# 64/63

 Ratio 64/63 Factorization 26 × 3-2 × 7-1 Monzo [6 -2 0 -1⟩ Size in cents 27.264092¢ Names septimal comma,Archytas' comma Color name r1, ru unison, Ru comma FJS name $\text{P1}_{7}$ Special properties square superparticular,reduced,reduced subharmonic Tenney height (log2 nd) 11.9773 Weil height (log2 max(n, d)) 12 Wilson height (sopfr (nd)) 25 Harmonic entropy(Shannon, $\sqrt{nd}$) ~4.36048 bits Comma size small S-expression S8 https://en.xen.wiki/w/File:Ji-64-63-csound-foscil-220hz.mp3[sound info] open this interval in xen-calc
English Wikipedia has an article on:

64/63, the septimal comma (also Archytas' comma, or sometimes in German Leipziger Komma), is a small 7-limit superparticular comma which separates 9/8 and 8/7 and has the eighth square number as a numerator.

## Temperaments

Tempering out this comma equates 9/8 and 8/7, and also equates 7/4 with 16/9, so that the just dominant seventh chord, 1-5/4-3/2-16/9, and the otonal tetrad, 1-5/4-3/2-7/4, are equated to the same chord. Equal divisions of the octave tempering out 64/63 include 12, 15, 22, 27, 37, 49 and 59.

The Archytas' comma is similar to the Didymus or syntonic comma, 81/80, in that when it is tempered out it makes a stack of four fifths equal a major third (octave equivalent). In the case of 81/80, however, the major third is 5/4, while with the Archytas comma, the major third is 9/7. (Note that porcupine, which tempers out 64/63, uses a minor tone as a generator and generally is considered to have 5/4 major thirds, so it doesn't depend on this equivalency.)

If you are using 9/7 major thirds, this also implies that the major third is split into two equal steps that represent both 9/8 and 8/7: If a stack of four fifths gets you to (octave-equivalent) 9/7, and a stack of two fifths gets you to 9/8, then the difference must be (9/7)/(9/8) = 8/7. The 8/7 and 9/8 intervals are equal, however, as a result of the generation process.

See Archytas family for the family of rank-3 temperaments where it is tempered out; see Archytas clan for the clan of rank-2 temperaments where it is tempered out.

## Approximation

If one wants to treat Archytas' comma as a musical interval in its own right as opposed to tempering it out, you will find that it acts as a sort of chroma – specifically, it functions as a septimal equivalent of 55/54, from which it differs by a keenanisma, or of 56/55, from which it differs by a Werckisma. In addition, its incredible proximity to 1/44th of the octave – to the point where the 44-64/63 comma is tempered out in edos as large as tens of thousands – enables the tuning of ruthenium temperament. As a result, the major second of 22edo is a good approximation to 17/15, due to it being the mediant of 9/8 and 8/7, so that the ~7:8:9 chord is much more accurately a 17/15–17/15 chord, with the outer dyad as 9/7, by tempering out 2025/2023.