Arrow of time across five centuries of classical music

The concept of time series irreversibility—the degree by which the statistics of signals are not invariant under time reversal—naturally appears in nonequilibrium physics in stationary systems which operate away from equilibrium and produce entropy. This concept has not been explored to date in the realm of musical scores as these are typically short sequences whose time reversibility estimation could suffer from strong finite size effects which preclude interpretability. Here we show that the so-called horizontal visibility graph method—which recently was shown to quantify such statistical property even in nonstationary signals—is a method that can estimate time reversibility of short symbolic sequences, thus unlocking the possibility of exploring such properties in the context of musical compositions. Accordingly, we analyze over 8000 musical pieces ranging from the Renaissance to the early Modern period and show that, indeed, most of them display clear signatures of time irreversibility. Since by construction stochastic processes with a linear correlation structure (such as $1/f$ noise) are time reversible, we conclude that musical compositions have a considerably richer structure, that goes beyond the traditional properties retrieved by the power spectrum or similar approaches. We also show that musical compositions display strong signs of nonlinear correlations, that nonlinearity is correlated to irreversibility, and that these are also related to asymmetries in the abundance of musical intervals, which we associate to the narrative underpinning a musical composition. These findings provide tools for the study of musical periods and composers, as well as criteria related to music appreciation and cognition.

Figure 1

Data extraction protocol. [Panel (a)] Scheme for the process of extracting one pitch sequence from each MIDI file. [Panel (b)] Samples of pieces from different authors (note that the $y$ axis has been shifted for illustration purposes).

Figure 2

Alphabet and sequence size. [Panel (a)] Histogram of the alphabet (number of different notes) for all the pieces analyzed (median $=30$ and kurtosis $=0.6$). [Panel (b)] Semilog histogram describing the amount of pieces that have a certain size $N$ (in number of notes). The majority of pieces have less than ${10}^3$ time steps (median $=456$ and kurtosis $=23$). The smallest piece has $N=35$ data points, whereas the largest has $N=15795$ data points.

Figure 3

Sample time series of $N=6$ data and its associated horizontal visibility graph (HVG) (see [29] for details). By assigning a temporal arrow to each of the links, the degree sequence splits into an in degree and an out degree sequence, such that flipping the time series (time reversal operation) is equivalent to interchanging the in and out degree sequences. Assessing time reversibility in the time series reduces in this context to spot differences in the statistics of the in and out degree sequences, such as comparing the in and out degree distributions.

Figure 4

[Panel (a)] Log-log plot of the minimum number of effective symbols needed to determine time irreversibility of a given musical composition, as a function of the size $N$ of the composition, using our HVG-based method (red plus) and using a standard 2-gram comparison in the note sequence (black crosses). We can see that the number of symbols in the method based on the note sequence is systematically in the same order of magnitude of total size of the sequence; hence finite size effects are too strong and make the method unapplicable. The method based on the HVG requires a substantially smaller number of symbols, orders of magnitude smaller than the sequence size, hence making this method useful in this data set. [Panel (b)] Histogram of the reduction factor $\rho$ (see the text) which compares the effective number of symbols needed in the HVG irreversibility method with respect to a standard method based on $n$-gram statistics. When we average over all the musical pieces, the average number of symbols is reduced 17 times, and this reduction can reach up to 75 times in some cases. This strongly reduces the finite size effects and hence makes the HVG method useful to explore time irreversibility in short sequences.

Figure 5

Irreversibility ratios ${\mathrm{IR}}_1$ for three theoretically HVG-reversible processes (uniform white noise, unbiased random walk, and pink noise) and one theoretically HVG-irreversible process [fully chaotic logistic map $x_{n+1}=4x_n(1-x_n)]$. In every case we generate a time series of $N={10}^4$ data points and symbolize with a vocabulary of $\left|\mathcal{V}\right|=100$ symbols. Red crosses correspond to the empirical irreversibility value of order 1 of the time series ${\text{KLD}}_1\left(\text{in}\right|\left|\text{out}\right)$, black dots correspond to the ensemble average irreversibility value $\langle {\text{KLD}}_1{\left(\text{in}\right|\left|\text{out}\right)\rangle }_{\text{null}\text{model}}$ of ${10}^3$ null models constructed by randomizing the empirical time series (see the text), and the brackets correspond to $\pm$ one standard deviation (note that the $y$ axis is in logarithmic scales). The irreversibility ratio of order 1 (see the text) decides whether the empirical time series is reversible (${\mathrm{IR}}_1\le 1$) or irreversible (${\mathrm{IR}}_1>1$), and correctly predicts the true nature of the process. Interestingly, the empirical irreversibility value ${\text{KLD}}_1\left(\text{in}\right|\left|\text{out}\right)$ of pink noise is notably larger than for white noise or random walk, which could mislead to the suggestion that pink noise is irreversible. The irreversibility ratio asserts that this is not the case, suggesting that the difference in the raw values is due to differences in the specific marginal distribution of the underlying processes, which only yield spurious effects on the determination of temporal irreversibility.

