225/224: Difference between revisions

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{{Infobox Interval

~~| Ratio = 225/224~~

| Name = marvel comma, septimal kleisma

~~| Monzo = -5 2 2 -1~~

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

~~| Cents = 7.71152~~

| Comma = yes

| Name = septimal kleisma~~, <br/> marvel comma~~

| Color name = ryy-2, ruyoyo negative 2nd

~~| FJS name = d-2~~<~~sup>25</sup><sub>7</sub~~>

| ~~Sound~~ =

}}

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]]).

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 [[~~Meantone family#Septimal meantone|Septimal~~ meantone]].

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 leads to the [[marvel family]] of temperaments, which ~~enables~~ [[~~marvel chords~~]].

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 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 ~~family]], the family of rank-3 temperaments where it is tempered out~~

* [[Marvel]]

* [[Marvel temperaments]], a collection of rank-2 temperaments where it is tempered out

* [[Marvel chords]]

* [[Small comma]]

* [[Gallery of just intervals]]

* [[List of superparticular intervals]]

* [[Wikipedia: Septimal kleisma]]

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

~~[[Category:Interval]]~~

~~[[Category:Ratio]]~~

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

~~[[Category:Superparticular]]~~

[[Category:Marvel]]

[[Category:Commas with unknown etymology]]

@@ Line 1: / Line 1: @@
 {{Infobox Interval
-| Ratio = 225/224
+| Name = marvel comma, septimal kleisma
-| Monzo = -5 2 2 -1
+| Color name = ryy-2, ruyoyo negative 2nd,<br> Ruyoyo comma
-| Cents = 7.71152
+| Comma = yes
-| Name = septimal kleisma, <br/> marvel comma
-| Color name = ryy-2, ruyoyo negative 2nd
-| FJS name = d-2<sup>25</sup><sub>7</sub>
-| Sound =
 }}
+{{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]]).
+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 [[Meantone family#Septimal meantone|Septimal meantone]].
+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 leads to the [[marvel family]] of temperaments, which enables [[marvel chords]].
+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 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 family]], the family of rank-3 temperaments where it is tempered out
+* [[Marvel]]
-* [[Marvel temperaments]], a collection of rank-2 temperaments where it is tempered out
-* [[Marvel chords]]
 * [[Small comma]]
 * [[Gallery of just intervals]]
 * [[List of superparticular intervals]]
-* [[Wikipedia: Septimal kleisma]]
-[[Category:7-limit]]
-[[Category:Interval]]
-[[Category:Ratio]]
-[[Category:Small comma]]
-[[Category:Superparticular]]
 [[Category:Marvel]]
+[[Category:Commas with unknown etymology]]