ constant, which is slightly less than unity. The values of μ_{t} and σ_{t} will then grow along the manifold keeping their ratio almost constant in time. Intuitively speaking, this slow manifold is formed because when \({\sigma }_{t}\ll {\mu }_{t}\) the beta distribution is almost symmetric and an update does not change μ_{t} significantly but increases σ_{t} and thus also σ_{t}/μ_{t} (Fig. 3(d)). When \({\sigma }_{t}\sim {\mu }_{t}\), the right peak of \({\tilde{\varphi }}_{t}\) dominantly influences the next distribution \({\varphi }_{t+1}\) and μ_{t} grows so much that σ_{t}/μ_{t} decreases (see Fig. 3(e) and Methods for more details). This is quantitatively shown in Fig. 3(f), where one can see that the curve representing the update of the ratio σ_{t}/μ_{t} intersects with the invariant line at similar points for varying μ_{t}. As can be observed in Fig. 3(f) and will be discussed more analytically in Methods, the constant value of σ_{t}/μ_{t} is smaller for larger β_{N}.
How the mean grows on the slow manifold can be understood from Fig. 3(g). One finds that, for various values of the mean, its growth rate is of the same order of magnitude. This indicates that the mean (and thus also the standard deviation) grows nearly exponentially over time. The comparison between different values of β_{N} in Fig. 3(g) shows that the growth rate is not very sensitive to the value of β_{N}.
The model’s dynamics for finite β_{T} and β_{N}—i.e. when both novelty and typicality are evaluated—are illustrated in Fig. 3(c). Generally, the standard deviation converges to a fixed point where the effects of the typicality and novelty terms balance; when the standard deviation is larger (smaller) than its asymptotic value the dynamics is similar to that of the typicality (novelty) term only. In particular, for small μ_{t} and σ_{t} (i.e. in the early stage of evolution), we again find a slow manifold where both the mean and standard deviation eventually increase while their ratio stays almost constant. If \({\sigma }_{t}\ll {\mu }_{t}\) when σ_{t} reaches the fixed point, then the value of μ_{t} is effectively frozen, leading to the emergence of marginally stable points.
One can also confirm the presence of a similar slow manifold in the case of β_{N} > 0 for other choices of ϕ_{t} with a boundary at θ = 0, i.e. the gamma and lognormal distributions (see Supplemental Material). This shows that it is a rather general phenomenon, as expected from the aforementioned intuitive argument.
Examining the Model with Experimental Data
Let us now compare the consequences of the present model with the observed data of music evolution. Among the aforementioned four statistical evolutionary laws, the first law (beta distribution) is naturally incorporated in the model. When the novelty term is present and the initial values satisfy \(\sigma < \mu \ll 1\), the dynamics of the model spontaneously lead to the phase where both the mean and standard deviation increase over time keeping their ratio almost constant and slightly less than unity (the second and third laws). This explains the origin of these laws, which are expected by the model irrespective of small changes in initial values. We have also shown that the last law (exponential growth) is also derived from the dynamics of the model in the early stage of evolution.
To illustrate the characteristics of the present model and to examine the model’s predictive ability, let us briefly discuss another evolutionary model, for which the evaluation function R_{t} is simply a function of θ, rather than a functional of ψ_{t} or ϕ_{t} as in the SCE model. Since the constancy of σ/μ suggests scaleinvariant dynamics, we consider an evaluation function with a log potential: R_{t} = lnθ. As a natural choice for ϕ_{t}, we here use the lognormal distribution instead of the beta distribution because it is kept invariant under the selection process and its shape is similar to that of the beta distribution (see Supplemental Material for a graphical comparison between these distributions). The creator’s model is then written as
where \({\tilde{\mu }}_{t}\) is the log mean and \({\tilde{\sigma }}_{t}^{2}\) is the log variance, which are related to the mean and standard deviation as \({\mu }_{t}=\exp ({\tilde{\mu }}_{t}+{\tilde{\sigma }}_{t}^{2}\mathrm{/2)}\) and \({\sigma }_{t}/{\mu }_{t}=\sqrt{\exp ({\tilde{\sigma }}_{t}^{2})1}\). We call this model the logpotential model.
