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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. It can be described as the difference between [[99/70]] and its [[octave complement]] [[140/99]]. | |||
In terms of superparticular commas, it is the difference between: | In terms of superparticular commas, it is the difference between: | ||
* [[99/98]] and [[100/99]] | * [[99/98]] and [[100/99]] | ||
* [[351/350]] and [[364/363]] | |||
* [[1716/1715]] and [[2080/2079]] | |||
* [[3025/3024]] and [[4375/4374]] | * [[3025/3024]] and [[4375/4374]] | ||
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* [[245/243]] and [[121/120]] | * [[245/243]] and [[121/120]] | ||
* [[245/242]] and [[81/80]] | * [[245/242]] and [[81/80]] | ||
* [[176/175]] and [[896/891]] | |||
It also 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 == | == 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 | [[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 == | == Etymology == | ||
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* [[List of superparticular intervals]] | * [[List of superparticular intervals]] | ||
== | == References == | ||
[[Category:Kalismic]] | [[Category:Kalismic]] | ||
[[Category:Commas named by translating something into another language]] | [[Category:Commas named by translating something into another language]] | ||
[[Category:Commas named after mathematicians]] | [[Category:Commas named after mathematicians]] | ||
Latest revision as of 01:55, 12 April 2026
| Interval information |
reduced
S33/S35
9801/9800, the kalisma, sometimes described as Gauss' comma, is an unnoticeable 11-limit comma measuring about 0.18 ¢. 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.
In terms of superparticular commas, it is the difference between:
It is also the difference between the following notable non-superparticular commas:
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 kal- ("beautiful")[1]. Gene Ward Smith, not aware of Margo's work, proposed gaussisma in 2004, reasoning that D. H. Lehmer claimed Carl Friedrich Gauss had mentioned the ratio[2].