Editing Equal temperament (section)

=== Equal temperament with a non-integral number of notes per octave ===
While traditional equal temperaments—such as 12‑TET, 19‑TET, or 31‑TET—divide the octave into an integral number of equal parts, it is also possible to explore systems that divide the octave into a non-integral (often irrational) number. In such temperaments, the interval between successive pitches is defined by the ratio 2^(1/N), where N is not an integer. This results in irrational step sizes, meaning their multiples never exactly equal an octave.

Such tunings are of interest because, by deliberately sacrificing the octave (i.e., the second harmonic), they can yield a system that offers an improved overall approximation of other intervals in the harmonic series.

For example, in a tuning system based on 18.911‑EDO, the step size is 1200⁄18.911 ≈ 63.45 cents. Approximating the just perfect fifth (with a ratio of 3:2, or about 701.96 cents) requires about 11 steps:

* ''11 steps × 63.45 cents ≈ 698.95 cents,''
yielding an error of roughly 3 cents.

Similarly, for the just major third (with a ratio of 5:4, or about 386.31 cents), 6 steps are used:

* ''6 steps × 63.45 cents ≈ 380.70 cents,''
resulting in an error of approximately 5.61 cents.

Thus, although a perfect octave is absent, the consonance of many other intervals in these systems can be significantly higher than in integer-based equal temperaments.