Musical composition based on skewed statistical distributions of stochastic processes

Abstract

The inclusion of skewed statistical distributions in the stochastic process for composing music contributes to clarify the effect of some pitch patterns for generating new pieces of music. The aim of this study was to compare and measure the effect of using skewed statistical distributions instead of using the common uniform distribution applied in Markov chains. We applied an explorative data analysis related to the Shannon entropy and the Monte Carlo approach, to generate and compare different stochastic realizations associated with a particular exponential and uniform distributions and two types of note selection based on intervals. Findings suggested that the presence of an exponential statistical distribution may generate a wide range of entropy values that could be associated with the diversity concept in complex systems. On the other hand, the presence of the uniform distribution may generate a narrow range of entropy values possibly associated with less diverse behaviors in simple systems. Therefore, the use of skewed statistical distributions, in particular the exponential, in the stochastic process for musical composition sets ground for the emergence of articulated musical patterns.

Reviewing Editor:

• Nomusa Makhubu, University of Cape Town, South Africa

Subjects:

• Composition & Orchestration
• Elements of Music
• Music Technology
• Theory of Music

Keywords:

• Stochastic process
• skewed statistical distributions
• musical composition
• Monte Carlo simulation
• Shannon entropy
• complex systems
• Python language

1. Introduction

It is a fact that there is not a unique way of composing a piece of music. Musicians have used combinations of different empirical strategies and theoretical structures across the history of music for composing. However, the coordination of these musical efforts is, at the end of the creative process, a decision depending on the subjectivity of the musician. In addition, the use of different types of technologies in music have affected the creative process and its final output (Urkevich, 2020; Webster, 2002). For example, the development of the metronome, from mechanical to digital devices (Bonus, 2014; Martin, 1988; Wotton, 1930), has increase the accuracy of measuring the beats per minute (bpm) and provided the accessibility for anyone who wants to practice music in almost any place. Nowadays, contemporary musicians have increased the use of digital technologies for generating simple and complex pieces of music across different musical styles. One of these implementations is the use of mathematical and computational models for composing music (Ames, 1987). In particular, the stochastic approach—one of other general approaches: rule-based and artificial intelligence systems (Maurer, 1999)—has been used by musicians and scientists for assisting the composition process and generating original pieces of music. This approach uses non-deterministic methods for generating a collection of a random sequence of notes based on a Markov chain (Ames, 1989). This process will select a note based on its current state, and this selection will be dependent on a set of closest notes weighted equally. These type of weights are related to the uniform distribution over some intervals (Serfozo, 2009). However, there are other statistical distributions that can be used in such a selection process. In this vein, we extend the stochastic approach of mathematical models for composing music to add highly skewed statistical distributions related to singular audio signals belonging to particular musicians.

Compared with other stochastic processes for generating a piece of music, we propose to add highly skewed statistical distributions to the transitional probabilities of the Markov chain. Based on the work of Lugo and Alatriste-Contreras (2024a), we aim to partially control the compositional process by weighting the selection of pitches using those types of statistical distributions, which were previously identified by a signal processing and data analysis. Therefore, our main questions are the following: What is the difference between using skewed and non-skewed statistical distributions for composing pieces of music in the stochastic process? are they significantly different from each other? what are the implications for such similarities or dissimilarities? Consequently, our research methodology is based on a theoretical and quantitative perspective in which we used the tuning system of the Western music to generate a collection of possible notes that follow different statistical distributions in the stochastic process. After generating such a collection, we analyzed each sequence of notes by an explorative data analysis that uses statistical analysis and model simulations for identifying arrangements of sounds related to melodies. In particular, we used the complex systems approach as a framework for collecting and using the music, and computer programing and statistical analysis for processing the data. As we mention above, based on the stochastic approach for composing music, we propuse a simple but significant change in the Markov chain that composes music following different transitional probabilities. In the field of computer programming, we use the Python computer language for writing code and share it in an open platform for transparency in science (Macleod et al., 2021). In the statistics sphere, we use an explorative data analysis for identifying patterns among different stochastic realizations based on values of the Shannon entropy. In particular, to validate our results, we used a sensitivity analysis based on a Monte Carlo simulation that estimates the probability of occurrence of such realizations. Therefore, we hypothesize that the identification of statistical patterns in different pieces of music assists in the process of composing new music with or without the use of computational tools such as machine learning algorithms. In particular, if we know the type of statistical distribution that best fit some particular audio waveform, for example the range of pitches and their frequencies, we can explore and compose simple pieces of music or audio samples—audio recordings—using the computer approach. Such samples can represent the initial inputs for developing any piece of music. By following this approach, we aim to contribute to identify the fundamentals of composition that explain and replicate aspects of the human creativity.

