64/63: Difference between revisions

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'''64/63''', the '''septimal comma''' (also '''Archytas' comma''', or more simply and systematically the '''archytas comma''' or '''archy comma'''), is a [[small comma|small]] [[7-limit]] [[superparticular]] [[comma]] which separates [[9/8]] and [[8/7]] and has the eighth square number as a numerator. It can be considered the [[2.3.7 subgroup]] equivalent of the [[~~81/80~~]] ~~comma~~, and seperates complex pythagorean intervals from simpler 7-limit ones. For example, it is the difference between [[32/27]] and [[7/6]], and the difference between [[81/64]] and [[9/7]]. Since its numerator is a power of 2, it is a [[Mersenne comma]].

'''64/63''', the '''septimal comma''' (also '''Archytas' comma''', or more simply and systematically the '''archytas comma''' or '''archy comma'''), is a [[small comma|small]] [[7-limit]] [[superparticular]] [[comma]] which separates [[9/8]] and [[8/7]] and has the eighth square number as a numerator. It can be considered the [[2.3.7 subgroup|2.3.7-]][[subgroup]] equivalent of the [[syntonic comma]], and seperates complex pythagorean intervals from simpler 7-limit ones. For example, it is the difference between [[32/27]] and [[7/6]], and the difference between [[81/64]] and [[9/7]]. Since its numerator is a power of 2, it is a [[Mersenne comma]].

== 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, [[36:45:54:64|1–5/4–3/2–16/9]], and the harmonic seventh chord, [[4:5:6:7|1–5/4–3/2–7/4]], are equated to the same chord. Equal temperaments 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.)~~

Archytas' comma is similar to Didymus' or the syntonic comma, 81/80, in that when it is tempered out it makes a stack of four fifths [[octave reduction|octave reduced]] equal a relatively consonant major third. In the case of 81/80, the major third is [[5/4]], while with Archytas' comma, the major third is [[9/7]].

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 [[Archytas family]] for the family of rank-3 temperaments where it is tempered out; see [[Archytas clan]] for the clan of rank-2 temperaments where it is tempered out.

== Approximation ==

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]].

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]].

== Notation ==

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 }}
 {{Wikipedia|Septimal comma}}
-'''64/63''', the '''septimal comma''' (also '''Archytas' comma''', or more simply and systematically the '''archytas comma''' or '''archy comma'''), is a [[small comma|small]] [[7-limit]] [[superparticular]] [[comma]] which separates [[9/8]] and [[8/7]] and has the eighth square number as a numerator. It can be considered the [[2.3.7 subgroup]] equivalent of the [[81/80]] comma, and seperates complex pythagorean intervals from simpler 7-limit ones. For example, it is the difference between [[32/27]] and [[7/6]], and the difference between [[81/64]] and [[9/7]]. Since its numerator is a power of 2, it is a [[Mersenne comma]].
+'''64/63''', the '''septimal comma''' (also '''Archytas' comma''', or more simply and systematically the '''archytas comma''' or '''archy comma'''), is a [[small comma|small]] [[7-limit]] [[superparticular]] [[comma]] which separates [[9/8]] and [[8/7]] and has the eighth square number as a numerator. It can be considered the [[2.3.7 subgroup|2.3.7-]][[subgroup]] equivalent of the [[syntonic comma]], and seperates complex pythagorean intervals from simpler 7-limit ones. For example, it is the difference between [[32/27]] and [[7/6]], and the difference between [[81/64]] and [[9/7]]. Since its numerator is a power of 2, it is a [[Mersenne comma]].
 == 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, [[36:45:54:64|1–5/4–3/2–16/9]], and the harmonic seventh chord, [[4:5:6:7|1–5/4–3/2–7/4]], are equated to the same chord. Equal temperaments 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.)
+Archytas' comma is similar to Didymus' or the syntonic comma, 81/80, in that when it is tempered out it makes a stack of four fifths [[octave reduction|octave reduced]] equal a relatively consonant major third. In the case of 81/80, the major third is [[5/4]], while with Archytas' comma, the major third is [[9/7]].
-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 [[Archytas family]] for the family of rank-3 temperaments where it is tempered out; see [[Archytas clan]] for the clan of rank-2 temperaments where it is tempered out.
 == Approximation ==
-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]].
+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]].
 == Notation ==