The Bohlen-Pierce Site

Quotation from

Authors: Max V. Mathews, John R. Pierce and Linda A. Roberts
Published in: Harmony and Tonality, Editor J. Sundberg, Royal Swedish Academy of Music
No. 54, 1987, pp. 59-84


The Pierce scale
The Pierce scale was actually first discovered by
H. Bohlen (1978), who published an article in
Acustica, in which he described the scale almost
exactly as we have done here. The work was not
known to us at the time we did our research or
prepared the draft for this article. However, now
that we have read the Bohlen paper, we believe that
it is clear that he proposed exactly the same scale
including the tempered form of the scale using the
13th root of 3 as the tempered factor and including
timbres which have only odd partials such as a
square wave for playing this scale. Bohlen derived
his scale from a theory of combination tones and, in
particular, combination tones involving 3*f 1 -
f 2 . We derived the scale based on experimental
measurements of the intonational sensitivity of the
3:5:7 and 5:7:9 chords. The two sources for the
scale, although different, are not in conflict, since
Bohlen's derivation was primarily theoretical and
ours was primarily experimental. We find it
heartening that we arrived at the same result from a
completely different direction. Obviously there is no
question about who derived the scale first, since
Bohlen published a decade sooner than we did.

End of quotation.

(Actually, there is a typo in this paper. The frequencies of the combination tones should read 2·f1-f2, rather than 3·f1-f2.)