64/63

Interval information

Ratio	64/63
Factorization	26 × 3-2 × 7-1
Monzo	[6 -2 0 -1⟩
Size in cents	27.26409¢
Names	septimal comma, Archytas' comma
Color name	r1, ru unison, Ru comma
FJS name	[math]\displaystyle{ \text{P1}_{7} }[/math]
Special properties	square superparticular, reduced, reduced subharmonic
Tenney norm (log2 nd)	11.9773
Weil norm (log2 max(n, d))	12
Wilson norm (sopfr(nd))	25
Comma size	small
S-expression	S8
[sound info]
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English Wikipedia has an article on:

Septimal comma

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. It is a Mersenne comma.

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 interval as 9/7, by tempering out 2025/2023.

Sagittal notation

In the Sagittal system, the downward version of this comma (possibly tempered) is represented by the sagittal  and is called the 7 comma, or 7C for short, because the simplest interval it notates is 7/1 (equiv. 7/4), as for example in G-F⁠ ⁠. The upward version is called 1/7C or 7C up and is represented by .

See also

• Septimal comma (disambiguation page)
• Gallery of just intervals
• List of superparticular intervals