IV Encuentro Nacional de Investigación en posgrados - ENIP 2009. Santafé de Bogotá, Colombia, 2009. ISBN 978-958-719-374-9 1 Controlling Chaotic Melodies Andrés E. Coca, Gerard O. Tost and Zhao Liang ABSTRACT This paper presents an algorithm for automatic composition of melodies by using nonlinear dynamical systems. The variables of the systems are used for extraction of components that constitute the musical melody (pitch, rhythm and dynamics) which can be adjusted according to specific musical input. It should be highlighted that the proposed algorithm not only can generate melodies with microtonal scales, but also is capable of having the whole set of possible scales and modes in the temperated system. Techniques for control of chaos are used to transform the chaotic attractor to a periodic or a fixed point attractor, generating melody of a repeated musical phrase, a series of consecutive notes (arpeggio) or a sustained musical note. Keywords— automatic composition, chaotic dynamical system, controlling chaos I. I NTRODUCTION main characteristics of chaotic systems are topological transitivity, dense unstable periodic orbits within its invariant set and sensitivity to initial conditions [7]. This last feature triggers much interests in automatic music composition. T He Chaotic music has its origin in the analysis carried out on the fractal structures presented in the classical music [11] and experiments realized in self-similar music [15]. This step gave the application of fractal structures in the automatic generation of music, creating a new genre in computer assisted composition (CAC) called “fractal music” [13]. One of the methods for the algorithmic generation of fractal objects, known as Iterated Function Systems (IFS) and a special case the L-system or Lindenmayer system, has been explored in musical experiments [16]. Due to the relationship between fractal geometry and chaotic attractor, in the late 80’s, some researchers have begun to explore the musical potential by nonlinear dynamical systems, especially chaotic systems [14]. Finally arriving the composition of melodies with contiguous or discrete chaotic systems of different dimensions. The main feature of these applications Andrés E. Coca: aecocas@unal.edu.co, Master in Engineering - Industrial Automation, National University of Colombia. Licenciate of Music, University of Caldas. Gerard O. Tost: golivart@unal.edu.co, Professor Dept. of Electronic Engineering and computing, National University of Colombia. Zhao Liang: zhao@icmc.usp.br, Professor Institute of Mathematics and Computer Science, University of São Paulo, São Carlos, Brazil is the possibility to obtain large quantities of different musical fragments with a slight change in initial conditions of the system [18]. Currently, the composition of chaotic systems is discussed from various perspectives, such as the application of chaos to change the variables of compositional elements of electroacoustic music [29], the design and construction of musical instruments including control of chaos [28], the study of complexity and cognition through the transformation of the dynamic characteristics of the circuit of Chua to the musical space [26], generation of polyrhythmic by coupled nonlinear oscillators [27], and the combination of grammatical rules with chaotic systems [25], among others. However, in some cases, the chaotic phenomenon is undesirable. The main reason is the critical dependence on initial conditions, which is never perfectly known in experimental practice. So the chaotic system produces exponential divergence between the evolution of the actual path temporary and theoretical trajectory, turning the system inherently unpredictable. Thus, sometimes chaotic dynamics should be controlled. Moreover, due to the fact that chaotic dynamics contains an infinite number of unstable periodic orbits that are visited ergodically in evolution, the application of control technique makes it possible to have an infinite number of types of dynamic behavior by using a single chaotic system [5]. In the musical context, these unstable periodic orbits may comprise an infinite number of musical phrases shaped by groups of highly similar semi-phrases melodies, which is key feature for many established musical forms. Moreover, if it takes the system to a periodic point, one can generate groups of consecutive notes that form various chords that are useful in the composition of accompanies. As a special case, the control strategy, which drives the system from chaotic state to a fixed point, ensures the transition in a melody from the unpredictable behavior to a sustained note that can be used in the formulation as a pedal note. All these help to expand the CAC tools, avoiding monotony and giving variety to the musical results. The first method to control chaos was proposed by Ott, Grebogi and Yorke in 1990, which is known as the OGY method [23]. This method has evoked great interest in studying controlling of chaos. Subsequently developed other methods such as the method of Time-Delayed Feedback (TDAS) [19], generally known as a method of Pyragas controller and method for induction into the fixed point (FPIC) [21]. 