This study is divided in five sections. The first section gives a brief overview of the literature related to the stochastic process for composing music. The second section presents the data used in the stochastic process and related materials of coding libraries. The third section shows the method that describes briefly the modification from the non-skewed to skewed statistical distributions in the Markov chain. Moreover, we describe the explorative data analysis for comparing such changes and identifying relationships between them. The fourth section shows the results. Finally, the fifth section presents the discussion and gives final comments.

1.1. Literature review

This section provides a brief literature review of the field of stochastic models for composing music. In particular, we are interested in showing the contributions of atonality ideas—the lack of a key—and the tonality system—a tonal hierarchy—for composing music based on musical intervals. However, when comparing such a type of model with the other approaches for algorithmic composition mentioned in the Introduction, we identify a key advantage of using it: reproducibility. It is worth to mention that whereas such general approaches delegate the creative decision to the computer (Cope, 1984; Hutson, 2018), the stochastic process assists the musicians in their creative process for musical composition. For example, based on the open practice of transparency, openness, and reproducibility in science (Nosek et al., 2015), we may provide an approach to ensure the replication of findings by providing data, key research materials, and using a predictable way of generating a sequence of notes based on probabilistic variables associated with well-known statistical distributions. Therefore, we can see the use of the stochastic model as a flexible formulation in which musicians and scientists can incorporate it into their processes of composition and analysis.

We come back to the point of atonality and tonality. The term of atonality in music is referred to as the lack of a key related to the twelve-tone equal temperament system (Forte, 1977; Rahn, 1980). Because of the lack of a tonal center, atonality does not consider a central triad, which is based on intervals related to their consecutive thirds, and the notes in the chromatics scales are not related to each other. Therefore, the term of atonality describes an alternative way of composing music without a tonal hierarchy. Examples of this approximation include the work of Claude Debussy and Igor Stravinsky just to mention a few (De Leeuw, 2005). It was not until the work of Arnold Schoenberg and the Viennese School that the term of atonality was used as we know it (The Editors of Encyclopaedia Britannica, 2023; De Leeuw, 2005). In addition, it is worth mentioning that the musical work of Ferrucio Busoni not only was other example of using an atonality approximation in his compositions, but also he contributes in the related field of microtones (Busoni, 1911). For example, Busoni considered a subdivision of semitones that increase the musical system—new intervals—of the twelve-tone equal temperament (Morgan, 1991). Using this subdivision, the musician can identify and use new notes. The understanding of these ideas is not trivial because there is a large number of traditional instruments that cannot reach such semitones. For example, notes related to such a subdivision of semitones can be located between the keys of a piano or, in a guitar, can be located between the frets. Then, the use of computers for identifying and playing such notes is the best and the most efficient tool for reaching such subdivision of notes. Therefore, atonality and tonality in music are interwoven systems based on intervals that can be used directly in the stochastic process. Both systems can generate sequences of random notes following a particular criterion of note selection.

In addition, in the context of mixing the music and science for composing music, the work of Iannis Xenakis has shown a trascendental contribution. In his book of Formalized Music (Xenakis, 1992), he provided an interdisciplinary and novel view for understanding and generating music. Like Busoni, he also considered the creation of sounds in a microsound level in which computer methods based on mathematics and statistics are the best option for generating sounds and using them as musical instruments. In addition, he suggested that ‘arts lead science’ because arts are partially inferential and experimental, and science is ‘entirely inferential and experimental’. For example, during the process of composing a piece of music, the musicians can follow different paths in which their emotional expressions are not only communicating based on a particular musical system, but also are based on their cultural and social contexts. Therefore, Xenakis provided the bases for composing music based on a different approximation in which the science can contribute to creating pieces of music by using novel combinations of sounds. Following these ideas, there are different references that research and promote the development of this particular approximation, for example Curtis and Strawn (1987), Curtis (1989), and Moore (1990) provided the historical perspective and basic concepts of the computer music, Curtis (2001) addressed the area of microsound and the way of using it for composing music and Taube (2005) and Manaris and Brown (2014) showed the programing perspective for using particular computer languages of musical composition and data analysis.

