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 | [math]\text{P1}_{7}[/math] |
| Special properties | square superparticular, reduced |
| Tenney height (log2 nd) | 11.9773 |
| Weil height (log2 max(n, d)) | 12 |
| Wilson height (sopfr (nd)) | 25 |
| Harmonic entropy (Shannon, [math]\sqrt{n\cdot d}[/math]) |
~4.25459 bits |
| Comma size | small |
| S-expression | S8 |
[sound info] | |
| open this interval in xen-calc | |
64/63, the septimal comma (also Archytas' comma, or sometimes in German Leipziger Komma), is a superparticular ratio which equates 9/8 and 8/7 if tempered out and has the eighth square number as a numerator. It 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 when 64/63 is tempered out. 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.
On the other hand, if one should be so bold as to treat the Archytas' comma as a musical interval in its own right, you will find that it acts as a sort of chroma – specifically, it functions as the septimal equivalent of 55/54, from which it differs by a keenanisma. In addition, its incredible proximity to 1/44th of the octave - to the point where 44-64/63 comma is tempered out in EDOs as large as tens of thousands - enables the tuning of ruthenium temperament.