2 In this paper, a new musical composition algorithm is proposed. It allows the generation of melodies from dynamical systems with continuous or discrete variable, in addition to the possibility of modifying the parameters and specifications of solving the system equations such as initial conditions, the time interval integration and the proper parameters of the model. It also has a feature to change some predefined musical specifications. Within these stand out the possibility of using microtonal musical scales without technical limits in tone division and the total number of scales in the temperated system. The latter is part of the results obtained by a combinatorial analysis on the scales and musical modes. Here it is worth noting that the total number of possible scales and modes using transpositions and combinations of tones, semitones and tone and a half is 11124 and 1584, respectively [24]. Thus, the algorithm covers a large part of the musical space. The paper is organized as follows. Section II describes the development of the musical compostion algorithm, Section III gives a quick review of the chaos control methods, Section IV presents the musical results by applying the control techniques. Section V concludes the paper. II. A LGORITHM FOR C OMPOSITION OF C HAOTIC M ELODIES The algorithm starts by using the numerical solution of a nonlinear dynamical system consisting of three variables x(t), y(t), and z(t). The first variable x(t) is assigned to the extraction of musical pitches (frequencies and MIDI notes), the second variable y(t) is attributed to the duration of each musical note (in seconds or in time units), and the third variable z(t) is the musical velocity (intensity and musical dynamics). Each pair of variables can be unconditionally exchanged in musical property representation. Data transformation of the variables are described in the following subsections [24]. II-A. Musical Specifications Below we describe the initial inputs of the chaotic musical composition algorithm which are selected according to the musical target or techniques of instrumentation. II-A.1. Musical Scale Specifications: 1. Number of Octaves k: indicates the extent of scale in octaves, represented as k ∈ N and within the range 0 < k ≤ 7. 2. Tonic Υτ,o : is the initial tone where one wants to start the scale. It is defined as the pair Υτ,o = Υ (τ, o), where τ : {τ ∈ N |1 ≤ τ ≤ 12 } is the tone and the o : {o ∈ N |o < k } is the number of octaves of the scale. 3. Mode m0 : is a value within the range 0 < m0 ≤ m, where m ∈ [0, 11] is the maximum number of possible modes for a given scale. It indicates the number of required shifts which a scale starts in a tone given by Υτ,o . 4. Name or Structure of the Scale ψ: the set of interval generators that form the architecture of the musical scale. It is represented by the set ψ = (s, t, tm ), where s ∈ [0, 12] is the number of semitones, t ∈ [0, 6] is the number of tones and tm ∈ [0, 4] the number of tone and a half that form the structure of the desired musical scale ξ with n notes. 5. Tone Division ∆: this specification is used only when you want to use a micro-tonal chromatic scale, in which the number of tone divisions of the ∆ ∈ N1 should be ∆ ≥ 2. ∆ = 2 is used as the tempered system. II-B. Extraction of Frequencies and Musical Notes The extraction of frequencies and musical notes is divided into three stages. The first step is to generate the membership binary vector and the scale intervals of the musical scale specified by ξ. In the second step, a normalization of the variable is performed, and the final step maps the normalized data to the scale intervals. II-B.1. Part 1: Scale Generation: The inverse of the number of tone divisions λ = 1/∆, is called tuning factor of the scale and is within the range λ ∈ (0, 0,5]. With the tuning factor and the number of octaves, a vector S of dimension 1 × (p + 1) is constructed. The variable p, equal to p = 6k/λ, is the number of notes containing a chromatic scale in