9801/9800: Difference between revisions
Expansion |
m Improve linking |
||
| (15 intermediate revisions by 9 users not shown) | |||
| Line 1: | Line 1: | ||
{{Infobox Interval | {{Infobox Interval | ||
| | | Name = kalisma | ||
| | | Color name = 1oorrgg-2, Bilorugu comma | ||
| Comma = yes | |||
| | |||
}} | }} | ||
'''9801/9800''', the '''kalisma''' | '''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 factors into two 13-limit superparticular commas | 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 [[wiktionary:%CE%BA%CE%B1%CE%BB%CF%8C%CF%82 #Ancient Greek|''kal-'' ("beautiful")]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_12989.html Yahoo! Tuning Group | ''Re: Kalisma/xenisma (new names?) -- JI tuning and Scala file'']</ref>. [[Gene Ward Smith]], not aware of Margo's work, proposed ''gaussisma'' in 2004, reasoning that {{w|D. H. Lehmer}} claimed {{w|Carl Friedrich Gauss}} had mentioned the ratio<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10130.html#10135 Yahoo! Tuning Group | ''Re: {{lbrack}}tuning-math{{rbrack}} Digest Number 1011'']</ref>. | |||
== See also == | == See also == | ||
* [[List of superparticular intervals]] | * [[List of superparticular intervals]] | ||
== Notes == | |||
[[Category:Kalismic]] | [[Category:Kalismic]] | ||
[[Category:Commas named by translating something into another language]] | |||
[[Category:Commas named after mathematicians]] | |||
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].