31 Note Equal Temperament

The division of the octave into 31 equal parts has some desirable characteristics and one fortunate coincidence. The desirable characteristics are the well tuned thirds, (2^(8/31) = 1.1958733 and^(10/31) = 1.2505655), which are much nearer just intonation than those of 12 note equal temperament. The perfect fourth and fifth are less good than ET12 but still acceptable (2^(18/31) = 1.4955179). The coincidence is that the 31 notes map, in a logical manner, onto the 35 note names of the Western notational system. The enharmonic equivalents Fbb = D##, Cbb = A##, E## = Gbb, B## = Dbb, not shown below, complete the mapping.

This tuning distinguishes between the diatonic semitone or minor second (3 steps) and the chromatic semitone or augmented unison (2 steps). However, it does not distinguish major tones (9/8 in just intonation) from minor tones (10/9), so ET31 is a mean tone system.

Steps = 31 * Log[base 2] f/f0 where f is the frequency in ET31