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

作者：Krantz Richard^*; Douthett Jack

来源：Journal of Mathematics and Music, 2011, 5(3): 171-194.

DOI：10.1080/17459737.2011.641765

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

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.

• 出版日期2011