On the Euler scale and the μEuclidean integer relation algorithm

Abstract

The equal-tempered 10-tone scale e n/10 (n=0,±1,±2, …), using the Euler number e=2.71828… as a pseudo-octave is shown to approximate well the prime number harmonics 2, 3, 5, and 11. Equal-tempered scales simultaneously approximating certain frequency ratios, shall be called tonal scales.1 Some of the properties of the Euler scale and its relation to other tonal scales are explored. The general mathematical problem of identifying tonal scales can be solved by investigating integer relations, using the μEuclidean algorithm, a modification of the PSLQ algorithm. If restricted to two numbers, the μEuclidean algorithm goes over identically into the ancient Euclidean algorithm, contrary to the PSLQ algorithm. The μEuclidean is able to solve a certain class of higher dimensional integer relations where the PSLQ (not the PPSLQ) algorithm breaks down. In general, the μEuclidean algorithm finds smaller integer relations than the (P)PSLQ algorithm. In an appendix, a simple alternative procedure is presented for determining tonal scales based on continued fractions.