Abstract
Western music is predominantly based on the equal temperament with a constant semitone frequency ratio of 21/12. Although this temperament has been in use since the nineteenth century and in spite of its high degree of symmetry, various musicians have repeatedly expressed their discomfort with the harmonicity of certain intervals. Recently it was suggested that this problem can be overcome by introducing a modified temperament with a constant but slightly increased frequency ratio. In this paper we confirm this conjecture quantitatively. Using entropy as a measure for harmonicity, we show numerically that the harmonic optimum is in fact obtained for frequency ratios slightly larger than 21/12. This suggests that the equal temperament should be replaced by a harmonized stretched temperament as a new standard.
Highlights
Musical intervals between two tones are perceived as harmonic if the corresponding frequencies are related by simple fractional ratios [1]
With respect to a given reference frequency f ref, where log2 z = ln z/ ln 2 denotes the logarithm to base 2. With this definition it is possible to translate frequency ratios into pitch differences. In music theory such pitch differences are usually measured in cents (¢), defined as 1/100 of a semitone in the standard equal temperament (ET)
As a convenient notation for what follows, we shall denote by ETx a stretched equal temperament with a semitone stretch of
Summary
Musical intervals between two tones are perceived as harmonic if the corresponding frequencies are related by simple fractional ratios [1]. With the increasing complexity of music, frequent key changes became more important This led to the fascinating development of so-called temperaments [2], i.e. tuning systems seeking for a reasonable compromise between harmonicity and invariance under transposition, attempting to reconcile the exponential structure of the scale with the linear organization of the partials. Capurso suggested what is known as the circular harmonic system (c.ha.sTM) [7, 8], where all intervals exhibit beats All these proposed chromatic temperaments belong to the same family in so far as they are perfectly equal, differing only in their semitone frequency ratio.
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