ABSTRACT
There is a serious discrepancy between the psychological representation of pitch, the mel scale, and the musical representation of pitch (the pitch helix). The mel scale is a monotonic, unidimensional mapping of the frequency of a pure tone (a sine wave); the helix is a cyclic, bi-dimensional mapping of the repetition rate of multi-harmonic tones (musical notes). The circular dimension of the pitch helix is tone chroma and the longitudinal dimension is tone height (see Ueda & Oghushi, 1988, for a review). Recently, Patterson (1989, 1990) has emphasized the bi-dimensionality of pitch by demonstrating that one can construct a sequence of notes in which tone height rises an octave while tone chroma remains fixed. Consider a sound composed of 20 harmonics of 100 Hz, and the perceptual change that occurs as the odd harmonics (100, 300, 500,…) are attenuated, as a group, by an ever increasing amount. The tone height rises smoothly from 100 to 200 Hz without any change in tone chroma. In retrospect, this is not surprising; when the attenuation is greater than about 20 dB, the odd harmonics are effectively removed, leaving a harmonic series that is the octave of the original note (200, 400, 600,…). What it demonstrates, however, is that the mel scale is completely inadequate as a representation of pitch; it cannot explain how we move continuously from a note to its octave without going through all the intervening tone chromas. The pitch helix has a separate dimension for tone height and, in this case, the new data can be accommodated simply by assuming that it is possible to move continuously along a line from a note to its octave, as well as around the chroma circle. The fact that notes exist between the circuits of the helix means that it is actually a helical cylinder rather than a helical wire, as suggested previously. But the helical cylinder is an obvious extension of the traditional representation.
