Zipf's Law, Music Classification, and Aesthetics

The connection between aesthetics and numbers dates back to pre-Socratic times. Pythagoras, Plato, and Aristotle worked on quantitative expressions of proportion and beauty such as the golden ratio. Pythagoreans, for instance, quantified " harmonious " musical intervals in terms of proportions (ratios) of the first few whole numbers: a unison is 1:1, octave is 2:1, perfect fifth is 3:2, perfect fourth is 4:3, and so on (Miranda 2001, p. 6). The Pythagorean scale was refined over centuries to produce well-tempered and equal-tempered scales (Livio 2002, pp. 29, 186). Galen, summarizing Polyclitus, wrote, " Beauty does not consist in the elements, but in the harmonious proportion of the parts. " Vitruvius stated, " Proportion consists in taking a fixed nodule, in each case, both for the parts of a building and for the whole. " He then defined proportion as " the appropriate harmony arising out of the details of the work itself; the correspondence of each given detail among the separate details to the form of the design as a whole. " This school of thought crystallized into a universal theory of aesthetics based on " unity in variety " (Eco 1986, p. 29). Some musicologists dissect the aesthetic experience in terms of separable, discrete sounds. Others attempt to group stimuli into patterns and study their hierarchical organization and proportions (May 1996; Nettheim 1997). Leonard Meyer states that emotional states in music (sad, angry, happy, etc.) are delineated by statistical parameters such as dynamic level, register, speed, and continuity (2001, p. 342). Building on earlier work by Vilfredo Pareto, Al-fred Lotka, and Frank Benford (among others), George Kingsley Zipf refined a statistical technique known as Zipf's Law for capturing the scaling properties of human and natural phenomena (Zipf 1949; Mandelbrot 1977, pp. 344–345). We present results from a study applying Zipf's Law to music. We have created a large set of metrics based on Zipf's Law that measure the proportion or distribution of various parameters in music, such as pitch, duration, melodic intervals, and harmonic consonance. We applied these metrics to a large corpus of MIDI-encoded pieces. We used the generated data to perform statistical analyses and train artificial neural networks (ANNs) to perform various classification tasks. These tasks include author at-tribution, style identification, and " pleasantness " prediction. Results from the author attribution and