Evolutionary Methods for Melodic Sequences Generation from Non-Linear Dynamic Systems Eleonora Bilotta1, Pietro Pantano2, Enrico Cupellini3, Costantino Rizzuti1 Department of Linguistics, University of Calabria, Cubo 17b Via P. Bucci, Arcavacata di Rende (CS) 87036, Italy
[email protected] [email protected] 2 Department of Mathematics, University of Calabria, Cubo 30b Via P. Bucci, Arcavacata di Rende (CS) 87036, Italy
[email protected] 3 Department of Mathematics, University of Torino, Via Carlo Alberto 10, Torino 10123, Italy
[email protected] 1
Abstract. The work concerns using evolutionary methods to evolve melodic sequences, obtained through a music generative approach from Chua s circuit, a non-linear dynamic system, universal paradigm for studying chaos. The main idea was to investigate how to turn potential aesthetical musical forms, generated by chaotic attractors, in melodic patterns, according to the western musical tradition. A single attractor was chosen from the extended gallery of the Chua s dynamical systems. A specific codification scheme was used to map the attractor s space of phases into the musical pitch domain. A genetic algorithm was used to search throughout all possible solutions in the space of the attractor s parameters. Musical patterns were selected by a suitable fitness function. Experimental data show a progressive increase of the fitness values.
Keywords: Chua s Attractor, Genetic Algorithm, Generative Music.
1 Introduction Mathematics and Music are strictly tied since the early foundation of Western culture. Since 80s, chaos and fractal geometry have strongly affected the development of a new field of musical research, with the use of non-linear dynamic systems for artistic purpose. Many musical researchers tried using non-linear dynamical systems as melodic pattern generators, able to generate variation or paraphrase like alteration of specified groups of events [1]. Bidlack [2] says that chaos is of potential interest to composers who work whit computers, because it offers a mean of endowing computer-generated music with certain natural qualities not attainable by other means . According to Harley [3], non-linear functions and Music exhibit a
comparable degree of self-similarity or autocorrelation. Lately, modern researches on complexity use sounds and Music trying to develop new methods supporting traditional mathematical ways in understanding emerging behaviours in complex and chaotic systems. In these years Rodet [4] used Chua s circuit for sound modeling, due to its non-periodical components; Bilotta et al. [5] used the same circuit for producing melodies. In the following paragraphs we show how evolutionary methods could be used in order to manage the problems related to generative creation of music. Evolutionary music is a growing discipline that reached surprising successes. Khalifa, and Foster [6], for example, made a composition tool using Genetic Algorithms that generate and combine musical patterns with two fitness functions based on melodic intervals and tonal ratios. The paper is organized as follows: Section 2 presents Chua s circuit and its attractors, describing also the codification process we used. Section 3 explains the Genetic Algorithm and fitness function formal aspects. Section 4 reports about the experimental results and musical analysis of these findings. Section 5 concludes the paper illustrating new directions of this work.
2 Chaos and generative music
2.1 Chua s oscillator Chua s oscillator is a non-linear circuit exhibiting a chaotic dynamical behaviour which provides a great family of strange attractors [7, 8]. Chua s circuit is a dynamical system with three grades of freedom and six control parameters:
α , β , γ , a, b, k ; its dimensionless equations are: .
x = kα [ y − x − f ( x)]
(7)
.
(7)
.
(7)
y = k ( x − y + z) z = k (− β y − γ z ) where:
f ( x) = bx +
1 ( a − b){| x + 1 | − | x − 1 |} 2
(7)
In the initial part of this research we chose to focus our attention on one of the
most famous Chua s attractor: the double scroll pattern (Figure 1). Table 1 shows the control parameter s values of this system.
Fig. 2. Three-dimensional graphic of Chua s double scroll attractor in the space of phases.
Table 3. Control parameters and initial values for the Chua s double scroll.
α= 9,3515908493 a = -1,1384111956
β= 14,7903198054 b = -0,7224511209
γ = 0,0160739649 k=1
2.2 Musical codification Generative music is based on two separate processes: an algorithm generating numerical sequences and a process for translating these sequences into melodic patterns. Since the coding process, also called musification, can be realized in a completely arbitrary manner, it is obvious that the quality of the musical rendering depends substantially on the choices which have been made. For this reasons, in defining a codification scheme, it s important to try realizing not only a mechanism that allows for a simple translation between numerical sequences and musical parameters, but also a codification system allowing for a meaningful transformations preserving the main characteristics of the system s main features in order to exploit, from a musical point of view, the potentials dynamical systems have (fractal nature, different kind of behaviour and so on). At the first stage we was concerned in defining a codification system to associate the solution x(t), y(t), z(t) of the system to a succession of musical notes Sk: Sk = {note1, note2, note3, ..., notek}.
(7)
We have used a simplified codification scheme taking into account only the pitch parameter. Using a linear mapping (Figure 2) we translated into musical pitches, according to the MIDI protocol, the X-axis coordinate in the space of phases (see Figure 1). Moreover we chose to create a new MIDI note every time the waveform,
ti m e related to the evolution of the X coordinate in the space of phases, reaches a maximum or a minimum point. Figure 2 shows a graphic schematisation of the used musical mapping; this kind of musical code can reveal, through the melodic pattern, some topological information about the dynamical system behaviour.
Fig. 2. Graphic schematisation of the musification process. The linear mapping transforms the numerical values of the attractor waveform in musical pitches.
3 Evolutionary process. In order to solve some of the problems related to the musification process, our system adopts a Genetic Algorithm (GA) search strategy to select melodic sequences obtained through a music generative approach. GAs are often cited as appropriate for exploring high dimensional parameter spaces as large regions of problems space can be covered quickly. Bilotta et al. [9] also presented a method based on a Genetic Algorithm to produce automatic music developing a fitness function based on consonance, which allows evaluating the pleasantness of a sequence of notes generated by an algorithm. In this work we chose to use the parameters (
α , β , γ , a, b ) of the Chua s oscillator as a genotype and the melodic patterns,
produced by mapping the X coordinate of the space of phases in a sequence of MIDI events, as phenotype. Table 2 shows the range in which every parameter can be varied: Table 3. Range of parameters defining the searching space. These intervals were chosen to maintain the attractor s stability changing only one parameter a time. However changing more parameters at the same time the system s divergence from the double scrolling attractor (over flow) can occur. 8.80< α