Figure 6

(a) Raw order-1 irreversibility value ${\text{KLD}}_1\left(\text{in}\right|\left|\text{out}\right)$ of a time series of size $N$ generated by a reversible white noise process $x_t=\eta ,\eta \sim \text{Uniform(0,1)}$ (red crosses). For comparison, the results of a null model (mean $\pm$ one standard deviation) where we randomize the original time series 200 times and compute the irreversibility value is shown in black solid circles. The raw irreversibility value decreases without bounds in a fashion similar to the null model, certifying that positive irreversibility values are here only due to finite size effects which vanish as $N$ increases. (b) Order-1 irreversibility ratio ${\mathrm{IR}}_1$ for the process depicted in panel (a), certifying that the time series is not distinguishable from a reversible process for any time series size. (c) Similar to (a), for a time series of size $N$ generated by an irreversible, fully chaotic logistic map $x_{t+1}=4x_t(1-x_t)$ (red crosses). The raw irreversibility value is always positive and stabilizes for $N>{10}^3$. The null model is by definition reversible, and its irreversibility value is only positive due to finite size effects, hence decreasing as $N$ increases. Distinguishability therefore increases with $N$. (d) Similar to (b), for the chaotic case, certifying that the process is irreversible and that the level of confidence increases without bounds.

Figure 7

[Panel (a)] Estimated normalized histogram of irreversibility ratios ${\mathrm{IR}}_m$ ($m=1,2,3$) for all pieces in our data set, in a semilog plot. The green area highlights the region where pieces are reversible. A large portion of the pieces are indeed generated by an irreversible process. [Panel (b)] Abundance per irreversibility confidence class. [Panel (c)] Estimated normalized histogram of raw irreversibility values ${\text{KLD}}_1\left(\text{in}\right|\left|\text{out}\right)$, for all pieces (black) and only those pieces which have previously been certified as irreversible (i.e., ${\mathrm{IR}}_1>1$) (orange). A linear binning of size 0.05 has been performed.

Figure 8

Evolution of irreversibility over different musical periods. ( a) Irreversibility ratio ${\mathrm{IR}}_1$ vs the composer's date of birth of each piece (crosses), for all pieces considered. The red solid line denotes the irreversibility threshold, suggesting that most of the pieces are statistically irreversible. (b) Similar to panel (a), but where a cubic function (green dashed line) has been fitted to data after a linear binning (in periods of 45 yr) has been performed. The fit highlights a modulating trend by which compositions net irreversibility confidence tend to vary over different periods. (c) Similar to the (a) panel, but where irreversibility ratio is averaged over all the pieces of the same composer (one data point per composer). The modulating trend is still present, and for reference we also include the different musical periods (Renaissance, Baroque, Classical, Romantic, and early Modern). (d) Similar to (a), but where we plot the raw irreversibility value ${\text{KLD}}_1\left(\text{in}\right|\left|\text{out}\right)$ as a proxy for the entropy production rate per piece. Qualitatively the same pattern as in (a) emerges. (e) Similar to (b) but where we plot ${\text{KLD}}_1\left(\text{in}\right|\left|\text{out}\right)$. Orange dots correspond to a linear binning and can be interpreted as the average entropy production rate of all pieces over each 45-yr period. This rate is maintained approximately constant over time. The fit is a local estimated scatterplot smoothing (LOESS). (f) Similar to (c), but averaging the value ${\text{KLD}}_1\left(\text{in}\right|\left|\text{out}\right)$ for all pieces of each artist separately. This gives us a proxy for an average entropy production rate per composer.

Figure 9

Nonlinearity index. [Panel (a)] Histogram of the nonlinearity index $\xi$ (see the text) for all pieces considered. [Panel (b)] Nonlinearity index $\xi$ as a function of the composer's date of birth.

