15625/15552: Difference between revisions

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'''15625/15552''', which is sometimes called the '''kleisma''', is the [[small comma|small]] [[5-limit]] [[comma]] that separates a stack of six classic minor thirds of [[6/5]] with [[3/1]], ~~and~~ is also the difference between the classic chromatic semitone ([[25/24]]) and the greater diesis ([[648/625]]). Like the [[magic comma]], instead of being tempered out in [[12edo]], it is inflated to an entire semitone—six 6/5's make an augmented eleventh (18 steps), while 3/1 is a perfect ~~tweltfth~~ (19 steps); conversely, 25/24 is mapped to a semitone while 648/625 is tempered out. Its [[square superparticular #Glossary|S-expression]] of S252 ~~× S26~~ directly implies that it is expressible as ([[25/24]])2/([[27/25]]).

'''15625/15552''', which is sometimes called the '''kleisma''', is the [[small comma|small]] [[5-limit]] [[comma]] that separates a stack of six classic minor thirds of [[6/5]] with [[3/1]], or a stack of five 6/5's with [[5/2]], or a stack of three diptolemaic chromatic semitones ([[25/24]]) with the Pythagorean whole tone ([[9/8]]). It is also the difference between the classic chromatic semitone ([[25/24]]) and the greater diesis ([[648/625]]), which is also known as the diminished comma.

Like the [[magic comma]], instead of being [[tempering out|tempered out]] in [[12edo]], it is inflated to an entire semitone—six 6/5's make an augmented eleventh (18 steps), while 3/1 is a perfect twelfth (19 steps); conversely, 25/24 is mapped to a semitone while 648/625 is tempered out. Its [[square superparticular #Glossary|S-expression]] of S252⋅S26 directly implies that it is expressible as ([[25/24]])2/([[27/25]]).

== Temperaments ==

~~This~~ comma ~~is tempered out~~ in the [[~~hanson~~]] temperament. See [[Kleismic family]] for the rank-2 temperament family where it is tempered out~~. See~~ [[~~Kleismic rank three family|~~Kleismic rank-3 family]] for the rank-3 temperament family where it is tempered out.

Tempering out this comma in the 5-limit leads to the [[kleismic]] temperament.

See [[Kleismic family]] for the rank-2 temperament family where it is tempered out and [[Kleismic rank-3 family]] for the rank-3 temperament family where it is tempered out.

== Etymology ==

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== See also ==

* [[Kleisma]] ~~(disambiguation page)~~

* [[Kleisma]]

== External links ==

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+{{Interwiki
+| en = 15625/15552
+| de = 15625/15552
+| es =
+| ja =
+}}
 {{Infobox Interval
 | Ratio = 15625/15552
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 {{Wikipedia|Kleisma}}
-'''15625/15552''', which is sometimes called the '''kleisma''', is the [[small comma|small]] [[5-limit]] [[comma]] that separates a stack of six classic minor thirds of [[6/5]] with [[3/1]], and is also the difference between the classic chromatic semitone ([[25/24]]) and the greater diesis ([[648/625]]). Like the [[magic comma]], instead of being tempered out in [[12edo]], it is inflated to an entire semitone—six 6/5's make an augmented eleventh (18 steps), while 3/1 is a perfect tweltfth (19 steps); conversely, 25/24 is mapped to a semitone while 648/625 is tempered out. Its [[square superparticular #Glossary|S-expression]] of S25<sup>2</sup> × S26 directly implies that it is expressible as ([[25/24]])<sup>2</sup>/([[27/25]]).
+'''15625/15552''', which is sometimes called the '''kleisma''', is the [[small comma|small]] [[5-limit]] [[comma]] that separates a stack of six classic minor thirds of [[6/5]] with [[3/1]], or a stack of five 6/5's with [[5/2]], or a stack of three diptolemaic chromatic semitones ([[25/24]]) with the Pythagorean whole tone ([[9/8]]). It is also the difference between the classic chromatic semitone ([[25/24]]) and the greater diesis ([[648/625]]), which is also known as the diminished comma.
+Like the [[magic comma]], instead of being [[tempering out|tempered out]] in [[12edo]], it is inflated to an entire semitone—six 6/5's make an augmented eleventh (18 steps), while 3/1 is a perfect twelfth (19 steps); conversely, 25/24 is mapped to a semitone while 648/625 is tempered out. Its [[square superparticular #Glossary|S-expression]] of S25<sup>2</sup>⋅S26 directly implies that it is expressible as ([[25/24]])<sup>2</sup>/([[27/25]]).
 == Temperaments ==
-This comma is tempered out in the [[hanson]] temperament. See [[Kleismic family]] for the rank-2 temperament family where it is tempered out. See [[Kleismic rank three family|Kleismic rank-3 family]] for the rank-3 temperament family where it is tempered out.
+Tempering out this comma in the 5-limit leads to the [[kleismic]] temperament.
+See [[Kleismic family]] for the rank-2 temperament family where it is tempered out and [[Kleismic rank-3 family]] for the rank-3 temperament family where it is tempered out.
 == Etymology ==
@@ Line 16: / Line 26: @@
 == See also ==
-* [[Kleisma]] (disambiguation page)
+* [[Kleisma]]
 == External links ==