9801/9800: Difference between revisions
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'''9801/9800''', the '''kalisma''', sometimes described as ''Gauss' comma'', is an [[11-limit]] [[ | '''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 [[99/98]] 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 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 factors into the two smallest 13-limit superparticular commas: 9801/9800 = 10648/10647 | It factors into the two smallest 13-limit superparticular commas: 9801/9800 = ([[10648/10647]])([[123201/123200]]). | ||
== Temperaments == | == Temperaments == | ||
Latest revision as of 11:21, 5 November 2024
| 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 99/98 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 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].