As shown in Methods, in this model the standard deviation and mean both grow exponentially over time with a fixed rate \({e}^{\beta {\tilde{\sigma }}^{2}}\) and their ratio σ_{t}/μ_{t} is kept constant. While the dynamics of the logpotential model is similar to that of the SCE model on the slow manifold, an important difference is the sensitivity to the initial values and selection coefficient. For the logpotential model the ratio σ_{t}/μ_{t} can be tuned arbitrarily by adjusting the initial condition, whereas for the SCE model it is driven to an asymptotic value spontaneously by the dynamics irrespective of the initial condition. The growth rate \({e}^{\beta {\tilde{\sigma }}^{2}}\) is very sensitive to the value of β in the logpotential model, whereas it is not very sensitive in the SCE model as we discussed above.
In Fig. 4(a,b), the time evolution of the frequency of tritones and that of nondiatonic motions in the classical music data is numerically compared with the solutions of the SCE model and the logpotential model. The models are initialized with the mean and standard deviation at the earliest time and the model parameters (β_{T} and β_{N} for the SCE model, and β for the logpotential model) are optimized to minimize the squared error of the means and standard deviations throughout the time period of the data. If the evolutions of these two features share the same mechanism, it is reasonable to use the same model parameters to fit both sets of data. The parameters are thus optimized to fit both sets of data simultaneously and the optimized values are given inside each figure.
We see that the SCE model can roughly fit both sets of data whereas the logpotential model can fit only one set of data. Quantitatively, the root mean squared errors for the tritone and nondiatonic motion data are 4.5 × 10^{−3} and 1.5 × 10^{−2} for the SCE model, and 7.2 × 10^{−3} and 6.1×10^{−3} for the logpotential model, respectively. This result indicates not only that the two sets of data can be explained/predicted by the mechanism described by the SCE model in a unified way but also that the prediction is not trivial. On the other hand, we also see some discrepancies between data and model predictions (e.g. in the values of σ/μ). Such small discrepancies can be explained in several ways: removing the simplifications assumed for the SCE model may bring small changes in model predictions, as discussed later, and they may be simply due to statistical/systematic error in the data. If we try to fit the two sets of data individually using different parameter values, the fitting error for the logpotential model is slightly smaller than that for the SCE model (see Supplemental Material).
To examine the ability of the model with different data, Fig. 4(c) illustrates results of another analysis on a different musical feature extracted from a different dataset. The dataset is a collection of enka music (a genre of Japanese popular music) compiled and published by a music publisher^{30,31}. Here we focus on the rhythms and use as a feature the frequency of “rare rhythms” that are defined as bigrams of note values whose ratio is not one of {1, 1/2, 2, 2/3, 3/2, 1/3, 3, 1/4, 4, 1/6, 6} (see Methods for details). Both the mean and standard deviation decrease over time, which is qualitatively different from the previous two cases. For this case, only the SCE model can reproduce the history of the mean and standard deviation. Predictions for the near future are also provided in Fig. 4(c). As expected from the decrease of the standard deviation, typicality plays a more influential role for data selection and the model makes a testable prediction that the mean will converge in the future. One interpretation of this result is that enka music is considered as a kind of “soul music”^{32} and the evaluators (listeners) would prefer a typical enka song over a novel one. On the other hand, the logpotential model predicts a linearlike decrease of the mean, which can be discriminated from the prediction of the SCE model in a near future. The results show the nontrivial ability of the SCE model to explain and predict the nonlinear evolution of the enka music and demonstrate that the model is relevant not only for Western classical music but also for music developed in a different cultural background.
Discussion
In conclusion, we have analyzed Western classical music data and found several statistical evolutionary laws, in particular, steady increase of the mean and standard deviation of frequencies of rare events and nearly constancy of their ratio, which indicate some driving force for the evolution of the music style. As a theoretical explanation of the phenomenon, we have formulated and analyzed SCE models in which creators and evaluators coevolve by influencing each other through a social selection process. The evaluation function for the social selection is constructed with the novelty and typicality terms representing cost required for obtaining and memorizing data in the process of information gathering. We have shown that when the creator’s and evaluator’s models are beta distributed as observed in real data and the novelty term is active, the system generally has a slow manifold in which both the mean and standard deviation grow almost exponentially while their ratio stays almost constant. This property and the fact that the system’s dynamics are relatively insensitive to the selection coefficients make the present model more predictive and distinct from a Darwinian evolutionary model with a logarithmic potential.
