The Mel Scale's disqualifying bias and a consistency of pitch-difference equisections in 1956 with equal cochlear distances and equal frequency ratios

Abstract

In 1956, Stevens ‘commissioned’ an experiment to equisect a pitch difference between two tones. Results appear to reveal a methodological flaw that would invalidate the Mel Scale (Stevens and Volkmann, 1940). Stevens sought to distinguish sensory continua, e.g., loudness and pitch, on various criteria. He expected that the pitch continuum would not exhibit ‘hysteresis’; i.e., that subjects dividing a pitch difference (Δf) into equal-appearing parts would not set dividing frequencies higher when listening to notes in ascending order than in descending order. Seven subjects equisected a pitch difference, between tones of 400 and 7000 Hz, into equal-seeming parts by adjusting the frequencies of three intermediate tones. All seven exhibited hysteresis, contrary to expectation. This outcome bears on other issues. Years prior, Stevens suggested that equal pitch differences might correspond to equal cochlear distances, but not to equal frequency ratios nor to equal musical intervals (Stevens and Davis, 1938; Stevens and Volkmann, 1940). In 1960 (reported now), both the 1940 Mel Scale and the equal pitch differences of 1956 were compared to equal cochlear distances, using a frequency-position function that fitted Békésy's cochlear map (Greenwood 1961, 1990). When ascending and descending settings were combined to contra-pose biases, equal pitch differences did coincide with equal distances — which the Mel Scale did not. Further, the biased ascending-order data coincided with the Mel Scale, suggesting the Mel Scale was similarly biased. Thus, the combined-order equal pitch differences of 1956 — but not the Mel Scale — are consistent with equal cochlear distances. However, since the map between 400 and 7000 Hz is nearly logarithmic, equal frequency ratios also approximate equal distances. Ironically, above 400 Hz, Békésy's map and Stevens' equal-distance hypothesis jointly imply that musical intervals will nearly agree with equal pitch differences, which Stevens thought he had disconfirmed. However, given Békésy's map, only near the cochlear apex will equal distances not approximate equal frequency ratios; and Pratt's (Pratt, 1928) bisections of Δfs greater than an octave indicated that equal pitch differences, on average, did agree with equal distances. However, they did so for only two of four subjects and coincided instead with equal frequency ratios for one musical subject. Historical distinctions suggest that between the parts of equisected Δfs subjective equivalence may be of two kinds — one linked to musical intervals, leading to equal frequency ratios; a second linked to ‘tone-height’ and ‘distance’, leading to deviations from equal frequency ratios near the apex, though not appreciably if equisected Δfs are less than an octave (or if perhaps subjects are musicians). Data of other kinds suggest that, if pure-tone pitch height were a function of place, the place could be the apical excitation-pattern edge, in any case not a maximum, which in neural data shifts and disappears with tone level.