225/224: Difference between revisions

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{{Infobox Interval

| Name = septimal kleisma~~, marvel comma~~

| Name = marvel comma, septimal kleisma

| Color name = ryy-2, ruyoyo negative 2nd,<br> Ruyoyo comma

| Comma = yes

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The interval of '''225/224''', the '''~~septimal kleisma~~''' or '''~~marvel comma~~''' is a [[7-limit]] [[superparticular]] ~~ratio~~. It pops up as the difference between ~~pairs of~~ 7-limit ~~ratios~~, ~~for example as (~~[[15/14]])/([[16/15]]) or ([[45/32]]~~)/(~~[[7/5]]). ~~Another useful relation is as~~ the difference between ~~the~~ [[25/24]], the classical chromatic semitone, and [[28/27]], the septimal third-tone.

The interval of '''225/224''', the '''marvel comma''', otherwise known as the '''septimal kleisma''', is a [[7-limit]] [[superparticular]] [[comma]]. It pops up as the difference between a 7-limit ratio and a 5-limit ratio. For example, it's the difference between [[16/15]] and [[15/14]], and between [[7/5]] and [[45/32]]. Moreover, it can be seen as the amount by which [[8/7]] exceeds a stack of two {{nowrap|[[16/15]]'s}}, or as the amount by which a stack of two {{nowrap|[[5/4]]'s}} exceeds [[14/9]]. It's also the difference between [[75/64]] and [[7/6]], and between [[25/24]], the classical chromatic semitone, and [[28/27]], the septimal third-tone.

It is ~~also~~ the difference between [[81/80]] and [[126/125]] and is tempered out alongside these two commas in [[septimal meantone]]. In the 11-limit, it factors neatly into ([[385/384]])([[540/539]]).

As a comma with a single power of 7 in it, it is tremendously useful in terms of bringing prime 7 into the framework of [[5-limit]] [[just intonation|JI]]; tempering it out maps [[7/4]] to the classic augmented sixth, [[225/128]] and enables all of the aforementioned equivalences.

In terms of commas, it is the difference between [[81/80]] and [[126/125]] and is tempered out alongside these two commas in [[septimal meantone]]. In the 11-limit it factors neatly into ([[385/384]])([[540/539]]), and in the 13-limit, ([[351/350]])([[625/624]]) or ([[325/324]])([[729/728]]).

== Temperaments ==

Tempering out this comma alone in the 7-limit leads to the [[marvel]] temperament, which enables [[marvel chords]]. See [[~~marvel~~ family]] for the family of rank-3 temperaments where it is tempered out. See [[~~marvel~~ temperaments]] for a collection of rank-2 temperaments where it is tempered out.

Tempering out this comma alone in the 7-limit leads to the [[marvel]] temperament, which enables [[marvel chords]]. See [[Marvel family]] for the family of rank-3 temperaments where it is tempered out. See [[Marvel temperaments]] for a collection of rank-2 temperaments where it is tempered out.

== Approximation ==

If we do not temper out this interval and instead repeatedly stack (and octave-reduce) it, we ~~get~~ [[311edo]]~~, where it is equal to 2 steps, meaning 311edo~~ is a circle of 225/224's. Note that this is not true for 226/225 or 224/223, the adjacent superparticulars, as they accumulate too much error to close into a circle in 311edo.

If we do not temper out this interval and instead repeatedly stack (and octave-reduce) it, we almost return to the starting point at the 311th step, meaning [[311edo]] is a [[consistent circle]] of 225/224's. Note that this is not true for 226/225 or 224/223, the adjacent superparticulars, as they accumulate too much error to close into a circle in 311edo.

== Etymology ==

''Marvel comma'' was named after the corresponding temperament, marvel, which was named by [[Gene Ward Smith]] in 2002–2003.

