Equal Temperament - Rational Semitone

Rational Semitone

For any semitone that is a proper fraction of a whole tone, exactly one equal division of the octave lets the circle of fifths generate all the notes of the equal division while preserving the order of the notes. (That is, C is lower than D, D is lower than E, etc., and F♯ is indeed sharper than F.) The number of divisions needed for the octave is seven times the number of divisions a whole tone minus twice the number of divisions of the semitone. The corresponding fifth spans a number of divisions equal to four whole tones minus one semitone. Hence, for a semitone of one-half of a whole tone, the corresponding equal temperament scheme is 12-EDO with a fifth of seven divisions. A semitone of one-third of a whole tone corresponds to 19-EDO with a fifth of eleven divisions.

12-EDO is the equal temperament with the smallest number of divisions that allows for a rational semitone to preserve the desired properties concerning note order and the circle of fifths. Plus it has the desirable property of making the semitone exactly one-half of a whole tone. These are additional reasons why 12-EDO became the predominant form of equal temperament.

While each rational semitone corresponds to only one equal temperament, the reverse is not the case. For example, both a semitone of one-seventh, and a semitone of eight-ninths both use 47-EDO, which is the smallest number of divisions that has two different semitones. However, they have different values for the fifth, as a semitone of one-seventh uses a fifth of twenty-seven divisions while a semitone of eight ninths uses a fifth of twenty-eight divisions.