31edo

31EDO, short for 31 equal divisions of the octave, is a musical temperament that divides the octave into 31 equal steps. Each step is about 38.71 cents, so the whole octave totals 1200 cents. In 31-EDO, pitches are represented by integers from 0 to 30, with octave equivalence meaning that pitch 31 is the same as pitch 0.

This temperament sits between the familiar 12-EDO system and more finely divided microtonal tunings. It is studied and used in theoretical and experimental contexts because its subdivision is dense enough to approximate several just intonation intervals more closely than 12-EDO, while remaining implementable in software and, to a lesser extent, in custom-built keyboards or electronic instruments.

Interval relationships in 31-EDO are defined by steps of equal size. For example, the major third in just intonation (5:4, about 386.31 cents) is well approximated by 10 steps (about 387.10 cents). The perfect fifth (3:2, about 701.96 cents) is commonly approximated by 18 steps (about 696.78 cents), which is a close, though imperfect, match. Other intervals also have characteristic approximations that differ from their 12-EDO values, giving different voice-leading possibilities and color.

In practice, 31-EDO is used in music theory, computer-assisted composition, and certain microtonal performance contexts. It is implemented in many digital tuners and synthesizers, where frequency values follow f0 · 2^(n/31). Notation can vary, with pitch classes numbered 0–30 and octave equivalence applied across registers.