Description of regular twelve-tone musical tunings.

A framework is presented for the systematic description of musical tunings, based on the music-theoretical and on the frequency-ratio concept of musical intervals. All intervals are described as sums of octaves and fifths. Since the octave is always pure (2:1) knowledge of the sizes of the fifths is sufficient to determine the sizes of other intervals. In regular 12-tone tunings, 11 out of the 12 fifths are equal in size, either pure or tempered. Rules are given by which the size of the consonant intervals, and other intervals, can be derived from a single variable, the main fifth. Tunings are evaluated with the help of mean temperings of the consonant intervals in the major triads, mean temperings of all the triads in a tuning, and mean temperings of triads weighted with respect to a certain key. This is a theoretical study, which should be supplemented by observations from practice and perceptual experiments.