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 2^(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 00
C# Augmented unison 277
C## Doubly augmented unison 4155
Dbb Diminished second 139
Db Minor second 3116
D Major second 5194
D# Augmented second 7271
D## Doubly augmented second 9348
Ebb Diminished third 6232
Eb Minor third 8310
E Major third 10387
E# Augmented third 12465
Fb Diminished fourth 11426
F Perfect fourth 13503
F# Augmented fourth 15581
F## Doubly augmented fourth 17658
Gbb Doubly diminished fifth 14542
Gb Diminished fifth 16619
G Perfect fifth 18697
G# Augmented fifth 20774
G## Doubly augmented fifth 22852
Abb Diminished sixth 19735
Ab Minor sixth 21813
A Major sixth 23890
A# Augmented sixth 25968
A## Doubly augmented sixth 271045
Bbb Diminished seventh 24929
Bb Minor seventh 261006
B Major seventh 281084
B# Augmented seventh 301161
Cb Diminished octave 291123
C Perfect octave 311200

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