53 equal temperament

In music, 53 equal temperament, called 53 TET, 53 EDO, or 53 ET, is the tempered scale derived by dividing the octave into 53 equal steps (equal frequency ratios). Play^ⓘ Each step represents a frequency ratio of   2^1 ∕ 53 , or 22.6415 cents (Play^ⓘ), an interval sometimes called the Holdrian comma.

53 TET is a tuning of equal temperament in which the tempered perfect fifth is 701.89 cents wide, as shown in Figure 1, and sequential pitches are separated by 22.642 cents.

The 53-TET tuning equates to the unison, or tempers out, the intervals ⁠ 32 805 / 32 768 ⁠, known as the schisma, and ⁠ 15 625 / 15 552 ⁠, known as the kleisma. These are both 5 limit intervals, involving only the primes 2, 3, and 5 in their factorization, and the fact that 53 TET tempers out both characterizes it completely as a 5 limit temperament: It is the only regular temperament tempering out both of these intervals, or commas, a fact which seems to have first been recognized by Japanese music theorist Shohé Tanaka. Because it tempers these out, 53 TET can be used for both schismatic temperament, tempering out the schisma, and Hanson temperament (also called kleismic), tempering out the kleisma.

The interval of ⁠ 7 / 4 ⁠ is closest to the 43rd note (counting from 0) and   2^{43 ∕ 53} = 1.7548   is only 4.8 cents sharp from the harmonic 7th   = ⁠ 7 / 4 ⁠ in 53 TET, and using it for 7-limit harmony means that the septimal kleisma, the interval ⁠ 225 / 224 ⁠, is also tempered out.