The Bohlen-Pierce Symposium
First symposium on the Bohlen-Pierce scale, Boston, March 7 – 9, 2010
Barlow: On the Harmonic Rationalization of Pitch Intervals of Known Size


For over two thousand years it has been known that two frequencies in a mutually small-number ratio form a harmonic interval. An example of this is the ratio 1:2, where one frequency is double the other, forming the interval of an octave. It is also undeniable that an interval with a large-number ratio close in intervallic size to a much more harmonic interval could be perceived as the latter. For example an interval with the frequency ratio 1000:1999 is only 0.07% or 1 cent smaller than an octave and would be taken for one. In such a case, the auditory perception mechanism “bends” the less harmonic interval to seem like its more harmonic neighbor.
Can one measure the harmonicity of an interval, given its ratio? Some work has been done here by various reseachers. This paper will describe my own work in the field, in which – by means of an algebraic formula for harmonicity, which I developed in 1978 – satisfactory ratio lists for pitch sets (e.g. for diatonic, n-tone equal tempered or the Bohlen-Pierce scales) are extractable, given only the pitches’ distance in intervallic units (cents) from a given starting point. Furthermore, the rationalizations derived provide a description of the harmonic properties and potential of the pitch sets not evident in the mutual distances of their elements.

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