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

Note Interval above C Steps Cents
C Perfect unison 0 0
C# Augmented unison 2 77
C## Doubly augmented unison 4 155
Dbb Diminished second 1 39
Db Minor second 3 116
D Major second 5 194
D# Augmented second 7 271
D## Doubly augmented second 9 348
Ebb Diminished third 6 232
Eb Minor third 8 310
E Major third 10 387
E# Augmented third 12 465
Fb Diminished fourth 11 426
F Perfect fourth 13 503
F# Augmented fourth 15 581
F## Doubly augmented fourth 17 658
Gb Diminished fifth 16 619
G Perfect fifth 18 697
G# Augmented fifth 20 774
G## Doubly augmented fifth 22 852
Abb Diminished sixth 19 735
Ab Minor sixth 21 813
A Major sixth 23 890
A# Augmented sixth 25 968
A## Doubly augmented sixth 27 1045
Bbb Diminished seventh 24 929
Bb Minor seventh 26 1006
B Major seventh 28 1084
B# Augmented seventh 30 1161
Cb Diminished octave 29 1123
C Perfect octave 31 1200