Musical Scales with Pythagorean Intervals

Abstract

The article examines the arithmetical qualities of musical scales tuned unequally with elementary intervals of the breadth of 0.98045 of a semitone (a "diminished" semitone) and 1.01955 of a semitone (an "augmented" semitone). It is demonstrated that each interval comprising a sum or a variety of a whole number of perfect fifths and the whole number of perfect octaves (in other words, a Pythagorean interval) may be presented as a sum of diminished and augmented semitones. Thereby, the Pythagorean major second is equal in sum to three augmented semitones, a minor Pythagorean third is equal in sum to two diminished semitones, a major Pythagorean third is equal in sum to four augmented semitones, a perfect fourth is equal in sum to three diminished and two augmented semitones, etc. Thereby, with the use of diminished and augmented semitones as elementary intervals of the scale the opportunity arises to build musical scales containing other Pythagorean intervals, besides the octave.