Figure 10

Irreversibility is closely related to nonlinearity. (a) Histogram of ${\mathrm{IR}}_1$ for all pieces considered, compared to a similar histogram applied on surrogate pieces where the linear correlation structure (power spectrum) is maintained and other (nonlinear) temporal correlations are removed. We observe that a very large percentage of the surrogate pieces are time reversible, in agreement with the theory. (b),(c) ${\mathrm{IR}}_1$ of each of the 8856 pieces considered, where we clearly see that many of these are confidently irreversible. Their surrogates, where only linear temporal correlations are maintained, are however largely reversible: the construction of the surrogates removes in most of the cases the irreversible character of the original pieces, further demonstrating that statistical irreversibility is a property that cannot be explained by linear correlations (power spectrum) and ought to be related to nonlinear traits.

Figure 11

Mutual information index ${\text{MI}}_{\mathrm{index}}$ between irreversibility, nonlinearity, and asymmetry. Significance for the mutual information between the three properties: irreversibility $[{\text{KLD}}_1\left(\text{in}\right|\left|\text{out}\right)]$ as defined in Eq. (2), nonlinearity $\left[\langle {\widehat{y}}^{\prime }\left(x\right)\rangle \right]$ defined in Eq. (5), and asymmetry ($D_{\uparrow \downarrow }$) as defined in Eq. (8). The red solid line determines the threshold above which there is a statistically significant correlation between indices (the larger ${\text{MI}}_{\mathrm{index}}$, the stronger). On the $x$ axis: (a-i) asymmetry vs irreversibility, (n-i) nonlinearity vs irreversibility, and (n-a) nonlinearity vs asymmetry. Triangles represent the MI significance for all the pieces, while squares display the same quantities computed only considering the pieces which are confidently irreversible (${\text{IR}}_1>1$). The error bars represent the 95% confidence intervals computed by bootstrapping with replacement.

Figure 12

Interval distributions. [Panel (a)] Distribution of the intervals for all the pieces in this study. [Panel (b)] Distribution of the intervals for the ensemble of surrogates constructed in the nonlinearity test.

Figure 13

Interval difference. [Panel (a)] Distributions of the values of $D_{\uparrow \downarrow }$ for all the pieces in the study. [Panel (b)] Mean value of the interval difference for each composer.

Figure 14

Irreversibility ratios of different order are correlated. [Panels (a),(b)] Linear-log scatter plots of irreversibility ratios ${\mathrm{IR}}_m$ vs ${\mathrm{IR}}_{m-1}$ for each piece in the data set. The red dashed line is the best linear fit, concluding that all three orders are strongly correlated. [Panels (c),(d)] Linear scatter plots of composer-averaged irreversibility ratios. The strong correlation observed suggests that, for the set of signals analyzed in this work, ${\mathrm{IR}}_1$ is sufficient to characterize irreversibility and no higher-order quantifiers are needed.

Figure 15

DFA and MDFA calculations. Three different examples (pieces) from the corpus, from top to bottom: Liszt (excerpt from Christmas tree suite), Bartok (excerpt from mikrokosmos), and Mozart (violin sonata no. 21). The panel columns are (a) DFA computation (red crosses) with 20 surrogates represented by the shaded area. (b) MDFA computation performed as in the DFA case. (c) Slope values—polynomial fit derivatives—for the DFA computation. (d) Slope values—polinomial fit derivatives—for the MDFA computation. Continuous curves are the polynomial fits of red crosses and straight lines come from linear regression, of red crosses within scaling ranges, vertically translated for visualization purposes.

Figure 16

Effect of piece length on the nonlinearity index $\xi$ [panel (a)], mean local slope $\langle {\widehat{y}}^{\prime }\left(x\right)\rangle$ [panel (b)], and asymmetry $D_{\uparrow \downarrow }$ [panel (c)].

Figure 17

Effect of piece length on ${\text{IR}}_1$ and ${\text{KLD}}_1\left(\text{in}\right|\left|\text{out}\right)$. As shown by $R^2$, there are subtle correlations of the irreversibility ratio and irreversibility value with piece length [panels (a),(b)], and this correlation is removed [panel (c)] if we only consider the irreversibility value ${\mathrm{KLD}}_1\left(\text{in}\right|\left|\text{out}\right)$, the pieces of which have been previously certified to be irreversible (${\mathrm{IR}}_1>1$).