== See also ==

* [[Marvel]]

* [[Small comma]]

* [[Gallery of just intervals]]

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[[Category:Marvel]]

[[Category:Commas with unknown etymology]]

@@ Line 1: / Line 1: @@
 {{Infobox Interval
-| Name = septimal kleisma, marvel comma
+| Name = marvel comma, septimal kleisma
 | Color name = ryy-2, ruyoyo negative 2nd,<br> Ruyoyo comma
 | Comma = yes
@@ Line 6: / Line 6: @@
 {{Wikipedia|Septimal kleisma}}
-The interval of '''225/224''', the '''septimal kleisma''' or '''marvel comma''' is a [[7-limit]] [[superparticular]] ratio. It pops up as the difference between pairs of 7-limit ratios, for example as ([[15/14]])/([[16/15]]) or ([[45/32]])/([[7/5]]). Another useful relation is as the difference between the [[25/24]], the classical chromatic semitone, and [[28/27]], the septimal third-tone.
+The interval of '''225/224''', the '''marvel comma''', otherwise known as the '''septimal kleisma''', is a [[7-limit]] [[superparticular]] [[comma]]. It pops up as the difference between a 7-limit ratio and a 5-limit ratio. For example, it's the difference between [[16/15]] and [[15/14]], and between [[7/5]] and [[45/32]]. Moreover, it can be seen as the amount by which [[8/7]] exceeds a stack of two {{nowrap|[[16/15]]'s}}, or as the amount by which a stack of two {{nowrap|[[5/4]]'s}} exceeds [[14/9]]. It's also the difference between [[75/64]] and [[7/6]], and between [[25/24]], the classical chromatic semitone, and [[28/27]], the septimal third-tone.
-It is also the difference between [[81/80]] and [[126/125]] and is tempered out alongside these two commas in [[septimal meantone]]. In the 11-limit, it factors neatly into ([[385/384]])([[540/539]]).
+As a comma with a single power of 7 in it, it is tremendously useful in terms of bringing prime 7 into the framework of [[5-limit]] [[just intonation|JI]]; tempering it out maps [[7/4]] to the classic augmented sixth, [[225/128]] and enables all of the aforementioned equivalences.
+In terms of commas, it is the difference between [[81/80]] and [[126/125]] and is tempered out alongside these two commas in [[septimal meantone]]. In the 11-limit it factors neatly into ([[385/384]])([[540/539]]), and in the 13-limit, ([[351/350]])([[625/624]]) or ([[325/324]])([[729/728]]).
 == Temperaments ==
-Tempering out this comma alone in the 7-limit leads to the [[marvel]] temperament, which enables [[marvel chords]]. See [[marvel family]] for the family of rank-3 temperaments where it is tempered out. See [[marvel temperaments]] for a collection of rank-2 temperaments where it is tempered out.
+Tempering out this comma alone in the 7-limit leads to the [[marvel]] temperament, which enables [[marvel chords]]. See [[Marvel family]] for the family of rank-3 temperaments where it is tempered out. See [[Marvel temperaments]] for a collection of rank-2 temperaments where it is tempered out.
 == Approximation ==
-If we do not temper out this interval and instead repeatedly stack (and octave-reduce) it, we get [[311edo]], where it is equal to 2 steps, meaning 311edo is a circle of 225/224's. Note that this is not true for 226/225 or 224/223, the adjacent superparticulars, as they accumulate too much error to close into a circle in 311edo.
+If we do not temper out this interval and instead repeatedly stack (and octave-reduce) it, we almost return to the starting point at the 311th step, meaning [[311edo]] is a [[consistent circle]] of 225/224's. Note that this is not true for 226/225 or 224/223, the adjacent superparticulars, as they accumulate too much error to close into a circle in 311edo.
+== Etymology ==
+''Marvel comma'' was named after the corresponding temperament, marvel, which was named by [[Gene Ward Smith]] in 2002–2003.
 == See also ==
+* [[Marvel]]
 * [[Small comma]]
 * [[Gallery of just intervals]]
@@ Line 22: / Line 28: @@
 [[Category:Marvel]]
+[[Category:Commas with unknown etymology]]