Cognition of Tonal Centers: A Fuzzy Approach
G. Vidyamurthy, Jaishankar Chakrapani · Computer Music Journal · 1992
Music is based on the notion of harmony. Hence a general theory of music has to cope deeply with formal theories of harmony. However, harmony is a very subjective issue. The term is inherently ambiguous, as it refers to a lower level where smoothness and roughness are evaluated and at the same time to a higher aesthetic level where harmony is functional in a given style. Furthermore, the judgment of harmony does not seem to have a natural basis, but appears to be a common response acquired by people in a certain cultural area (Cazden 1945). Therefore opinions on the subject may vary widely depending on sociocultural backgrounds, and many an attempt to formalize the concept has been inadequate (Costere 1962). Nevertheless, it is consoling to note that harmony/ consonance does have a saving grace. We observe that while there is a difference of opinion on what constitutes harmony, there is general agreement on the relative order of consonance of music intervals. Numerological theories of consonance have attempted to capture this aspect (Cazden 1962), but here again a lot is left to the imagination as the theory does not clearly delineate what constitutes the order of simplicity of musical intervals. In this article, we look at the numerological theory of consonance from a physical point of view and, based on a few other observations, arrive at a relative ordering of musical intervals. The order arrived at tallies well with what is generally agreed upon. Based on the ordering of consonance of musical intervals we then introduce the notion of approximating a sequence of notes to its harmonically compatible note or tonal center. Tonal centers can be thought of as an approximation of the melody describing its flow. It becomes evident from our results that the tonal center is the counterpoint of the melody shifted by an octave. We present a method to approximate a sequence of notes to their tonal centers. This method uses fuzzy formalism (Mamdani 1981) and is posed as an optimization problem based on the physiological factors relevant to hearing music. Our approach is significant because it does not adopt any heuristics. Furthermore, we demonstrate that different variations of a music piece approximate to similar series of tonal centers. Right from the Elizabethan period when lutenist composers chose a popular melody and embellished it in their own particular style, to the current-day jazz musicians who compose varied melody lines for a given harmony, improvisation of music pieces has been an established practice. Our feature extraction technique provides a strong basis for the algorithmic composition of variations and improvisation of music pieces. We now present some concepts fundamental to the understanding of the article. An auditory event can be characterized for our purposes by four parameters: pitch, duration, timbre, and loudness. Pitch can be defined as the auditory property of a note that is conditioned by its frequency relative to the other notes. The range of musical pitch has been defined as the range within which the interval of an octave can be perceived (Watt 1917). This has been found to correspond roughly to the range of the piano (Guttman 1962), that is, from 27 Hz to 4,000 Hz. From this continuum of frequencies, a set of discrete frequencies is selected in such a way that the frequencies bear a definite interval relationship to one another. So, pitch in the musical sense corresponds to a frequency that is selected Computer Music Journal, Vol. 16, No. 2, Summer 1992, ? 1992 Massachusetts Institute of Technology.