9801/9800: Difference between revisions

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{{Infobox Interval

| ~~Icon~~ =

| Name = kalisma

| ~~Ratio = 9801/9800~~

| Color name = 1oorrgg-2, Bilorugu comma

~~| Monzo~~ = ~~-3 4 -2~~ -2 2

| Comma = yes

~~| Cents = 0.17665~~

~~| Name = kalisma~~, ~~ Gauss'~~ comma

| ~~Color name =~~

~~| FJS name =~~

~~| Sound~~ =

}}

'''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]])2 and [[9/8]], or between ([[176/175]])2 and [[2048/2025]].

~~'''9801/9800'''~~, the ~~'''kalisma''' or '''Gauss' comma''', is an~~ [[~~11-limit~~]] [[~~unnoticeable comma~~]] ~~measuring about 0.18 cents. It is the smallest 11-limit~~ [[~~superparticular~~]] ~~interval. It can be described as the difference between~~ [[99/98]] and [[~~100~~/99]]~~, and between~~ [[99/70]] and ~~its octave complement,~~ [[~~140~~/99]].

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]]

~~Tempering it out leads to~~ [[~~kalismic temperaments~~]]~~, which splits the~~ [[~~octave~~]] ~~into two equal parts, each representing 99~~/~~70~140~~/~~99. Odd edos cannot temper it out.~~

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 two 13-limit superparticular commas ~~which are also very small~~: 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 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 ==

This comma was named ''kalisma'' by [[Margo Schulter]] in 2000 from the Greek root [[wiktionary:%CE%BA%CE%B1%CE%BB%CF%8C%CF%82 #Ancient Greek|''kal-'' ("beautiful")]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_12989.html Yahoo! Tuning Group | ''Re: Kalisma/xenisma (new names?) -- JI tuning and Scala file'']</ref>. [[Gene Ward Smith]], not aware of Margo's work, proposed ''gaussisma'' in 2004, reasoning that {{w|D. H. Lehmer}} claimed {{w|Carl Friedrich Gauss}} had mentioned the ratio<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10130.html#10135 Yahoo! Tuning Group | ''Re: {{lbrack}}tuning-math{{rbrack}} Digest Number 1011'']</ref>.

== See also ==

* [[Kalismic temperaments]]

* [[Unnoticeable comma]]

* [[List of superparticular intervals]]

~~[[Category:11-limit]]~~

== References ==

~~[[Category:Unnoticeable comma]]~~

~~[[Category:Ratio]]~~

~~[[Category:Superparticular]]~~

[[Category:Kalismic]]

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

[[Category:Commas named after mathematicians]]

@@ Line 1: / Line 1: @@
 {{Infobox Interval
-| Icon =
+| Name = kalisma
-| Ratio = 9801/9800
+| Color name = 1oorrgg-2, Bilorugu comma
-| Monzo = -3 4 -2 -2 2
+| Comma = yes
-| Cents = 0.17665
-| Name = kalisma, <br>Gauss' comma
-| Color name =
-| FJS name =
-| Sound =
 }}
+'''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''' or '''Gauss' comma''', is an [[11-limit]] [[unnoticeable comma]] measuring about 0.18 cents. It is the smallest 11-limit [[superparticular]] interval. It can be described as the difference between [[99/98]] and [[100/99]], and between [[99/70]] and its octave complement, [[140/99]].
+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]]
-Tempering it out leads to [[kalismic temperaments]], which splits the [[octave]] into two equal parts, each representing 99/70~140/99. Odd edos cannot temper it out.
+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 two 13-limit superparticular commas which are also very small: 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 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 ==
+This comma was named ''kalisma'' by [[Margo Schulter]] in 2000 from the Greek root [[wiktionary:%CE%BA%CE%B1%CE%BB%CF%8C%CF%82 #Ancient Greek|''kal-'' ("beautiful")]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_12989.html Yahoo! Tuning Group | ''Re: Kalisma/xenisma (new names?) -- JI tuning and Scala file'']</ref>. [[Gene Ward Smith]], not aware of Margo's work, proposed ''gaussisma'' in 2004, reasoning that {{w|D. H. Lehmer}} claimed {{w|Carl Friedrich Gauss}} had mentioned the ratio<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10130.html#10135 Yahoo! Tuning Group | ''Re: {{lbrack}}tuning-math{{rbrack}} Digest Number 1011'']</ref>.
 == See also ==
-* [[Kalismic temperaments]]
-* [[Unnoticeable comma]]
 * [[List of superparticular intervals]]
-[[Category:11-limit]]
+== References ==
-[[Category:Unnoticeable comma]]
-[[Category:Ratio]]
-[[Category:Superparticular]]
 [[Category:Kalismic]]
+[[Category:Commas named by translating something into another language]]
+[[Category:Commas named after mathematicians]]