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. | ||
If one wants to treat Archytas' comma as a musical interval in its own right as opposed to tempering it out (which in higher-accuracy contexts causes significant damage to the [[7-limit]]), you will find that it acts as a sort of chroma – specifically, it functions as a septimal equivalent of [[55/54]], from which it differs by a [[385/384|keenanisma]], or [[56/55]], from which it differs by a [[441/440|Werckisma]]. In addition, its incredible proximity to 1/44th of the octave - to the point where [[Septimal ruthenia|44-64/63 comma]] is tempered out in EDOs as large as tens of thousands - enables the tuning of [[ruthenium]] temperament. As a result, the major second of [[22edo]] is a good approximation to [[17/15]], due to it being the [[mediant]] of [[9/8]] and [[8/7]], so that the ~7:8:9 chord is much more accurately a 17/15 - 17/15 chord, with the outer dyad as 9/7, by tempering [[2025/2023]]. | |||
== See also == | == See also == | ||
Revision as of 22:38, 30 July 2024
| 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.
If one wants to treat Archytas' comma as a musical interval in its own right as opposed to tempering it out (which in higher-accuracy contexts causes significant damage to the 7-limit), you will find that it acts as a sort of chroma – specifically, it functions as a septimal equivalent of 55/54, from which it differs by a keenanisma, or 56/55, from which it differs by a Werckisma. In addition, its incredible proximity to 1/44th of the octave - to the point where 44-64/63 comma is tempered out in EDOs as large as tens of thousands - enables the tuning of ruthenium temperament. As a result, the major second of 22edo is a good approximation to 17/15, due to it being the mediant of 9/8 and 8/7, so that the ~7:8:9 chord is much more accurately a 17/15 - 17/15 chord, with the outer dyad as 9/7, by tempering 2025/2023.