Quantifying the rationality of rhythmic signals

Rhythms and vibrations represent the quintessence of life, they are ubiquitous (systemic) in all living systems. Recognising, unfolding these rhythms is paramount in medicine, for example in the physiology of the heart, lung, hearing, speech, brain, the cellular and molecular processes involved in biological clocks. The importance of the commensurability of the frequencies in different rhythms has been thoroughly studied in music. We define a log-frequency correlation measure on spectral densities that gives the temporal evolution of the distribution of frequency ratios (rational or irrational) in between two signals, using analytic wavelets. We illustrate these concepts on numerical signals (sums of sine functions) and voice recordings from the Voice-Icar-Federico II database. Finally, with a second correlation operation from two of these ratio distributions (a reference one, the other from the voices) we introduce another quantity that we call \emph{sonance}, measuring the ``harmony'' (rationality) of two voices sung together as a function of a pitch transposition.