Plato’s Timaeus and optimal pentatonic scales

Abstract

After a short review of Pythagorean theory of harmonic ratios and musical scales as it is described in Plato’s Timaeus treatise, the concept of ‘optimality of a sequence of (real) numbers with respect to Pythagorean ratios’ is defined and main theorem of this article proves that there are only three optimal sequences of length 6, which correspond to three well-known pentatonic scales which are used in many musical traditions (including Chinese, Japanese and others). It is also noted that a definition similar to our optimal scales has appeared in a treatise by Sadi-al-Din Urmavi, a thirteenth century Iranian musicologist.

GRAPHICAL ABSTRACT

KEYWORDS:

• Optimal scale
• Pythagorean ratios
• Timaeus
• Pentatonic scale
• Safi-al-Din al-Urmavi

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 It may be thought that optimal scales can be constructed by simply choosing first notes in the circle of fifths but it is not the case: the first seven notes in the circle of fifths are Do, Sol, Re, La, Mi, Si, Fa# and it can be easily checked that the corresponding scale is not optimal.