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 comma, ru unison
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
[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 you 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, and is perhaps among the smallest intervals capable of this kind of function – specifically, it functions as the septimal equivalent of 55/54, from which it differs by a keenanisma.

See also

• Septimal comma (disambiguation page)
• Archytas family
• Archytas clan
• Small comma
• Gallery of just intervals
• List of superparticular intervals
• Wikipedia: Septimal comma

External links

• Wikipedia: Komma (Musik) #Septimales Komma (German)