Chaos structures in Gregorian Chant

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2 Chaos structures in Gregorian Chant PIETRO DI LORENZO, Seconda Università degli Studi di Napoli 2

3 Abstract Music signals are approximately periodic in micro and macro forms. Everybody is familiar with the fact that musical sounds are a superposition of a number of separate free elementary periodic vibrations. In this approach musical sound signals should have a deterministic behaviour. Also a musical piece seems to have a recurrent structure. On the other hands, there is a widespread opinion that music is a complex, chaotic, nonlinear guided by bifurcation phenomena. Is it possible to recognize the above mentioned properties? If determinism=rationality and chaos=irrationality, then I hope to be successful. Musicologists believe that macrophorm, text distribution, orchestration, polyphonic structures are rational but rhythms, melodic developments, are irrational structures. Empirically, Frova (1999) states (without proving it) that musical signals are complex. In fact, there are a lot of components related by feedback relations, similar to self-organised systems. Are they random (or noise) effects or actually the appearance of complexity? It is possible to indagate nondeterministic phenomena via nonlinear dynamic tools. My aim is to study music (gregorian chant repertory) from the point of view of signals processing. I encode each melody in a numerical array of pitchs. Then I recognize and analyse orbits, phases space, power spectrum, Poincaré sections, correlation functions, and I estimate some of significant analytic parameters (fractal dimension via correlation and capacity dimension, Lyapunov exponents). The results allow us to stating that music is a chaotic phenomenon. In fact: the orbits occupy only a well limited region in the phase space; they have a non simple geometrical pattern (strange attractor); the power spectrum is neither with spikes (as in periodic or quasi-periodic signal) nor a wide band (as in white or colored noise), the Lyapunov exponents are positive, the dimension is fractal. Introducion : linear analysis of time series J. Fourier (1807) stated that it is possible to write f(x) (generally assumed square integrable on (a,b) or, weakly, integrable) as a series of trigonometric functions φ k f (x) = = c k k 0 φ k where c f (x) φ (x) dx. k = b a k P.G.L. Dirichlet (1829) stated a first convergence criterion for Fourier Series: if f(x) is a function (with a finite number of maxima and minima over the period and everywhere continuous, except at a finite number of points where it may have discontinuities of the first kind), then it converges for all x (and moreover at points of discontinuity) at the sum of series. Subsequently, this assertion was extended to arbitrary function of bounded variations (C.Jordan, 1881). These results are also applied in discrete analysis. Physical aspects Music signals are represented by amplitude of pressure wave versus time. But human perception is not built up on the amplitude. It represents forte and piano that are less important than pitch in a hierarchical perception (and pitch corresponds to frequency). Thus, a sound signal in time is a record marked (at the extremes) by the persisting absence of a signal or by one of its values considered 3

4 nonvoluntary or meaningless (sound noise/noise). Physically speaking and on the basis of the theory of signals, I suggest the following definition of melody: a fragment of signal that is a temporal window marked by other sounds or by voluntary pauses. It is clear that the marking takes place on linguistic and Gestaltic choices, i.e. a self-contained sound pattern, complete and self-contained features, and the absence of a macro-structural redundancy. In brief, something, which reminds necessary and sufficient conditions familiar to mathematicians. One isolated note is a periodic wave Everybody is familiar with the fact that musical sounds are a superposition of a number of separate free elementary periodic vibrations, according Fourier Theory (Jeans, 1937). In this meaning musical sound signals should have a deterministic behaviour. Also a piece seems to have a recurrent structure. But, Frova (1999) states that musical signals are complex. In fact, there are a lot of components with feedback relations, similar to self-organised systems and small variations that modify the periodic regularity. Are they random (or noise) effects or actually the appearance of complexity? In fact, the creative process of music is irrational. There is the romantic idea of the composer imprisoned by his own unstructured creativity, generated by a spontaneous creative raptus. This is in contrast to the Chomskian identity of music as language, subjected to grammar or syntax rules. Baroni - Dal Monte - Jacoboni (1999) stated deterministic and rational grammar of music with the aim of accurate statistical measurement of the recurrent macro and micro structural characteristics. But, the idea of irrationality of music is wrongly rooted in collective fantasy. In fact, every philologist knows that it is not always true: behind a signed manuscript there is always a propelling and creative work but also a work of rational restructuring. Non linearmodel to analyse time series The objective of experimental data analysis is to find a pattern or structure that models the data. In recent years the theory of Dynamical Systems has shown that the solution of simple, nonlinear equations can exhibit complicated (complex) temporal behaviour with strange attractor (that is chaotic and fractal). A part of the Dynamical System Theory studies the Nonlinear Time Series using recent mathematical tools. There is not a complete and univocal characterization of chaos in a data series. Nevertheless, the spreadly used indicators are: the existence of a strange attractor, the power low in the power spectrum, the fractal dimension of the attractor, the positive Lyapunov exponents, the Hurst exponent, LZ and Entropy index. This model can analyse every signal. For music, I think that it is particularly appropriate to: 1) "pure" melodies: melodies like those shown in Selfridge-Field (1998) (mixed, selfaccompanied, submerged, wandering and widespread) are not correctly computable; 2) monodic music without accompaniment. 3) 3omorithmic pieces; 4) melodies characterised by melodic pattern with joint tone interval or small interval. Melodic encoding Here, melodies are coded into numerical code: every note corresponds to a integer numerical bit according to rule that C=1, C#=2, D=3 etc. Note durations are conventionally set at a basic music figure (here a Quarter). Every longer figure is represented with a copy of the same code repeated as many times as it covers up duration. This code is justified by perception: men distinguish the intervals in a melody that is 4

