9801/9800: Difference between revisions
+where "Gauss' comma" comes from |
Gauss' comma is more of a description than a name |
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{{Infobox Interval | {{Infobox Interval | ||
| Name = kalisma | | Name = kalisma | ||
| Color name = 1oorrgg-2, Bilorugu comma | | Color name = 1oorrgg-2, Bilorugu comma | ||
| Comma = yes | | Comma = yes | ||
}} | }} | ||
'''9801/9800''', the '''kalisma''' | '''9801/9800''', the '''kalisma''', sometimes described as ''Gauss' comma'', is an [[11-limit]] [[unnoticeable 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. Tempering it out also means that [[10/9]] and [[11/7]] are 600 cents apart, as well as are [[11/10]] and [[14/9]]. | 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. Tempering it out also means that [[10/9]] and [[11/7]] are 600 cents apart, as well as are [[11/10]] and [[14/9]]. | ||
Revision as of 10:43, 25 October 2024
| Interval information |
reduced
S33 / S35
9801/9800, the kalisma, sometimes described as Gauss' comma, is an 11-limit unnoticeable 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. Tempering it out also means that 10/9 and 11/7 are 600 cents apart, as well as are 11/10 and 14/9.
It factors into the two smallest 13-limit superparticular commas: 9801/9800 = 10648/10647 × 123201/123200.
Temperaments
Tempering it out leads to the kalismic temperament, which splits the octave into two equal parts, each representing 99/70~140/99. Odd-numbered edos cannot temper it out. See Rank-4 temperament #Kalismic (9801/9800) for some technical details.
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].
See also
- Rank-4 temperament #Kalismic (9801/9800)
- Kalismic temperaments, a collection of rank-3 temperaments where it is tempered out
- List of superparticular intervals