64/63: Difference between revisions

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{{Interwiki

| en = 64/63

| de = 64/63

}}

: ''"Septimal comma" redirects here. For non-idiomatic usages, see [[Septimal]] and [[Comma]].''

{{Infobox Interval

~~| Icon =~~

| Name = septimal comma, Archytas' comma

~~| Ratio = 64/63~~

| Color name = r1, ru unison, rM, ruma

~~| Monzo = 6 -2 0 -1~~

~~| Cents = 27.264092~~

| Name = septimal comma, ~~ ~~Archytas' comma

| Color name = r1, ru ~~comma~~, ~~ru unison~~

~~| FJS name = P17~~

| Sound = Ji-64-63-csound-foscil-220hz.mp3

| Comma = yes

}}

'''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.

'''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 leads to [[superpyth]] temperament (sometimes called ''archy'' in the 2.3.7-subgroup), which equates 9/8 and 8/7, and also equates [[7/4]] with [[16/9]]. This means 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, 17, 22, 27, 37, 49 and 59 }}.

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 one is 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 (octave-reduced) reaches the interval 9/7, and a stack of two fifths reaches 9/8, then the difference must be (9/7)/(9/8) = 8/7. The 8/7 and 9/8 intervals are equated, 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.

== Comma pumps ==

The septimal version of the common vi–ii–V–I progression, which uses the 6:7:9 subminor and 14:18:21 supermajor triads, requires that 64/63 be tempered out in order to avoid shifting the root. If 64/63 is not tempered out and intervals are kept pure, the root in the final I chord will be 64/63 higher than the root in the vi chord.

~~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 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~~.)

== Notation ==

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.

~~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~~.

=== 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| |) }}.

~~On the other hand, if you should be so bold as~~ to treat ~~the~~ Archytas' comma as a musical interval in its own right, you will find that it acts as a sort of chroma~~, and is perhaps among the smallest intervals capable of this kind of function~~ – specifically, it functions as ~~the~~ septimal equivalent of [[55/54]], from which it differs by a [[385/384|keenanisma]].

== 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]].

== See also ==

* [[Septimal comma]] (disambiguation page)

* [[Archytas family]]

* [[Archytas clan]]

* [[Small comma]]

* [[Gallery of just intervals]]

* [[List of superparticular intervals]]

* [[Wikipedia: Septimal comma]]

~~[[Category:7-limit]]~~

~~[[Category:Small comma]]~~

~~[[Category:Definition]]~~

~~[[Category:Interval]]~~

~~[[Category:Ratio]]~~

~~[[Category:Superparticular]]~~

~~[[Category:Listen]]~~

~~~~

[[Category:Commas named for their regular temperament properties]]

[[de:~~64/63~~]]

[[Category:Commas named after polymaths]]

@@ Line 1: / Line 1: @@
+{{Interwiki
+| en = 64/63
+| de = 64/63
+}}
+: ''"Septimal comma" redirects here. For non-idiomatic usages, see [[Septimal]] and [[Comma]].''
 {{Infobox Interval
-| Icon =
+| Name = septimal comma, Archytas' comma
-| Ratio = 64/63
+| Color name = r1, ru unison,<br/>rM, ruma
-| Monzo = 6 -2 0 -1
-| Cents = 27.264092
-| Name = septimal comma, <br/>Archytas' comma
-| Color name = r1, ru comma, <br/>ru unison
-| FJS name = P1<sub>7</sub>
 | Sound = Ji-64-63-csound-foscil-220hz.mp3
+| Comma = yes
 }}
-'''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.
+{{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|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 leads to [[superpyth]] temperament (sometimes called ''archy'' in the 2.3.7-subgroup), which equates 9/8 and 8/7, and also equates [[7/4]] with [[16/9]]. This means 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, 17, 22, 27, 37, 49 and 59 }}.
+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 one is 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 (octave-reduced) reaches the interval 9/7, and a stack of two fifths reaches 9/8, then the difference must be (9/7)/(9/8) = 8/7. The 8/7 and 9/8 intervals are equated, 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.
+== Comma pumps ==
+The septimal version of the common vi–ii–V–I progression, which uses the 6:7:9 subminor and 14:18:21 supermajor triads, requires that 64/63 be tempered out in order to avoid shifting the root. If 64/63 is not tempered out and intervals are kept pure, the root in the final I chord will be 64/63 higher than the root in the vi chord.
+{{todo|add sound example}}
-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 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.)
+== Notation ==
+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.
-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.
+=== 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| |) }}.
-On the other hand, if you should be so bold as to treat the Archytas' comma as a musical interval in its own right, you will find that it acts as a sort of chroma, and is perhaps among the smallest intervals capable of this kind of function – specifically, it functions as the septimal equivalent of [[55/54]], from which it differs by a [[385/384|keenanisma]].
+== 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]].
 == See also ==
-* [[Septimal comma]] (disambiguation page)
-* [[Archytas family]]
-* [[Archytas clan]]
-* [[Small comma]]
 * [[Gallery of just intervals]]
 * [[List of superparticular intervals]]
-* [[Wikipedia: Septimal comma]]
-[[Category:7-limit]]
-[[Category:Small comma]]
-[[Category:Definition]]
-[[Category:Interval]]
-[[Category:Ratio]]
-[[Category:Superparticular]]
-[[Category:Listen]]
-<!-- interwiki -->
+[[Category:Commas named for their regular temperament properties]]
-[[de:64/63]]
+[[Category:Commas named after polymaths]]