64/63: Difference between revisions
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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. | 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 also == | |||
* [[Gallery of just intervals]] | |||
* [[Comma]] | |||
* [[63/62]] | |||
* [[65/64]] | |||
== External links == | == External links == | ||
* [https://en.wikipedia.org/wiki/Septimal_comma Septimal comma - Wikipedia] | * [https://en.wikipedia.org/wiki/Septimal_comma Septimal comma - Wikipedia] | ||
* [https://de.wikipedia.org/wiki/Komma_(Musik)#Septimales_Komma Komma (Musik) - Wikipedia #Septimales_Komma] | * [https://de.wikipedia.org/wiki/Komma_(Musik)#Septimales_Komma Komma (Musik) - Wikipedia #Septimales_Komma] | ||
Revision as of 14:11, 21 June 2020
| Interval information |
Archytas' comma
ru unison
reduced,
reduced subharmonic
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.