k octaves following the given tuning factor. The vector S containing a equally tempered scale intervals is generated by the geometric progression as follows iλ S(i+1) = 2 6 , 0 < i ≤ p (1) With the structure ψ = (s, t, tm ), the components of the membership vector V of a scale can be determined as follows  1, if ξi ∈ S (2) Vi = 0, if ξi ∈ /S The membership vector of a scale V acts as a filter on the vector S, keeping only those intervals generators of ξ and eliminating the others. The result of this operation is the vector Ei = Vi · Si . The elements in the vector E, which are equal to zero, are eliminated, we obtain the vector E′ . II-B.2. Part 2: Variable Normalization: The selected variable x(t) of the nonlinear dynamical system to represent the frequencies of musical notes should be normalized with respect to the vector E′ , because the range of the variable x(t) is different from the range of the data vector E′ . This normalization process, defined as x′ (t) = γ (x(t), E′ ), consists of a scaling and a translation of the numerical 3 solution of variable x(t) adapted to the data vector E′ , obtaining the normalized x′ (t), as follows: x′ (t) = αx(t) + β (3) where α is called scaling factor and is calculated by the following formula: α= 2k − 1 máx (x(t)) − mı́n (x(t)) (4) Similarly, the variable β is called a translation factor and is determined by the following equation: β = −α mı́n (x(t)) + mı́n (E′ ) = −α mı́n (x(t)) + 1 (5) II-B.3. Part 3: Mapping to the Closest Value: Once the variable is normalized, we proceed to determine for each value x′ (t) the closest value in the vector E′ , getting a match between data x′ (t) with the notes of the musical scale specified in ξ. This mapping is defined as Lx = Φ (x′ (t), E,µ). Then we will build a matrix D of dimension cx′ (t) × n, such that cx′ (t) represents the number of elements of a piece of numerical solution of x′ (t) and n is the number of musical notes.  0, if |x′ (t)j − E′ i | ≤ µ Dj,i = (6) x′j if |x′ (t)j − E′ i | > µ λ The threshold value µ, calculated by µ = 2 6 − 1, is equal to the minimal interval generator. Then we generate a new vector Lx of size cx′ (t) × 1, in which it holds the position of the minimum values for each row of the matrix D. The knowledge of tonic Υτ,o is required for the conversion ′ of the variable the appli x (t) to the musical2 space. Define cation ϕ : Υτ,o → fτ,o ; Υτ,o ∈ N ; fτ,o ∈ R1 to perform the conversion of frequency of a musical tone τ in the octave o as follows, fτ,o = ϕ (Υτ,o ) = 55 · 2 τ +12o−10 12 0 < i ≤ cx′ (t) Here we wish to relate the variable y(t) of the nonlinear dynamical system to rhythmic values in units of time by using the normalization process y ′ (t) = γ (y(t), R), which is made with respect to a vector R of size (1 × 7) that contains the appropriate numeric values representing the musical rhythm patterns, where 0 represents the sixty-fourth note, 1 represents the thirty-second note, ..., and 6 represents a whole note. In this step, the application of the mapping is unnecessary due to the relationship between the normalized y ′ (t) and the rhythmic values, applying the round function [30], i.e., Y = ℜ+∞ (y ′ (t)). The vector Y is used to generate the midi file. For generating a audio file, the converting of the rhythmic values contained in Y to time in second is required, as follows Y′ = 60 2Y · tp 24 (10) II-D. Variables for Dynamics For the transformation of the variable z(t) in musical velocity, it follows a procedure similar to that conducted in the previous transformations. The normalization of the variable now is made with respect to the predefined velocity contained in the vector U, i.e., z ′ (t) = γ (z(t), U). The vector U is initialized with constant values that represent the intervals of velocity of musical dynamics. After the normalization it takes out of the new mapping between the normalized z ′ (t) and vector U, using the threshold value µ = 10; represented as Lz = Φ (z ′ (t), U,µ). Thus the position of the minimum of each row Li,1 is obtained and we get the vector Z of size 1×cz′ (t) containing the velocity values generated by the variable z(t) of the nonlinear dynamical system. To generate the audio file, elements of the the vector Z are normalized with respect to the maximum: (7) With indices of L and frequency of tonic fτ,o , we calculate the frequencies of musical notes corresponding to the variable x′ (t) as follows Fi = fτ,o · E′ Lxi II-C. Variable for Rhythm Z′ = Z max (Z) (11) With the matrices X, Y and Z, one obtains a note matrix M to create the midi file, and F, Y′ and Z′ , a suitable matrix to generate the wav file by audio synthesis. (8) Making the conversion to standard MIDI, with the values of frequencies F, we obtain a vector X of dimension 1 × cx′ (t) , which contains the numbers of tones generated by x(t).   