The spiral model of pitch: Interrelations with musical and psychoacoustic scales and with cochlear parameters

Abstract

A spiral model of pitch can represent pitch chroma and pitch height. Its arclength provides the frequency-position map of the cochlea, and, when scaled by the number of auditory-ganglion cells, provides the Mel scale. Pitch-like scales fall into three groups. Group I is typified by the Cent scale, which can be expressed as a pitch spiral. Other Group-I scales are musical notation, logarithmic scales of frequency, Fechners Law for complex tones, and the pitch helix. Group II is typified by Greenwoods cochlear map, which is simply related to the arclength of the pitch spiral. Other Group-II scales are the sums of the number of (a) equivalent-rectangular bandwidths (NERBs); (b) difference limens for frequency (NDLFs); and (c) inner-hair cells (NIHCs) and are consistent with critical ratios. Group III is typified by the Mel Scale, which can be derived from the pitch spiral and the distribution of auditory-ganglion cells. Other Group-III scales are the sums of the number of (a) auditory ganglion cells (NSGCs), (b) frequency-modulation-difference limens (NFMDLs), and (c) Barks (NBARKs). The pitch spiral seems a basic concept from which other pitch-like scales can be derived.