9801/9800: Difference between revisions
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== 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. | 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. | ||
Revision as of 08:37, 9 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, or between (35/33)2 and 9/8.
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].