Sensory consonance of two simultaneous sine-tones
Reinhart Frosch · The Journal of the Acoustical Society of America · 2013
In Chapter 4 of my book “Musical Consonance and Cochlear Mechanics” (vdf, Zurich, 2012), four psychoacoustic experiments on the sensory consonance of two simultaneous sine-tones are described. In each of those experiments, the deeper-tone frequency fd was kept fixed, at fd = 132, 264, 528, or 1056 Hz. Each experiment was done twice, at sound-pressure levels of 50 and 70 dB (SPL). The resulting consonance curves (sensory consonance versus higher-tone frequency fh) exhibit consonance minima at beat-rates bm.d. = fh - fd (where “m.d.” = “most dissonant”) ranging from 13 Hz (at fd = 132 Hz) to 39 Hz (at fd = 1056 Hz). In Section 15.1 of the above-mentioned book, these most dissonant beat rates are shown to agree well with the following empirical law: bm.d. = (1.07s-0.5) * sqrt(favg), where favg = fd + bm.d./2. The present study was prompted by the comments of a reader; the just described empirical law is unsatisfactory because in the underlying experiments the deeper-tone frequency fd [rather than the average frequency (fd + fh)/2] was kept constant. It was found that the data agree equally well with the following modified empirical law: bm.d. = (1.09s-0.5) * sqrt(fd). This modification does not affect the validity of the complex-tone consonance theories described in Chapters 15 and 16 of the mentioned book.