Rank-2 temperaments with the generator close to a fifth and period an octave

In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET (31 tone ET) or 31-

EDO Edo ( ja, , , "bay-entrance" or "estuary"), also romanized as Jedo, Yedo or Yeddo, is the former name of Tokyo. Edo, formerly a ''jōkamachi'' (castle town) centered on Edo Castle located in Musashi Province, became the ''de facto'' capital of ...

(equal division of the octave), also known as tricesimoprimal, is the tempered scale derived by dividing the

octave In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...

into 31 equal-sized steps (equal frequency ratios). Each step represents a

frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from '' angular frequency''. Frequency is measured in hertz (Hz) which is ...

ratio of , or 38.71 cents (). 31-ET is a very good approximation of quarter-comma meantone temperament. More generally, it is a

regular diatonic tuning A regular diatonic tuning is any musical scale consisting of " tones" (T) and "semitones" (S) arranged in any rotation of the sequence TTSTTTS which adds up to the octave with all the T's being the same size and all the S's the being the same s ...

in which the tempered perfect fifth is equal to 696.77 cents, as shown in Figure 1. On an isomorphic keyboard, the fingering of music composed in 31-ET is precisely the same as it is in any other syntonic tuning (such as 12-ET), so long as the notes are spelled properly — that is, with no assumption of

enharmonicity In modern musical notation and musical tuning, tuning, an enharmonic equivalent is a Musical note, note, interval (music), interval, or key signature that is Equivalence class (music), equivalent to some other note, interval, or key signature bu ...

History and use

Division of the

octave