5 distance from notes (physically these are logarithmic ratio of frequencies). Silent, rests, pause or 0 codifies unintentional signal absence. Experimental data: analysed melodies and variants I select three pieces of monodic repertoire (Stabat Mater, Dies Irae, Victime Pascali) from Gregorian chant (Gregorian Chant here and in the following is generically used to call every early western Christian monodic music without specification) to point out an application of model. That is for several important reasons. The pieces selected are in syllabic style (a neuma for every note). In a now-days transcription with modern figure the neuma are all codified with the same symbol (usually Quaver). The transcription does not force us to quantize the sound during or the metric accent. In the Gregorian chant every melody is built up using just one selected scale. In Middle Age Music Theory there was only eight musical modes: each individual chant had to be assigned to a proper mode. Then, in every melody no modulation appears since there is no possibility to move form a scale to another in the same melody. Melodic simplicity. The intervals are small, there are just a few little jumps. Data elaboration I have analysed the experimental data via CDA (Chaos Data Analyzer, 1995, by the American Institute of Physics, J.C.Sprott-G.Rowlands). CDA analyses data in the form of a numerical time series dx (t) X(t) and plots the attractors in the phase-space (constructed with components X(t) and ). Then dt CDA computes the power spectrum (that shows the frequencies of the signal), the Lyapunov exponents (that is a quantitative characterization of chaos linked to the sensitivity to initial conditions), the fractal dimension and the Hurst exponent (that is the slope of the root-mean-square displacement of the signal versus time). But the resolutive proof to recognize chaos is to build up a surrogate data-test. CDA computes a new artificial data series that has the same Fourier-transform with the randomized phases, as you would shuffle a deck of cards. Then CDA computes again the data. If the results are the same you should suspect the veracity of deterministic chaos in your data. Otherwise, if it results a random pattern of the attractor then the data are typically deterministic chaos. Results The CDA analysis of the three series provides the following results (Stabat see figure, Dies Irae and Victime Pascali plots are very similar to Stabat). The attractor in the phase-space is chaotic (figures a1), but with a low-medium dimension (d=7, d=4, d=6). The figure a2 shows the power spectrum (log-log) that has a power low (that it is tipically link to chaos). 5

6 The fractal dimensions (computed via capacity dimension) are D a =0,55±0,17; D b =4,69±0,18; D c =0,41±0,14. The Lyapunov exponents are all positive and (for each series) the greatest are λ Ma =0,19±0,12; λ Mb =0,37±0,08; λ Mc =0,13±0,2. Chaotic orbits have at least one positive Lyapunov exponent. The Hurst exponents are H a =0,18; H a =0,27; H c =0,24. If the motion is a random walk (Brownian motion) than H=0,5. Exponents greater than 0,5 indicate persistence (past persist into the future), whereas exponents less than 0,5 indicate antipersistence (past trend tend to reverse in the future). Lz is the Lempel-Ziv complexity computed via symbolic dynamics. It is a measure of the algorithmic complexity of the time seris. Maximal complexity (randomness) has a value of 1.0, and perfect predictability has value of 0. It results that: LZ a =0,65; LZ b =0,61; LZ c =0,63. Entropy is the sum of the positive Lyapunov exponents (base e). The entropy is a measure of the disorder in the data. The computed entropies are: E a =0,8; E b =0,5; E c =0,8. The figure a4 shows surrogate data attractor (after a shuffle into the phases of the spectrum). I point out the great difference between figures 3 and 4: it is an important marker that the randomization of the phases of the spectrum destroys the determinism of the data series. In fact, the attractor fills a great part of the plane. Discussion The usefulness of chaos theory is controversial where the system in not manifestly deterministic. In particular, evidence for chaotic behaviour in field measurements has been claimed and disputed in many areas of sciences. However, chaos theory has inspired a new set of useful time series tools and provide a new language to formulate time series problems and to find their nonclassical and nonlinear solutions. In this gregorian chant music analysis the results are clear and reliable. The application of nonlinear analysis shows that the parameters are typical of deterministic chaos. A further proof of determinism in data series results from surrogate data. Conclusions Gregorian Chant melody is a complex phenomenon neither deterministic nor completely random, as you can see from patterns of attractor and form other dynamical parameters. 6

7 Address for correspondence: PIETRO DI LORENZO Dipartimento di Matematica Seconda Università degli Studi di Napoli 81100, Caserta, Italia -Via Antonio Vivaldi, (0)823/ fax +39+(0)823/ pietro.dilorenzo@unina2.it 7

8 References Baroni, M. - Dalmonte, R. - Jacoboni, C. (1999): "Le regole della musica", Torino, EDT. Bendat, J. S. Piersol, A. G. (1970): Random data, Wiley. Frova, A. (1999): Fisica acustica, Bologna, Zanichelli. The Gregorian Association (London, England) written by Peter Wilton. Jeans, J. (1937): Science and music, Cambridge University Press. Selfridge-Field, E. (1998): Conceptual and Representational Issues in Melodic Comparasion in Meoldic similarity ed. Hewlet, W.B. Selfidge-Field, E. Schreiber, T.-Schmitz, A. (2000): Surrogate time series, Physica D142,346(2000) Schreiber, T. (1999): Interdisciplinary application of nonlinear time series methods, Phys.Rep.308,2(1999). Sprott, J.C.-Rowlands,G. (1995): Chaos Data Analyzer, The professional version American Institute of Physics, North Carolina State University. 8