It has been demonstrated that the present model can predict the evolution of the Western classical music data better than a scaleinvariant evolutionary model (logpotential model). The present model had the ability to fit the two kinds of data (frequency of tritones and that of nondiatonic motions) in a unified way, whereas the logpotential model could only fit the data individually. From the perspective of the present model, the observed evolution of the mean and standard deviation of frequencies of musical features that were once rare is a consequence of pursuing novelty. For the dataset of enka music, both the mean and standard deviation of the frequency of rare rhythms were found to be decreasing, which indicated that typicality has more importance than novelty in the selection process. Predictions for the evolution of this feature that are testable in the next few decades have been made.
In the evolutionary process studied here, the balance between novelty and typicality (i.e. content dissimilarity and style conformity with respect to past data) plays an essential role. As we saw in the classical music data and enka music data, relative values of β_{N} and β_{T} can influence the direction and speed of evolution. We also found that the ratio σ/μ of the standard deviation and mean is an important metric of evolutionary dynamics, which can be used to infer from data the relative importance of novelty and typicality in the process of social selection/evaluation. Once these parameters are determined, the SCE model can be used to predict the evolution of musical features. Such predictive ability opens the possibility of new technologies such as hit song prediction^{33} and automatic composition systems that go beyond the ability of simply imitating the style of fixed training data^{19,20} and generate nextgeneration music.
Since the novelty and typicality biases represent the effective amount of cost necessary for the evaluator to gather information and are not dependent on particular features of music, they can be important for other types of culture, and the present model can be useful for analyzing not only music data but also other cultural data. Evolutionary dynamics of language^{34}, other genres of music^{6}, scientific topics^{35}, and sociological phenomena^{36,37} are among topics currently under investigation. Another relevant topic is the evolution of bird songs, where selectionbased learning is considered important^{38}. Evolutionary dynamics of bird songs have been studied based on dynamical systems that describe interaction between generators (singing birds) and imitators^{39}, which is similar to the noveltytypicality bias in this study.
Several remarks are made before closing the paper. First, there are multiple possible ways of extending and relaxing the condition of the minimal model analyzed in this study. Relaxing the assumption that both the creator and evaluator learn from the same data can lead to time displacement of their models. For example, if the evaluator learns its model ψ_{t+1} directly from the data X_{t}, instead of being biased by the evaluation function, then ψ_{t+1} = ϕ_{t} holds. We can also introduce overlaps between generations or dependence on data created by more than one past generation. These extensions can change the consequences of the model quantitatively and can possibly explain the small discrepancies between model prediction and data in Fig. 4. Systems with multiple creators and evaluators would also be important for investigating the diversification and specification of cultural styles.
Second, a way to test the present model is to observe the exponential growth of a relevant feature. However, this is not easy for music data because of the size of each data chunk (musical piece) is relatively small. A musical piece typically consists of 10^{2} to 10^{3} notes and thus observing the evolution of a frequency of musical events across some orders of magnitude is difficult due to data sparseness. It would be possible to alleviate the problem by extending creator’s and evaluator’s models in a Bayesian manner. Another direction for experimentally testing the model is to directly examine the evaluation function by means of music data with social rankings, by an evolutionary experiment involving humans as evaluators^{40}, or by psychological experiments^{41}. It would also be possible to infer the form of the evaluation function from such data by machinelearning techniques.
Third, one might think that music styles are transmitted via a set of rules (often called music theories) and the SCE model does not accord with the reality. It has been argued that traditional composition rules are not sufficient to describe the actual composition process from a computational viewpoint^{19}, and in fact traditional music theories tell little about the quantitative nature of music styles^{42,43}. In addition, recent studies on music informatics have suggested that traditional composition rules can be acquired from data via statistical learning^{21,44}. Based on these observations, our view is that although those composition rules may influence the transmission of music styles, the effect of statistical learning is essential for understanding the evolution of music styles.
