64/63: Difference between revisions
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{{Infobox Interval | {{Infobox Interval | ||
| Ratio = 64/63 | | Ratio = 64/63 | ||
| Monzo = 6 -2 0 -1 | | Monzo = 6 -2 0 -1 | ||
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| Sound = Ji-64-63-csound-foscil-220hz.mp3 | | Sound = Ji-64-63-csound-foscil-220hz.mp3 | ||
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{{Wikipedia|Septimal comma}} | |||
'''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. | '''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. | ||
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* [[Gallery of just intervals]] | * [[Gallery of just intervals]] | ||
* [[List of superparticular intervals]] | * [[List of superparticular intervals]] | ||
[[Category:7-limit]] | [[Category:7-limit]] | ||
[[Category:Small comma]] | [[Category:Small comma]] | ||
[[Category:Superparticular]] | [[Category:Superparticular]] | ||
[[Category: | [[Category:Octave-reduced subharmonics]] | ||
[[Category:Pages with internal sound examples]] | |||
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Revision as of 00:48, 26 December 2021
| Interval information |
Archytas' comma
ru unison
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.