Musical ratios in geometrical spacing of outer hair cells in the cochlea: strings of an underwater piano?

Abstract

Questions have arisen about the ability of conventional traveling wave theory to adequately describe the active cochlea. For example, if a sensitive microphone is inserted in the ear canal of a healthy human ear, it is likely to detect one or more faint, pure tones called spontaneous otoacoustic emissions. This study assumes that these tones represent the ringing of the cochlea's resonant elements and, revisiting the resonance theory of Helmholtz, puts forward a physical model of what these ringing elements could be: ripples reverberating between the rows of active outer hair cells. The model resembles a surface acoustic wave (SAW) resonator familiar in solid state elec-tronics, but in the biological case the interdigital electrodes, which alternate in polarity, are the rows of outer hair cells. In effect, resonant cavities are formed between cells in neigh-bouring rows. Because micrographs show that outer hair cells lie in a regular crystal-like lattice, the SAW model sees spacings between cells as forming cavities of defined lengths. Like an organ pipe, the frequency of each resonant cavity is the inverse of its length, and this work has investigated the geometrical properties of those lengths. Given a typical unit cell specified by a diagonal of 19°, a number of musically significant ratios naturally arise at partic-ular alignments of cells, notably the semitone as 1/cos19° = 1.06 and, 5 cells along, the octave as 1/cos [artan (5 tan 19°)] = 1.99. Closer investigation of the lattice geometry shows that, using unit cell parameters not atypical of those observed in micrographs, an approximation to the entire 12-tone scale is possible. The implication is that primary musical analysis may arise within the cochlea itself. Moreover, Helmholtz's resonance theory is a more apt description of the cochlea than traveling wave theory. Pythagoras was right: music is geometry.