Choosing the representative tones of an abstract Pythagorean scale with regard to just intonation

Abstract

Hellegouarch [1999a. “Gammes naturelles.” Gazette des Matematiciens 81:25–39.] relies on a remark that Euler made to state that the way of choosing the representatives of an abstract Pythagorean scale should be based on the fact that the sense of hearing tends to identify with a single ratio all the ratios which are only slightly different from it. For this reason, the representatives were chosen as the simpler ratios among the possible tones of the Pythagorean scale. However, this could be a misinterpretation of Euler's remark, since there may be single ratios much simpler in the vicinity of the Pythagorean ratios. This paper is only a brief contribution on the discussion about how to select these representatives and shows that if Pythagorean tuning is to emulate or replace syntonic just tuning, it is better to adapt the scale to the key.

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