In the same vein, the present study may continue these efforts for not only communicating and distributing the development of musical composition, but also for promoting the analysis of audio signals that provide a complement data for discover novel patterns in music. In the next section, we shall mention preliminary studies for considering in the stochastic process, describe materials related to the Python libraries, and present our method.

2. Materials and methods

Composing music using computers is not trivial. This process may follow a number of sequential steps that mix musical theory and programming skills. Nowadays, there are at least two possible approaches for composing music using computers. The first is the common way of composing material based on different audio softwares and divises for example, the Pro Tools software (https://www.avid.com/es/pro-tools) and the AKAI products (https://www.akaipro.com/). Such devices are generally used by musicians and people related to the music industry. On the other hand, the second is the scientific approach based on the use of programming languages. In this approach, the emphasis lies in applying scientific methods in musical composition in which mathematical and statistical methods are the bases of the composition and data analysis. Then, the selection of the programming language is one of the keys for translating basic and complex concepts of music into the code. Up until now, there are different computer languages that show variable approximations for composing music. In this study, we used the Python programming language and its third party libraries for composing music. In particular, we used the Musicpy library (https://musicpy.readthedocs.io/en/latest/) because it not only provides the tools for composing music, but also for analyzing music through the music theory logic. Moreover, we used the numpy (https://numpy.org/), matplotlib (https://matplotlib.org/), and scipy (https://scipy.org/) libraries. It is important to mention that the code, data, and results are openly available to replicate our study in the following Open Science Framework project: Music composition and statistical analysis (see the section of Availability of data, page 11).

In addition to the computer language as material for composing music, we also suggest that composing music using computers requires an ex-ante data analysis based on a signal processing. In particular, if we understand and visualize similarities of statistical patterns between different types of music, composers, and performers, we can go ahead with strong support for composing music based on a scientific background. For example, the work of Lugo and Alatriste-Contreras (2024a,b) identified similarities between different audio signals. In particular, based on the spectrum of audio signals, they identified the best-fit statistical distributions related to electric guitar and viola da gamba players. An important result of these references suggested that highly skewed distributions are the most common statistical attributes in the audio signal of pleasing music. Therefore, we selected one of the most representative cases of their results for setting the initial condition in our composing process: the viola da gamba, played by Jordi Savall and composed by Marin Marais. This case is related to the audio work of the ‘Second Livre’ of the ‘Pièces de viole des Cinq Livres’. The distribution that best describes this audio work was the exponential distribution (Figure 1).

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.

Therefore, using the audio signal of any other musician, we can identify statistical patterns that enhance our understanding of the music. Such statistical attributes may be used as inputs for composing different type of music genres. In the next section, we present the use of the these materials based on the stochastic models for composing music.

2.1. Method

Studies in stochastic models for composing music are primordially based on a collection of possible notes—i.e. a range of notes related to the scientific pitch notation in which the name of the note is associated with its number of octave—for generating a sequence of random notes that follows particular criteria for selection (Xenakis, 1992). The most elementary criterion is to select notes that are equally likely—i.e. independent and identical distributed. Based on this criterion, we can generate a sequence of notes that show nothing in common with the current and next notes—this description exemplifies the atonality formulation.

There is another process for generating a sequence of random notes that depends on the current note. This process is called a random walk following a Markov property (Ames, 1989; Gentle, 2009). The basic process starts by setting the initial note. Then, the next note is half-step higher or lower depending on the current note. The selection of this note is equally likely—i.e. the selection depending on the probability of flipping a fair coin. Following this basic idea, there are extensions that may vary the selection of the next note. For example, we can add a tone in our posible selection of semitones, or we can select notes that are related to triads in a particular key—the root, third and perfect fifth. In particular, we can use the atonality or tonality system. Even thought there exists different combinations for selecting the next note, the process commonly uses the uniform distribution where every outcome has the same probability of occurrence (Levin et al., 2008). Therefore, the sequence of random notes generated by this type of random walks depends on the rule of note selection and the type of statistical distribution for such a selection.

