64/63: Difference between revisions
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The interval 64/63, called '''septimal comma''' or '''Archytas' comma''' or (in German) [http://de.wikipedia.org/wiki/Komma_%28Musik%29#Leipziger_Komma Leipziger Komma], is a [[superparticular|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 interval '''64/63''', called '''septimal comma''' or '''Archytas' comma''' or (in German) [http://de.wikipedia.org/wiki/Komma_%28Musik%29#Leipziger_Komma Leipziger Komma], is a [[superparticular|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 a 7-limit comma with monzo {{Monzo| 6 -2 0 -1 }}. It is | The Archytas comma is a 7-limit comma with monzo {{Monzo| 6 -2 0 -1 }}. It is 27.264092 [[cent]]s in size. | ||
It 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 family|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. | |||
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. | |||
== External links == | == External links == | ||
Revision as of 18:21, 9 October 2018
The interval 64/63, called septimal comma or Archytas' comma or (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 a 7-limit comma with monzo [6 -2 0 -1⟩. It is 27.264092 cents in size.
It 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.