Comparing octaves, frequency ranges, and cochlear-map curvature across species

Comparisons of the cochlear maps of various species might be more revealing if the form and parameter values of the functions fitted to the maps were taken into more explicit account. One empirical frequency-position function (Greenwood, 1961), the form of which fits several species, was recently reviewed (Greenwood, 1990). It was shown that mechanical and physiological data from human, cat, guinea pig, chinchilla, and monkey, are well fitted by an almost-exponential function. An exponential term, the argument of which is normalized position, x, on the cochlear partition (x = 0 at apex, 1 at base), is first reduced by a small term, k < or = 1, before the quantity, (exponential - k), is multiplied by a third parameter, A, to yield the frequency associated with a given position, x. Since the normalized coefficient, alpha, of the exponential's argument is about the same, 2.1, in several species, there are some very simple but noteworthy consequences. The quantity (exponential - k) is thus nearly the same function of x (if k is about equal and in any case as x nears 1) among those species, despite differences in cochlear lengths. Therefore among these species, differences in frequency range are related only to the multiplier, A. Moreover, the function's form implies that only the exponential term (and k) determine the proportion of cochlear length occupied by an octave. Thus, if the exponential's coefficient and k are equal for some species, corresponding octaves (highest, next highest, etc.) correspond in these species to equal percentages of cochlear length, independent of length and frequency range (and they must differ if the coefficient differs). Further, these percentages diminish nearer the apex if cochlear maps (log-frequency versus position) are apically curved (k > 0). But to determine the presence or absence of curvature, cochlear maps must include points from the apical 25% of the cochlea; if not, a simple exponential (k = 0) will probably suffice to fit data in the basal 75%. The apical curvature may have functional value and that some degree of it may be typical is consistent with models which show that curvature and tapering-viscosity effects combine to reduce apical reflections and standing waves, smoothing cochlear impedance (Puria and Allen, 1991a, b).