Based on this process, we propose to modify the selection of notes based on highly skewed distributions related to Figure 1. In particular, we used the following algorithm based on the tonality system for generating a list of notes (Figure 2).

Figure 2. Flowchart of the musical composition.

The first process is associated with the selection of the key that defines the group of pitches that forms the musical composition. Instead of using the atonality approach, we used the tonality system associated with Western music due to its relevance and practical utility for regulating contemporary music. The second process is related to the criteria for note selection in the Markov property. Instead of using the uniform distribution for selecting a random sequence of notes, we propose to use highly skewed distributions. In particular, we used the exponential distribution mentioned in Figure 1. However, it is possible to use the uniform distribution for selecting random notes and to generate a list of notes as a way of comparing different results. These random selections are based on two criteria for choosing the next note. First, depending on the current note, the next note can be half-step higher or lower. Second, the next note can be half- or one-step higher or lower (Table 1). The third process generates the list of notes. In this process, we used the standard or concert pitch for setting the initial note, A4. Because this pitch is approximately located at the middle of the sounds related to the scientific pitch notation, we ensure that the starting process is balance or unbiased. The number of random notes generated by the process is based on the tempo—bpm. Then, we selected 120 notes per minute, bpm = 120, in each list of random notes. This tempo represents two beats per second that is sufficient to compose a piece of melody. Therefore, we used this algorithm for generating sequences of notes based on the exponential and uniform distributions. In each case, we used the first and second criteria for selecting the next note.

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

After generating the lists of notes, we used an explorative data analysis (Leek & Peng, 2015). In particular, we are interested in quantifying the amount of uncertainty associated with each random process of notes. We applied a model validation that compares the uniform with the exponential selection (Gentle, 2009; Oberkampf & Roy, 2010). For this purpose, we computed the Shannon entropy for each list and compared them with each other (Shannon, 1948). Mapping the values of the Shannon entropy by the type of statistical distributions and the next note criteria in a frequency histogram, we can identify which of them are the most or less uncertain. In other words, which combinations of such criteria show variability of notes related to an apparent melody or which ones show regularity of notes related to a stable pattern. Consequently, we used a sensitivity analysis based on a Monte Carlo simulation that estimates the probability of occurrence of note patterns (Johansen, 2010). Finally, we computed the Spearman correlation coefficient between the exponential and uniform distributions for testing similarities or dissimilarities (Scipy, 2023).

3. Results

This section presents our findings based on the explorative data analysis. As mentioned earlier, our model validation is based on generated lists of random notes related to different types of statistical distributions and the next note criteria. A crucial part of our model validation process is the sensitivity analysis that used a Monte Carlo simulation approach. Therefore, we shall display the collection of entropy values based on frequency histograms and the estimated Spearman correlation, as well as, the descriptive statistics for characterizing our results.

3.1. Model validation and sensitivity analysis

Our model validation compares the use of skewed and non-skewed statistical distributions in the stochastic process for music composition. In particular, we are interested in exploring the differences or similarities in choosing exponential or uniform random notes for generating a list of them. In addition, we consider two criteria for selecting the next note: half-step higher or lower notes and half-step and one-step higher or lower notes (Table 1). Then, combining the type of statistical distribution and those criteria, we used a sensitivity analysis for determining their relative influence in our model output.

After iterating 10, 000 times each combination of statistical distributions in the key of E minor and their note criteria, we computed their entropy values and plotted their probability histogram (Figure 3). Each subfigure compared the exponential and uniform distributions based on the criterion of half-step higher or lower note and the criterion of half-step and one-step higher or lower notes respectively (Table 1).

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.

Figure 3 displays the range of how probable low and high values of entropy are associated with each case. Subfigure (a) shows a variety of entropy values related to similar shapes of histograms. However, this subfigure displays a smaller range of entropy values associated with the stochastic realization based on the uniform distribution. On the other hand, entropy values associated with the exponential distribution show a wider range of values. Subfigure (b) reveals a variety of entropy values associated with disimilar shapes of histograms. This subfigure also shows a smaller range of entropy values associated with the stochastic realization based on the uniform distribution. Meanwhile, the values related to the exponential distribution show a different shape of histogram, resulting in a wider range of values. Moreover, for the exponential distribution, there is a higher probability of finding lower values of entropy which gives evidence of the presence of more diversity and the possible emergence of interesting patterns of notes. These results together with Table 2 may imply that in both cases, there are significant differences between the random realization based on the type of the stochastic proceses.

