Musical composition based on skewed statistical distributions of stochastic processes

Figures & data

Figure 1. Probability histogram of the best fit distribution related to the ‘Second Livre, Pièces de viole des Cinq Livres’. Figure based on the work of Lugo and Alatriste-Contreras (2024a). Based on the reported results of this reference, we used the estimated parameters related to the ‘Second Livre’ for reproducing this Figure. The size of the sample was equal to 10, 000. In addition, using the Anderson-Darling test for testing the presence of an exponential case, we find an statistic of 0.39409108950530936. Because this value is smaller than the significance level of 1%, the null hypothesis cannot be rejected, then the data come from the exponential distribution.

Figure 2. Flowchart of the musical composition.

Table 1. Distributions and criteria for note selection.

Case	Distribution	Note criterion
1	exponential	half-step higher or lower
2	exponential	half- or one-step higher or lower
3	uniform	half-step higher or lower
4	uniform	half- or one-step higher or lower

Figure 3. Probability histogram of entropy values. Each random realization is a list of 10, 000 entropy values in the key of E minor. Each value was the output of executing the stochastic process based on the type of statistical distribution and the criteria of note selection. The label of each subfigure is related to the type of statistical distribution used in each random realization (Table 1). See Table 2 for the descriptive statistics of each random realization.

Table 2. Descriptive statistics for case, entropy values.

Case^a	Mean	Median	Variance	skewness	kurtosis
1	2.710576502882607	2.7542476321387994	0.0718492151401425	−0.6990760795112066	0.22861060110614106
2	2.065821504302886	2.0506608249726996	0.27074636502742727	0.038281102044002	−0.8147185342872585
3	2.8104555711915187	2.8509478205456267	0.05432357949222283	−0.7177184126686955	0.31027285197057575
4	2.742780630107914	2.8163974757219528	0.10271883531814868	−1.3552274407654334	2.373126047521664

^a Each case is related to a stochastic note list of n = 10, 000.

Table 3. Spearman correlation coefficient between random realizations.

Comparison	Statistic	p-value
Cases 1 and 3^a	−0.02540861009730572	0.011055059822352071
Cases 2 and 4^b	−0.005155401564168692	0.6062182459299612

^aComparison between the exponential and uniform statistical distributions based on the half-step higher or lower note selection.

^bComparison between the exponential and uniform statistical distributions based on the half-step and one step higher or lower note selection.

Lugo, I., & Alatriste-Contreras, M. G. (2024a). Pièces de viole des Cinq Livres and their statistical signatures: the musical work of Marin Marais and Jordi Savall. arXiv:2404.18355 [cs.SD].

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Data availability statement

The data and the code that support the findings of this study are openly available in [‘Music composition and statistical analysis’] at https://osf.io/nqg7u/?view only = 6d546bec469049b68fe08704ec97b6ad.