64/63: Difference between revisions
m Moving from Category:Small comma to Category:Small commas using Cat-a-lot |
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{{Infobox Interval | {{Infobox Interval | ||
| Name = septimal comma, Archytas' comma | |||
| Name = septimal comma, | |||
| Color name = r1, ru unison, <br/>Ru comma | | Color name = r1, ru unison, <br/>Ru comma | ||
| Sound = Ji-64-63-csound-foscil-220hz.mp3 | | Sound = Ji-64-63-csound-foscil-220hz.mp3 | ||
| Comma = yes | |||
}} | }} | ||
{{Wikipedia|Septimal comma}} | {{Wikipedia|Septimal comma}} | ||
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* [[Gallery of just intervals]] | * [[Gallery of just intervals]] | ||
* [[List of superparticular intervals]] | * [[List of superparticular intervals]] | ||
Revision as of 16:40, 25 October 2022
| Interval information |
Archytas' comma
Ru comma
reduced,
reduced subharmonic
[sound info]
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.