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.

To clarify the results of Figure 3, we show the Spearman correlation test (Table 3). This test shows an association or relation between two sets of data, and it is particularly helpful for answering the question of how significantly similar or different are from each other the particular stochastic realizations associated with the entropy values. According to Table 3, it is evident that the values of the correlation coefficient in both cases are close to zero. Therefore, our selected realizations for comparison showed no evidence of similarities. They are different from each other.

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.

Therefore, the results observed with the distributions and the correlations suggest that the used of an exponential statistical distribution in the stochastic process of musical composition is significantly different from a random realization based on a uniform distribution. We highlight that for the exponential we observe a wider range of entropy values in each random realization and higher probabilities for lower entropies indicating that the output of the stochastic process for musical composition may indicate that complex patterns of notes emerge. These patterns are related to an apparent melody or music that is pleasing. On the other hand, a narrower range of such values supports the idea that the output of the stochastic process may generate regular patterns of notes associated with noisy sounds.

4. Discussion

The stochastic process is one option for composing music assisted by the use of computers. The use and exploration of this process have to consider the selection of an statistical distribution for generating the random realizations. In particular, the used of skewed statistical distributions may generate different outputs possibly associated with complex patterns of notes related to harmonic or melodic intervals. Therefore, it is not trivial to choose any type of statistical distribution in the stochastic process for composing music.

In this study, we generated and compared random realizations related to the stochastic process of the Markov chain for composing music based on skewed and non-skewed statistical distributions and the next note criteria. We found that the use of exponential or uniform statistical distributions in the stochastic process matters for composing music. In particular, based on our findings, we interpreted the wider range of entropy values and higher probabilities for lower entropies as desired statistical attributes related to melody or pleasing music. On the other hand, narrower range of entropy values and higher probabilities for higher entropies can be related to noisy sounds. This study therefore indicates that the benefits from using skewed statistical distributions against the use of non-skewed statistical distributions in the stochastic process may address the musical needs for composing music according to intervals of scales, harmonies, and melodies.

Most notably, this study exemplifies the presence of the emergence property in complex systems when showing how an small but significant change in the stochastic process affects the relationship between the note selection, and then its final output. In particular, the implications of dissimilarities between the random realizations associated with the use of the exponential or uniform statistical distributions are the following: (a) Composing music based on a particular frequency of pitches associated with a singular skewed distribution may generate branches of patterns related to pleasing pieces of music; and (b) it is highly probable that the creativity process, consciously or unconsciously, associated with most of the music may follow skewed statistical distributions for pitch selection.

Some advantages and disadvantages of our study are worth noting. The advantage of our method is the algorithm for composition because it is simple to execute and can be extended in different directions. The disadvantages for generating such an algorithm is to apply interdisciplinary ideas and tools associated with music theory and computer languages. In any case, any person who followed our proposal can compose simple patterns associated with melodies or chords progressions, as well as identify attributes of music that may unveil underdeveloped patterns in music. Future work should therefore include the use of other next note criteria, for example the use of interval quality based on the perfect Major/Minor intervals—i.e. root, 4th, 5th, and 8ve—and Major/Minor intervals—i.e. 2nd, 3rd, 6th, and 7th. In addition, we should include the use of other methods for musical composition, for example a cellular automaton based on music theory.

4.1. Conclusion

To use of skewed or non-skewed statistical distributions in the stochastic process generates different outputs. By using the former distributions, for example the exponential, we can generate sequence of notes that can be related to organized arrangements of sounds, also known as music. Meanwhile using the latter distributions, for example the uniform, generates note outputs related to random sounds that do not accord with intervals of scales, harmonies, and melodies.

Disclosure statement

No potential conflict of interest was reported by the author(s).

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.

Additional information

Notes on contributors

Igor Lugo

Igor Lugo is interested in an interdisciplinary approach of science. He is interested in the fields of economic geography, complexity, transportation systems, system of cities, ancient civilizations, musical composition, computational modeling, and programming.

Martha G. Alatriste-Contreras

Martha G. Alatriste-Contreras is interested in the interdisciplinary scientific research on complex systems. In particular, she is interested in the study of the properties and dynamics of economic systems, network and statistical analyses, modeling, and programming.