Music, Complexity, Information

“Two impulses struggle with each other within man: the demand for repetition of pleasant stimuli, and the opposing desire for variety, for change, for a new stimulus.” With these words, Arnold Schoenberg introduced the two fundamental principles which cast musical form. Repetition of perceptual elements – melodic motifs, rhythmic patterns, harmonic progressions– brings coherence to the musical structure, which is the basis of its comprehensibility. Variation, in turn, is necessary to avoid monotony and dullness. “Variety is the mother of delight in Music,” as Schoenberg’s fellow composer Giovanni Maria Bononcini had put it three centuries earlier. In the search for a satisfactory combination of intelligibility and aesthetic substance, music conveys a subtle balance of reiteration and change, of redundancy and novelty, of recurrent shapes and fresh figures.

Arnold Schoenberg (Selfportrait, 1925) conceptualized the complementary roles of repetition and variation in musical composition.
This delicate equilibrium between uniformity and diversity evokes the nature of the class of entities that we call complex systems. An intermediate degree of internal organization, which makes a system elude incoherent behaviour but, at the same time, allows for rich dynamics and functional flexibility, is the key signature of complexity. The disordered, random-like motion of gas molecules, or the periodic, unrelenting ticking of a clock, hardly qualify as a complex system’s outcome. On the other hand, even the most elementary function of a modest bacterium reveals the underlying complexity of the living organism.

Is it possible to quantify the degree of complexity of a system by measuring its distance to both randomness and order? A first empirical answer to this question was advanced in the 1930s by the philologist G. K. Zipf, who studied the statistics of word repetitions in transcriptions of long speeches and in written texts. He discovered a strong regularity in the relative frequencies of word occurrences –now known as Zipf’s law– which is vastly widespread over different authors, styles, and languages: the number of repetitions of the n- th most used word is approximately proportional to the inverse of the rank n. If, for instance, the 10th most used word in a text occurs 300 times, Zipf’s law predicts that the 100th most used word will appear some 30 times. From the work by M. G. Boroda, B. Manaris, and collaborators, we now know that a regularity similar to Zipf’s law holds for musical compositions. In this case, the role of words in the statistics of repetitions can be replaced by single notes –each one defined by its pitch and duration– or composite items such as note duplets and triplets, interval successions, and chords.

Two decades after Zipf’s work, social scientist H. Simon pointed out that Zipf’s law can be quantitatively explained by assuming that, as a text is generated, the usage frequency of a word increases proportionally to its previous appearances. This very simple dynamical rule for the reinforcement in the usage of words during text generation, which statistical mathematicians call a multiplicative process, was enough for Simon to derive the inverse relation between number of occurrences and rank. More recently, it has been shown that the same rule explains the relative frequency in the usage of single notes in musical pieces, suggesting a strong affinity between the process of text generation and music composition. Simon’s model can be conceptually interpreted as representing the progression of the author’s choices along the creative process –grammatical, morphological, semantic in language; melodic, harmonic, dynamical in music– which shape the work’s intelligibility. It captures a basic mechanism of reinforced use of certain perceptual elements, whose recurrence is essential to elicit lasting neurophysical and psychological responses in the listener’s brain, from creation and evocation of memories to association with images, sensations, and feelings. Of course, real literary texts and musical compositions are created as organic entities, not just as series of isolated decisions. The outcome, nevertheless, is an ordered sequence of events conveying information –a meaningful message. As the message flows, a cognitive frame supporting its coherence –the context– emerges spontaneously, favoring in turn the appearance of some elements at the expense of others. From this viewpoint, Simon’s model for Zipf’s law furnishes a unification of the concept of context in language and music. At the same time, variations in the form of Zipf’s law for musical compositions of different authors make it possible to discern between their specific choices. The intentional lack of definition of a tonality context in Schoenberg’s atonal pieces, for instance, can

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