12 |F| (9) X = 69 + log log 2 440 III. R EVIEW OF THE M ETHODS OF C ONTROLLING C HAOS In this section, wwe briefly reviews some classical methods of controlling of chaos, which ar applied to own musical composition system. 4 III-A. Method of Control by Time-Delayed Autosynchronization (TDAS) The TDAS method was proposed by Pyragas in 1992 [19]. This method considers the problem of stabilizing an unstable τ -periodic orbit embedded in the nonlinear system by means of the following control law u (t) = K (x (t) − x (t − τ )) (12) where K is the the gain of perturbation and τ is the delay time. If τ is the period of a periodic solution embedded in the variable x(t) without control (u = 0), by applying the control law, the variable will converge to this periodic orbit [20]. To apply the Pyragas method, it should be delay the variable and feeds it to the original system in such a way that stabilizes the desired orbit. III-B. Method of Fixed Point Induction Control (FPIC) This method was first proposed in 2004 [21] and was developed specially for controlling discrete systems. It is based on the theorem of continuity of eigenvalues. This control technique has the great advantage that it is allowed to stabilize chaotic orbit to periodic of orbits long period. It requires no prior knowledge of state variable but the fixed points of the system. With the fixed point, one designs a control strategy that drives the system to evolve in the specified fixed point [22]. The control technique FPIC can be applied to autonomous systems, non autonomous and systems with a period of delay. III-B.1. Autonomous Systems: Given a set of difference equations in the following form: x (n + 1) = f (x (n)) (13) where x ∈ Rn and f : Rn 7→ Rn . We want to control the system to a fixed point, which has a fixed point x∗ that complies with the conditions x∗ = f (x∗ ) |λi (J)| = λi δf δx x∗
(14) > 1 ∃i (15) Then the system can be controlled by adjusting the equation of the system. x (n + 1) = f (x (n)) + N x∗ N +1 (16) In this way, it is guaranteed to stabilize the fixed point for some positive N , which is determined according to the restriction: N > máx |λi (J)| − 1 (17) III-C. OGY Method This method uses Unstable Periodic Orbit (UPO) embedded in the chaotic attractor in order to take the trajectories sufficiently close to the unstable periodic point. Then the control is activated by applying small perturbations in some control parameter [23]. The OGY method is based on determining a set of unstable orbits from the Poincaré section of the chaotic attractor and then select one of these. IV. E XPERIMENTAL R ESULTS This section presents experimental results by using own melody generation algorithm and applying controlling of chaos. IV-A. Composition of Melodies with a Continuous System Controlled by the Pyragas Method The Pyragas method has been applied to control periodic orbits of τ -period inherent in the chaotic Chen attractor. The system with control is described as follows: ẋ = a (y − x) ẏ = (c − a) x − xz + cy + u (t) ż = xy − bz u (t) = K (y (t) − y (t − τ )) (18) With the parameters a = 35, b = 3 and c = 28. Where u (t) is the control law, K is the gain of perturbation and y (t − τ ) is the delayed output signal. When K = 0, the control law is eliminated and the attractor retains its chaotic behavior. The design of Pyragas method consist of finding the value of τ , which is the period τi of an unstable periodic orbit and its associated perturbation gain when the control action goes to zero u (t) = 0. This is equivalent to having the 2 perturbation time equal to zero D2 (t) = [y (t) − y (t − τ )] . These periods τ correspond to local minima of the graphical mean perturbation D2 (t) , by varying τ and kipping K a constant. To calculate τ graphically, we set K = 0,1 and prints D2 (t) for different values of τ . Figure 1 presents the resulting graphic. The local minimum is τ = 0.62, 1.22, 1.84, corresponding to the periods one, two and three, respectively. The next step is the pick of the time averaged perturbation for different values of K, leaving τ fixed in each the values found. Figure 2 shows the local maximum corresponding to the associated perturbation gains with the values of τ . The phase portrait of the controlled system is presented in Fig. 3. The control law makes the system to remain in a limit cycle whose equivalent music is a melody