Spiral model of pitch

Abstract

A spiral model of pitch interrelates tone chroma, tone height, equal temperament scales, and a cochlear map. Donkin suggested in 1870 that the pitch of tones could be well represented by an equiangular spiral. More recently, the cylindrical helix has been popular for representing tone chroma and tone height. Here it is shown that tone chroma, tone height, and cochlear position can be conveniently related to tone frequency via a planar spiral. For this ‘‘equal-temperament spiral,’’ (ET Spiral) tone chroma is conceived as a circular array with semitones at 30° intervals. The frequency of sound on the cent scale (re 16.351 Hz) is represented by the radius of the spiral defined by r=(1200/2π)θr, where θr is in radians. By these definitions, one revolution represents one octave, 1200 cents, 30° represents a semitone, the radius relates θ to cents in accordance with equal temperament (ET) tuning, and the arclength of the spiral matches the mapping of sound frequency to the basilar membrane. Thus, the ET Spiral gives tone chroma as θ, tone height as the cent scale, and the cochlear map as the arclength. The possible implications and directions for further work are discussed.