Acoustical numerology and lucky equal temperaments
Donald Elliott Hall · American Journal of Physics · 1988
Equally tempered musical scales with N steps per octave are known to work especially well in approximating justly tuned intervals for such values as N=12, 19, 31, and 53. A quantitative measure of the closeness of such fits is suggested, in terms of the probabilities of coming as close to randomly chosen intervals as to the justly tuned targets. When two or more harmonic intervals are considered simultaneously, this involves a Monte Carlo evaluation of the probabilities. The results can be used to gauge how much advantage the special values of N mentioned above have over others. This article presents the rationale and method of computation, together with illustrative results in a few of the most interesting cases. References are provided to help relate these results to earlier works by music theorists.