225/224: Difference between revisions

(42 intermediate revisions by 12 users not shown)

Line 1:

~~<span style~~=~~"display: block; text~~-~~align: right;">Other languages: [[:de:225/224 Deutsch]]~~<~~/span~~>

{{Infobox Interval

| Name = marvel comma, septimal kleisma

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

| Comma = yes

}}

[~~http:~~//en.~~wikipedia~~.~~org~~/~~wiki~~/~~Septimal_kleisma http:~~//en.~~wikipedia~~.~~org~~/~~wiki~~/~~Septimal_kleisma~~] [[Category:~~interval~~]]

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.

[[Category:~~ratio~~]]

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.

== 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]]

* [[Small comma]]

* [[Gallery of just intervals]]

* [[List of superparticular intervals]]

[[Category:Marvel]]

[[Category:Commas with unknown etymology]]

@@ Line 1: / Line 1: @@
-<span style="display: block; text-align: right;">Other languages: [[:de:225/224 Deutsch]]</span>
+{{Infobox Interval
+| Name = marvel comma, septimal kleisma
+| Color name = ryy-2, ruyoyo negative 2nd,<br> Ruyoyo comma
+| Comma = yes
+}}
+{{Wikipedia|Septimal kleisma}}
-[http://en.wikipedia.org/wiki/Septimal_kleisma http://en.wikipedia.org/wiki/Septimal_kleisma]      [[Category:interval]]
+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.
-[[Category:ratio]]
+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.
+== 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]]
+* [[Small comma]]
+* [[Gallery of just intervals]]
+* [[List of superparticular intervals]]
+[[Category:Marvel]]
+[[Category:Commas with unknown etymology]]