Fourth, there are potential sources of social influence that could affect the evolution of musical styles other than the novelty and typicality biases studied here. These sources include effect of random copying (neutral drift) in a finite population^{27}, interdependence among evaluators’ decision^{45}, active role of creators^{46}, indirect biases (e.g. publicity) independent of the data content^{2}, and psychological biases related to specific musical features^{33}. Our results do not exclude the relevance of these sources to the studied phenomena of music evolution. The contribution of this work is to propose another possible source of cultural evolution that can be particularly important when statistical learning is involved and to provide theoretical results that help identify its relevance in observed data. Further research is necessary to study the consequences of the SCE model when it is extended to incorporate those other sources of mutation and social selection and to identify their individual roles in evolutionary dynamics. It is also important to seek for a fundamental model that can explain an evolutionary origin of the novelty and typicality biases in Eq. (3) and can validate the assumption of the Boltzmann distribution in Eq. (2) as well as the beta distributions observed in the data.
Methods
Data analysis
A collection of Western classical musical pieces is used for the data analyses in Figs 1 and 4. The dataset consists of MIDI files of 9,727 pieces by 76 composers downloaded from a public web page (http://www.kunstderfuge.com). This dataset is compiled in order to cover a longer period of time than the datasets used in previous studies^{5,11} and to enable noiseless symbolic music analysis. (The dataset used in^{11} consists of audio data and the Peachnote corpus used in^{5} contains symbolic music data that are obtained by scanned sheet music by using music optical character recognition (OCR) software and thus contain noise.) The 76 composers are those with the largest number of available pieces and obvious duplications of two or more files for the same piece are avoided by looking at file names. Files with less than 100 musical notes are also removed. Each MIDI file is parsed and a sequence of pitches represented by integers in units of semitones is extracted; pitches are ordered according to their appearance in the file. To extract information on music styles that are irrelevant of superficial features such as pitch range and absolute key, the sequence of pitchclass intervals is obtained. Pitches are converted to pitch classes by applying a modulo operation of divisor 12. Then pitchclass intervals are obtained by taking the difference between adjacent pitch classes. Note that these intervals include both melodic intervals and harmonic intervals, which are not distinguished in our analysis. Finally, zero intervals, which correspond to successions of the same pitch or octave transitions, are dropped because they dilute other relevant features. Each musical piece is now represented as a sequence of pitchclass intervals denoted by x = (x_{1}, …, x_{N}). Since zero intervals are excluded, there are 11 types of unigrams and 121 = 11^{2} types of bigrams for pitchclass intervals.
In Figs 1(a,b) and 4(a), the frequency of tritones for each piece is defined as #{nx_{n} = 6}/x, where x denotes the number of elements in x. Although melodic tritones and harmonic tritones are not distinguished in our analysis, both their uses were once severely restricted in the medieval period and became increasingly common in later time periods^{15,43}. In Fig. 4(b), the frequency of nondiatonic motions is defined as \(\#\{n({x}_{n},{x}_{n+1})\in C\}/{\bf{x}}\), where the set of nondiatonic motions C consists of the following 20 elements: (1, 1), (1, 3), (1, 8), (1, 10), (2, 11), (3, 1), (3, 8), (4, 4), (4, 9), (4, 11), (8, 1), (8, 3), (8, 8), (9, 4), (9, 11), (10, 1), (11, 2), (11, 4), (11, 9), and (11, 11). It can be shown by direct calculation that these bigrams of pitchclass intervals represent nondiatonic motions that cannot be realized by note transitions on a diatonic scale^{18}. Each of the 121 bigram probabilities in Fig. 1(c) is similarly defined as the frequency of each possible pair (x_{n}, x_{n+1}).
For the result shown in Fig. 4(c), a dataset consisting of 761 songs of Japanese enka music is used^{30,31}. Each musical piece is first notated in the MusicXML format and then the sequence of note values (note lengths written in musical scores) is obtained. The ratio between adjacent note values often has simple ratios such as 1, 1/2, 2, 2/3, and 3/2. Similarly as we look at rare pitch events like tritones and nondiatonic motions for the Western classical music data, we observe the frequency of rare rhythms, which are defined as bigrams of note values whose ratio is not one of {1, 1/2, 2, 2/3, 3/2, 1/3, 3, 1/4, 4, 1/6, 6}.