composed by similar phrases. In order to check this, we applied different similarity measures between the first three semi-phrases that make up 5 Table 1 Measures of melodic similarity of the first three phrases of the Chen’s Mean perturbation melody controlled by Pyragas method Melodic contour Delay time Phrase A B C A 1 Phrase A B C A 1 C 0,8787 0,9066 1 Distribution of intervals B C 0,5492 0,6014 1 0,8975 1 A 1 A 1 Mean perturbation Fig. 1 Average time of perturbation for different periods of Chen’s equation B 0,8150 1 Distribution de pitch class B C 0,8196 0,8161 1 0,8634 1 Distribution of durations B C 0,6190 0,6216 1 0,7920 1 Time in seconds Gain of perturbation the total melodic phrase [2]. The results are presented in Table 1. There is a high similarity in melodic contour and the pitch class distribution, but not so high in terms of the distribution of duration and intervals. Figure 4 shows the piano roll melody generated by Chen’s chaotic system. Figure 5 shows the melodic contour of the first three phrases of the Chen’s melody and Fig. 6 shows the first phrase of the generated melody. Fig. 4 Piano roll melody generated by controlled Chen’s equation. Midinote Fig. 2 Average time of perturbation for different K values of Chen’s equation Time (beats) Fig. 5 Melodic contour of the first three phrases of the Chen’s melody. Y X Fig. 3 2D phase portrait of Chen’ equation after applying the control method of Pyragas Fig. 6 Melody generated by the attractor of Chen’s equation controlled by the Pyragas method. 6 IV-B. Melody Composition by Controlling Discrete System with FPIC We applied the FPIC (Fixed Point Induced Control) technique to control nonlinear quadratic map, described by Eq. (19), from chaotic attractor to a fixed point. xn+1 = b − ax2n period 2, as shown in Fig. 9. The system is stabilized after 60 iterations. A periodic point of period 2 corresponds to a melodic interval (relationship between 2 notes) and this continues until the end of the musical phrase. (19) With the parameters a = 4 and b = 0,5, the fixed points and eigenvalues are given by x∗ = (0,25, −0,5) , vp = (4, −2) (20) The control design requires N > 4 − 1, and we obtain b − ax2n + 1 (21) 5 By iterating the transformed map, we observe that the system quickly stabilizes the fixed point 0,25 (Fig. 7). xn+1 = Iterations Fig. 9 Iterations of the Gaussian map controlled by OGY method. In the score of the controlled Gaussian melody by OGY method (Fig. 10), we observe that the fourth quarter of fourth bar enters the compound major second (C3-D5) that covers a range of 26 semitones and is repeated several times to finish the melody. Iterations Fig. 7 Time Evolution of the nonlinear quadratic map controlled by FPIC. Figure 8 shows the melodic score generated by controlling the quadratic map with FPIC. It is observed that the melody stays in note B5 since the fourth time of the first bar. This note can be used in the composition of a pedal note (drone effect). Fig. 8 Melody generated by the nonlinear quadratic map controlled by FPIC. IV-C. Composition of Melodies with Discrete Systems Controlled OGY Method The OGY control method is applied to non-linear Gaussian map. The parameter is controlled to take the system to a periodic point of period 2. The control involves the parameter k when the iterations are more than u. ( 2 e−αxn + β n≤u xn+1 = (22) −αx2n k·e +β n>u By iterating the map with k = 0,1, which is activated after u = 60, it controls the system at a periodic point of Fig. 10 Melody generated with the Gaussian map controlled by OGY method. V. C ONCLUSIONS AND D ISCUSSIONS With the application of a method for controlling chaos, a chaotic dynamical system evolves to a periodic orbit or a fixed point. When a continuous chaotic system has to evolve to a periodic orbit, the system behaves as a limit cycle and this feature is reflected in the musical space in the form of phrases with high melodic similarity that can be used in some forms of music that form the fundamental structure of musical works. Moreover, when the system evolves to a fixed point, the period of stabilization can be selected in a convenient way to generate a range, an arpeggio or just that the melody remains in a note on a sustained bass. As this is of great importance in the development of a piece of music, it helps the musical consistency, avoid monotony, or can serve as an accompaniment to fund the main melody. We have also developed a methodology to control the melodic contour [24]. This methodology is based on the classical control theory. 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