9801/9800: Difference between revisions

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'''9801/9800''', the '''kalisma''', sometimes described as ''Gauss' comma'', is an [[unnoticeable comma|unnoticeable]] [[11-limit]] [[comma]] measuring about 0.18{{cent}}. It is the smallest 11-limit [[superparticular]] interval. It can be described as the difference between [[99/70]] and its [[octave complement]] [[140/99]], between ([[35/33]])<sup>2</sup> and [[9/8]], or between ([[176/175]])<sup>2</sup> and [[2048/2025]].

~~'''9801/9800'''~~, the ~~'''kalisma''', sometimes described as ''Gauss' comma'', is an~~ [[~~unnoticeable comma|unnoticeable~~]] [[~~11-limit~~]] [[~~comma~~]] ~~measuring about 0.18{{cent}}. It is the smallest 11-limit~~ [[~~superparticular~~]] ~~interval.~~

In terms of superparticular commas, it is the difference between:

* [[99/98]] and [[100/99]]

* [[351/350]] and [[364/363]]

* [[1716/1715]] and [[2080/2079]]

* [[3025/3024]] and [[4375/4374]]

It ~~can be described as~~ the difference between [[99/98]] and [[~~100~~/99]]~~, and between~~ [[99/70]] and ~~its octave complement,~~ [[~~140~~/99]]~~. It is also the difference between~~ [[~~245~~/~~243~~]] and [[~~121~~/~~120~~]]~~, and a stack of two 11/7's and 81/80 against 5/2.~~

It is also the difference between the following notable non-superparticular commas:

* [[245/243]] and [[121/120]]

* [[245/242]] and [[81/80]]

* [[176/175]] and [[896/891]]

It factors into the two smallest 13-limit superparticular commas: 9801/9800 = ([[10648/10647]])([[123201/123200]]).

It also factors into the two smallest 13-limit superparticular commas: 9801/9800 = ([[10648/10647]])⋅([[123201/123200]]).

== Temperaments ==

[[Tempering out]] this comma leads to the '''kalismic temperament''', which splits the [[octave]] into two equal parts, each representing 99/70~140/99. Tempering it out also means that [[10/9]] and [[11/7]] are ~~1/2-octave~~ apart, as well as are [[11/10]] and [[14/9]]. Odd-numbered edos cannot temper it out. See [[Rank-4 temperament #Kalismic (9801/9800)]] for some technical details. See [[Kalismic temperaments]] for a collection of rank-3 temperaments where it is tempered out.

[[Tempering out]] this comma leads to the '''kalismic temperament''', which splits the [[octave]] into two equal parts, each representing 99/70~140/99. Tempering it out also means that the [[pythagorean comma]] is split into two [[2835/2816]] halves, [[10/9]] and [[11/7]] are a semioctave apart, as well as are [[11/10]] and [[14/9]]. Odd-numbered edos cannot temper it out, as they do not have a semioctave.

See [[Rank-4 temperament #Kalismic (9801/9800)]] for some technical details. See [[Kalismic temperaments]] for a collection of rank-3 temperaments where it is tempered out.

== Etymology ==

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* [[List of superparticular intervals]]

== ~~Notes~~ ==

== References ==

[[Category:Kalismic]]

[[Category:Commas named by translating something into another language]]

[[Category:Commas named after mathematicians]]

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 | Comma = yes
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+'''9801/9800''', the '''kalisma''', sometimes described as ''Gauss' comma'', is an [[unnoticeable comma|unnoticeable]] [[11-limit]] [[comma]] measuring about 0.18{{cent}}. It is the smallest 11-limit [[superparticular]] interval. It can be described as the difference between [[99/70]] and its [[octave complement]] [[140/99]], between ([[35/33]])<sup>2</sup> and [[9/8]], or between ([[176/175]])<sup>2</sup> and [[2048/2025]].
-'''9801/9800''', the '''kalisma''', sometimes described as ''Gauss' comma'', is an [[unnoticeable comma|unnoticeable]] [[11-limit]] [[comma]] measuring about 0.18{{cent}}. It is the smallest 11-limit [[superparticular]] interval.
+In terms of superparticular commas, it is the difference between:
+* [[99/98]] and [[100/99]]
+* [[351/350]] and [[364/363]]
+* [[1716/1715]] and [[2080/2079]]
+* [[3025/3024]] and [[4375/4374]]
-It can be described as the difference between [[99/98]] and [[100/99]], and between [[99/70]] and its octave complement, [[140/99]]. It is also the difference between [[245/243]] and [[121/120]], and a stack of two 11/7's and 81/80 against 5/2.
+It is also the difference between the following notable non-superparticular commas:
+* [[245/243]] and [[121/120]]
+* [[245/242]] and [[81/80]]
+* [[176/175]] and [[896/891]]
-It factors into the two smallest 13-limit superparticular commas: 9801/9800 = ([[10648/10647]])([[123201/123200]]).
+It also factors into the two smallest 13-limit superparticular commas: 9801/9800 = ([[10648/10647]])⋅([[123201/123200]]).
 == Temperaments ==
-[[Tempering out]] this comma leads to the '''kalismic temperament''', which splits the [[octave]] into two equal parts, each representing 99/70~140/99. Tempering it out also means that [[10/9]] and [[11/7]] are 1/2-octave apart, as well as are [[11/10]] and [[14/9]]. Odd-numbered edos cannot temper it out. See [[Rank-4 temperament #Kalismic (9801/9800)]] for some technical details. See [[Kalismic temperaments]] for a collection of rank-3 temperaments where it is tempered out.
+[[Tempering out]] this comma leads to the '''kalismic temperament''', which splits the [[octave]] into two equal parts, each representing 99/70~140/99. Tempering it out also means that the [[pythagorean comma]] is split into two [[2835/2816]] halves, [[10/9]] and [[11/7]] are a semioctave apart, as well as are [[11/10]] and [[14/9]]. Odd-numbered edos cannot temper it out, as they do not have a semioctave.
+See [[Rank-4 temperament #Kalismic (9801/9800)]] for some technical details. See [[Kalismic temperaments]] for a collection of rank-3 temperaments where it is tempered out.
 == Etymology ==
@@ Line 20: / Line 30: @@
 * [[List of superparticular intervals]]
-== Notes ==
+== References ==
 [[Category:Kalismic]]
 [[Category:Commas named by translating something into another language]]
 [[Category:Commas named after mathematicians]]