Pitch Structure of Melodic Lines: An Interface between Physics and Perception

Pitch Structure of Melodic Lines: An Interface between Physics and Perception Jorge E. Useche (jeusecher@unal.edu.co) Conservatorio de Musica & Departamento de Fisica, Universidad Nacional de Colombia, Carrera 30 No. 45-03 Bogota, Colombia Rafael G. Hurtado (rghurtadoh@unal.edu.co) Departamento de Fisica, Universidad Nacional de Colombia, Carrera 30 No. 45-03 Bogota, Colombia Abstract This work studies melodic lines of western art music tradition from frequency of occurrence of their constitutive elements. The model of analysis uses pairs of successive sounds as the minimal structural elements of a melody. Each pair of possible sounds in a musical instrument is associated to a quantity related to the difference of acoustic energies of the sound waves. This quantity expresses consonance properties that have been studied in experiments about the perception of combinations of sounds. We find statistical distributions of this quantity that show the existence of preferences for certain elements in a given melodic line. This preference can be interpreted as a consequence of the use of both formal musical theory rules and the creativity of the composer in order to create pleasant sensations in the listener. Keywords: Consonance; Melody; Music composition; Perception; Cognition; Tempered scale. Introduction George Kingsley Zipf explored the relationship between the usefulness of a word and its frequency of occurrence in a text (Zipf, 1932). He defined rank (r) as the position of a word in an ordered list that goes from 1, for the most frequent word, to the number of words that differ “phonetically”. The relation between rank and frequency of occurrence (f) of a word in the Ulysses from James Joyce resulted into a power law with exponent -1, that can be expressed as where c is a constant. Zipf observed the same phenomenon in other human systems and enunciated a law of human behavior driven by a Principle of Least Effort (Zipf, 1949) that he obtained in analogy to mechanics in physics. Zipf also studied the case of melodic lines and reported a power law relationship between the length of musical intervals, defined as the number of notes between two notes that are played successively (Zipf, 1949), and the frequency of occurrence of each interval. In other words, the probability of finding an interval in a melodic line is inversely proportional to its length. He found the same result for ascending and descending intervals and combined both of them in the analysis. Benoit Mandelbrot (1966) extended Zipf’s work and found a more general form of power law that also describes other phenomena as Pareto’s rule. Most recently, Statistical mechanics has been used to describe the statistical properties of many systems, ranging from financial markets and wealth distribution to complex networks, which present either exponential or power law distributions. Power law distributions are frequently associated to self-organized criticality and scale invariance, while exponential distributions are associated to equilibrium processes and to the occurrence of an absolute scale that is similar to temperature in Statistical physics. Further studies of musical pieces using probabilistic aspects of musical styles and uncertainty in musical communication have been done in Information theory (Abdallah & Plumbley, 2009; Cohen, 1962; Cox, 2010; Meyer, 1957). These studies capture fundamental aspects of perception as sensation and meaning. This perspective needs some basic mathematical assumptions such as stochasticity, ergodicity, stationary and Markov consistency of the source (i.e. the piece of music itself), as well as an infinite memory capacity of the encoder (i.e. the human brain) (Cohen, 1962). These studies use quantities such as the mean and the variability of statistical distributions of sequences of sounds. Recent studies show that some statistical properties of musical compositions can be treated using concepts and analytical tools developed for studying complex systems by applying Statistical physics. Liu, Tse and Small (2010) found that successive notes in musical pieces can be represented as complex networks that exhibit power law distributions for some connectivity properties. Gunduz and Gunduz (2010) studied melody formation in musical pieces and found that during the progress of a melody the entropy grows with each new note until it takes a limiting value that is smaller than the entropy of a random composition. This paper explores the relation between microscopic properties of consonance in a melodic line and the macroscopic properties resulting from the composition process. Its aim is to establish links between pleasant sensations at the microscopic level with the ordering process which is needed to compose a musical piece. It presents the physical model for describing consonance, probability distributions for pitch structures and a discussion about some implications. Paths between cognition and physics A piece of music is a mentally constructed entity that is usually described as segmented in unites of all sizes. In order to ascribe some sort of reality to the internal structure of musical pieces one must treat them as mental products imposed or inferred from physical signals (Lerdahl &