Algorithmic and computational approaches to pure-tone approximations of equal-tempered musical scales

Abstract

We develop formalism for generating pure-tone systems that best approximate the modulation/transposition properties of equal-tempered scales. We define six measurements to determine the closeness of scales generated by pure intervals to equal-tempered scales. Two measures apply to scales generated by single intervals and rely heavily on continued fraction analysis. We show that generated scales can be optimized to preserve the pure-tone character of the scale by compromising the modulation/transposition properties by a least amount. The other four measures are generalizations that apply to scales generated by multiple intervals. These measures are directly applied to individual scales and numerically compared. We apply this formalism to pure intervals of import in antiquity and show that this approach yields the historically important just 12-tone system as an optimum scale. Finally, we show how this formalism can be applied to the analysis of non-standard scales.

Keywords:

• musical scales
• spectra
• spectral variation
• continued fractions
• computational algorithms
• equal temperament
• just temperament
• well-tempered scales

2000 Mathematics Subject Classifications:

• 00A65

Notes

In 1 we referred to Δ _c (I) as the width of the spectrum of I, but because of the definitions that follow in this paper, variation is more appropriate.

For rational generators statements 2 and 3 are not equivalent.

In our first paper 1, the last half of Statement 4 was inadvertently left out, and the statement read ‘Spectral variations of the scale are all equal’. Without the last half of Statement 4, the fifth generated 6-tone scale would be in agreement with the statement (see in Section 3.3), but this scale meets none of the other criteria of P2. With Statement 4 as it appears above, the statement becomes equivalent to the others in P2.