64/63: Difference between revisions

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== Temperaments ==

Tempering out this comma equates 9/8 and 8/7, and 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. Equal divisions of the octave tempering out 64/63 include {{EDOs| 12, 15, 22, 27, 37, 49 and 59 }}.

Tempering out this comma equates 9/8 and 8/7, and 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. Equal divisions of the octave tempering out 64/63 include {{EDOs| 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.)

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If one wants to treat Archytas' comma as a musical interval in its own right as opposed to tempering it out, 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 of [[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 the [[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 interval as 9/7, by tempering out [[2025/2023]].

== Sagittal notation ==

== Notation ==

In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented by the sagittal {{sagittal | !) }} and is called the '''7 comma''', or '''7C''' for short, because the simplest interval it notates is 7/1 (equiv. 7/4), as for example in ~~G-F~~{{nbhsp}}{{sagittal | !) }}. The upward version is called '''1/7C''' or '''7C up''' and is represented by {{sagittal| |) }}.

This interval is significant in the [[Functional Just System]] and [[Helmholtz-Ellis notation]] as the septimal formal comma which translates a Pythagorean interval to a nearby septimal interval.

=== Sagittal notation ===

In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented by the sagittal {{sagittal | !) }} and is called the '''7 comma''', or '''7C''' for short, because the simplest interval it notates is 7/1 (equiv. 7/4), as for example in G–F{{nbhsp}}{{sagittal | !) }}. The upward version is called '''1/7C''' or '''7C up''' and is represented by {{sagittal| |) }}.

== See also ==

@@ Line 13: / Line 13: @@
 == Temperaments ==
-Tempering out this comma equates 9/8 and 8/7, and 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. Equal divisions of the octave tempering out 64/63 include {{EDOs| 12, 15, 22, 27, 37, 49 and 59 }}.
+Tempering out this comma equates 9/8 and 8/7, and 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. Equal divisions of the octave tempering out 64/63 include {{EDOs| 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.)
@@ Line 24: / Line 24: @@
 If one wants to treat Archytas' comma as a musical interval in its own right as opposed to tempering it out, 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 of [[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 the [[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 interval as 9/7, by tempering out [[2025/2023]].
-== Sagittal notation ==
+== Notation ==
-In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented by the sagittal {{sagittal | !) }} and is called the '''7 comma''', or '''7C''' for short, because the simplest interval it notates is 7/1 (equiv. 7/4), as for example in G-F{{nbhsp}}{{sagittal | !) }}. The upward version is called '''1/7C''' or '''7C up''' and is represented by {{sagittal| |) }}.
+This interval is significant in the [[Functional Just System]] and [[Helmholtz-Ellis notation]] as the septimal formal comma which translates a Pythagorean interval to a nearby septimal interval.
+=== Sagittal notation ===
+In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented by the sagittal {{sagittal | !) }} and is called the '''7 comma''', or '''7C''' for short, because the simplest interval it notates is 7/1 (equiv. 7/4), as for example in G–F{{nbhsp}}{{sagittal | !) }}. The upward version is called '''1/7C''' or '''7C up''' and is represented by {{sagittal| |) }}.
 == See also ==