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. | '''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: | |||
* [[99/98]] and [[100/99]] | |||
* [[3025/3024]] and [[4375/4374]] | |||
It is also the difference between the following notable non-superparticular commas: | |||
* [[245/243]] and [[121/120]] | * [[245/243]] and [[121/120]] | ||
* [[245/242]] and [[81/80]] | * [[245/242]] and [[81/80]] | ||
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 == | ||
Revision as of 14:57, 22 February 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.
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 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.
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].