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 | 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 |

Gbb | Doubly diminished fifth | 14 | 542 |

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 |

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