Computational Approach to Musical Consonance and Dissonance.

In sixth century BC, Pythagoras discovered the mathematical foundation of musical consonance and dissonance. When auditory frequencies in small-integer ratios are combined, the result is a harmonious perception. In contrast, most frequency combinations result in audible, off-centered by-products labeled "beating" or "roughness;" these are reported by most listeners to sound dissonant. In this paper, we consider second-order beats, a kind of beating recognized as a product of neural processing, and demonstrate that the data-driven approach of Recurrence Quantification Analysis (RQA) allows for the reconstruction of the order in which interval ratios are ranked in music theory and harmony. We take advantage of computer-generated sounds containing all intervals over the span of an octave. To visualize second-order beats, we use a glissando from the unison to the octave. This procedure produces a profile of recurrence values that correspond to subsequent epochs along the original signal. We find that the higher recurrence peaks exactly match the epochs corresponding to just intonation frequency ratios. This result indicates a link between consonance and the dynamical features of the signal. Our findings integrate a new element into the existing theoretical models of consonance, thus providing a computational account of consonance in terms of dynamical systems theory. Finally, as it considers general features of acoustic signals, the present approach demonstrates a universal aspect of consonance and dissonance perception and provides a simple mathematical tool that could serve as a common framework for further neuro-psychological and music theory research.