The statistics obtained from the two datasets used to create Figs 1 and 4 are available at https://evomusstyle.github.io/. Although the raw music data cannot be published due to the copyright issue for both datasets, the lists of pieces used for the analysis are provided, from which one can in principle reproduce the exact data we have.
Model formulation
In the discussion above Eq. (3), it is postulated that the novelty term is described as the amount of the cost of obtaining data chunks containing “similar” information in the generated data X_{t} = ϕ_{t}. To express this mathematically, we introduce a function G(θ, θ′) that measures the similarity between data chunks θ and θ′. Assuming that obtaining each data chunk requires the same amount of cost, the total cost of obtaining data chunks similar to θ, denoted by Novelty(θ), is given as
where we have assumed translational invariance in the last expression. When evaluators can discriminate musical features with infinite precision, G(0, θ) is proportional to the delta function δ(θ). In this case, we have
Since the integral is constant with respect to θ, we have shown that the novelty term is proportional to \({\varphi }_{t}(\theta )\).
Model analysis
The parameters a_{t} and b_{t} of the beta distribution are in onetoone correspondence with the mean μ_{t} and standard deviation σ_{t} as follows.
We focus on the case 1 < a_{t} < b_{t}, which leads to μ_{t} < 1/2 and σ_{t} < μ_{t}, and in particular the regime where μ_{t} is small, as in the main text. When \({\mu }_{t}\ll 1\), Eq. (8) indicates that \({a}_{t}\ll {b}_{t}\), and then \({\mu }_{t}\simeq {a}_{t}/{b}_{t}\), \({\sigma }_{t}\simeq \sqrt{{a}_{t}}/{b}_{t}\), and \({\sigma }_{t}/{\mu }_{t}\simeq \mathrm{1/}\sqrt{{a}_{t}}\)
The behaviour of the SCE model defined in Eqs (1) and (4) for the case β_{T} > 0 and β_{N} = 0 can be understood from the following analysis. Equation (4) yields the following equations.
The fact that σ_{t+1} < σ_{t}, which is intuitively trivial, can be formally checked by differentiating the following quantity with respect to β_{T}:
where we have used Eq. (9). We can then show ∂h/∂β_{T} < 0 for β_{T} > 0. By noting that the transformation in Eqs (11) and (12) for any finite β_{T} can be realized by iterating infinitesimal transformations, it has been shown that σ_{t+1} < σ_{t}. By recursively applying Eqs (11) and (12) and substituting the result into Eq. (9), we can also see that
We can show μ_{t+1} < μ_{t} directly in a similar manner. Alternatively, one can understand this by looking at the mode (peak position)
We can easily show that k_{t} < μ_{t} and k_{t+1} = k_{t}, which means that the peak position is invariant under the dynamics. Since the difference μ_{t} − k_{t} decreases as the standard deviation σ_{t} decreases, the result σ_{t+1} < σ_{t} indicates μ_{t+1} < μ_{t}. The mean μ_{t} will converge to the mode k_{t} since σ_{t} → 0 as t → ∞.
For the case of β_{N} > 0 and β_{T} = 0, we gave an intuitive argument in the main text that a slow manifold is formed where σ/μ is almost constant and slightly less than unity. As discussed there, this slow manifold is formed because σ/μ is decreased by an update when it is close to unity, which is in turn due to the boundary at zero and the resulting asymmetric shape of the beta distribution. Here we provide some mathematical analyses to support this intuitive argument. First, we show that the heights of the two peaks of \({\tilde{\varphi }}_{t}={\varphi }_{t}{e}^{{\beta }_{N}{\varphi }_{t}}\) are equal. The position of the right and left peaks (denoted by θ_{+} and θ_{−}) are obtained by solving the following equation:
This yields \({\varphi }_{t}({\theta }_{\pm }\mathrm{)=1/}{\beta }_{N}\). Substituting this back into \({\tilde{\varphi }}_{t}\), we have \({\tilde{\varphi }}_{t}({\theta }_{\pm })=\mathrm{1/(}e{\beta }_{N})\), which is the height of the peaks. Thus the contributions of the two peaks for the determination of \({\varphi }_{t+1}\) are characterized by their position (mean) and width. As can be seen in Fig. 3(e), when σ/μ is close to unity the width of the left peak is so much less than that of the right peak because of the boundary at zero that the \({\varphi }_{t+1}\) is determined dominantly by the right peak.
