From: The World In Your Head by Steven Lehar
Fig. 11.5 Plot of perceived pitch P, in units of octaves above middle C, as a function of frequency F, expressed in terms of Hz/261.626, which is the frequency of middle C. The pitch is perceived to rise in equal intervals with each octave from C to C’ to C’’, etc., whereas the frequency doubles with each octave. The linear rise of frequency in integer units from 1 to 2 to 3 creates finer subdivisions of the pitch scale at frequencies 1, 1.585, 2, 2.322, 2.585, etc., and these subdivisions correspond to the subdivision of the octave interval in musical scales. Different tempered scales are proposed to subdivide the octave into rational intervals, in an attempt to impose a periodicity on the pitch dimension. This urge for order in the pitch dimension reflects a hierarchical tendency in search of periodicity of periodicity.