Next, for the regime of parameters of our interest (\({\mu }_{t}\ll 1\)), \({\sigma }_{t}/{\mu }_{t}\simeq \mathrm{1/}\sqrt{{a}_{t}}\) holds. This means that the ratio σ_{t}/μ_{t} is smaller if a_{t} is larger. That is, the gradient of the beta distribution near zero is smaller. As we see in Fig. 3(e), when the nextgeneration creator’s model \({\varphi }_{t+1}\) is dominantly determined by the right peak of \({\tilde{\varphi }}_{t}\), a_{t+1} > a_{t} generally holds. This shows that σ_{t+1}/μ_{t+1} < σ_{t}/μ_{t} when σ_{t}/μ_{t} is close to unity. Moreover, one finds from the relation \({\tilde{\varphi }}_{t}({\theta }_{\pm })=1/(e{\beta }_{N})\) that θ_{+} increase as β_{N} becomes larger, which in turn indicates that a_{t} becomes larger. Thus, for larger β_{N}, the asymptotic value of σ_{t}/μ_{t} tends to be smaller.
For the case both β_{N}, β_{T} > 0, a slow manifold is formed in the regime where μ_{t} and σ_{t} are small. To understand this, first note that \({a}_{t}\ll {b}_{t}\), \({\mu }_{t}\simeq {a}_{t}/{b}_{t}\), and \({\sigma }_{t}\simeq \sqrt{{a}_{t}}/{b}_{t}\) hold for \({\mu }_{t}\ll 1\). The effect of the typicality term can be written as a_{t+1} = a_{t} + β_{T}(a_{t} − 1) and \({b}_{t+1}\simeq \mathrm{(1}+{\beta }_{T}){b}_{t}\) from Eqs (11) and (12), which indicates the following.
This means that if \({\sigma }_{t}/{\mu }_{t}\ll 1\) the typicality term decreases this ratio by a constant factor of \(\mathrm{1/}\sqrt{1+{\beta }_{T}}\). On the other hand, the novelty term increases it by a factor that becomes larger for smaller σ_{t} (one can confirm this for example in Fig. 3(b)). Thus, the effect of the novelty term dominates over that of the typicality term for sufficiently small σ_{t}, in which case σ_{t}/μ_{t} increases. When \({\sigma }_{t}/{\mu }_{t}\simeq 1\), the effect of the typicality term is negligible, so the slow manifold is formed due to the effect of the novelty term. On the slow manifold, the typicality term acts on the mean μ_{t} as in Eq. (18), which has the effect of a constant reduction factor, and the novelty term acts as illustrated in Fig. 3(g), which has a larger effect for smaller μ_{t}. Thus, for sufficiently small μ_{t}, the dynamics is dominated by the novelty term, leading to a nearly exponential growth on the slow manifold.
The dynamics of the logpotential model can be solved as follows. Substituting Eq. (5) and R_{t} = lnθ into Eq. (2), we have
By solving for the parameters of \({\varphi }_{t+1}\) with the same mean and standard deviation, we find that
Using the relations between \(({\tilde{\mu }}_{t},{\tilde{\sigma }}_{t})\) and (μ_{t}, σ_{t}) given in the main text, we obtain
as claimed in the main text.
Data availability
The datasets generated and analyzed during the current study are available in the GitHub repository, https://evomusstyle.github.io/. See also Data Analysis section above.
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Acknowledgements
The authors would like to thank Masahiko Ueda and Nobuto Takeuchi for useful discussions and Kazuyoshi Yoshii for sharing the enka music data. This work was in part supported by GrantinAid for Scientific Research on Innovative Areas No. 17H06386 from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan and GrantsinAid for Scientific Research Nos 16J05486, 16H02917, 16K00501, and 19K20340 from Japan Society for the Promotion of Science (JSPS). The work of E.N. was supported by the JSPS Postdoctoral Research Fellowship.
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Both authors designed the work and wrote the main manuscript text. E.N. conducted numerial experiments and prepared all the figures. Both authors reviewed the manuscript.
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Nakamura, E., Kaneko, K. Statistical Evolutionary Laws in Music Styles. Sci Rep 9, 15993 (2019). https://doi.org/10.1038/s41598019523806
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