289/288: Difference between revisions
m →Semitonic: having two different sets of generators one of which used by the mapping seems unnecessarily confusing (and using reduced generators usually makes the numbers in the mapping simpler which is useful musically) |
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{{Infobox Interval | {{Infobox Interval | ||
| Name = semitonisma | | Name = semitonisma, septendecimal semitones comma, septendecimal 6-cent comma | ||
| Color name = 17oo2, soso 2nd,<br>Soso comma | | Color name = 17oo2, soso 2nd,<br>Soso comma | ||
| Comma = yes | | Comma = yes | ||
}} | }} | ||
'''289/288''', the '''semitonisma''', also described as the ''septendecimal semitones comma'' and ''septendecimal 6-cent comma'', is a [[small comma|small]] [[17-limit]] (also 2.3.17 [[subgroup]]) [[superparticular]] [[comma]] of about 6 [[cent]]s. It is the difference between [[17/16]] and [[18/17]], the two septendecimal semitones, as well as between [[17/12]] and [[24/17]], the two septendecimal tritones. It is also the difference between two stacked [[17/16]] and [[9/8]]. | |||
'''289/288''', the '''semitonisma''', also described as the ''septendecimal semitones comma'' and ''septendecimal 6-cent comma'', is a [[17-limit]] (also 2.3.17 [[subgroup]]) [[superparticular]] [[comma]] of about 6 [[cent]]s. It is the difference between [[17/16]] and [[18/17]], the two septendecimal semitones, as well as between [[17/12]] and [[24/17]], the two septendecimal tritones. It is also the difference between two stacked [[17/16]] and [[9/8]]. | |||
== Commatic relations == | == Commatic relations == | ||
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== Temperaments == | == Temperaments == | ||
Tempering out this comma results in the rank- | [[Tempering out]] this comma in the full 17-limit results in the rank-6 '''semitonismic''' temperament, or in the 2.3.17 subgroup, the rank-2 '''semitonic''' temperament. In either case the [[octave]] is split into two equal parts, each representing 17/12~24/17, and thus so is the whole tone, each representing 17/16~18/17. Odd edos cannot temper it out. | ||
=== Semitonic === | === Semitonic === | ||
[[ | See [[No-fives subgroup temperaments #Semitonic]]. | ||
=== Semitonismic === | === Semitonismic === | ||
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| ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]] | | ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]] | ||
|} | |} | ||
: mapping generators: ~17/12, ~3, ~5, ~7, ~11, ~13 | |||
: | [[Optimal tuning]]s: | ||
* [[WE]]: ~17/12 = 600.1471{{c}}, ~3/2 = 701.9563{{c}}, ~5/4 = 385.7246{{c}}, ~7/4 = 968.2366{{c}}, ~11/8 = 550.4343{{c}}, ~13/8 = 839.6439{{c}} | |||
* [[CWE]]: ~17/12 = 600.0000{{c}}, ~3/2 = 702.0260{{c}}, ~5/4 = 385.9237{{c}}, ~7/4 = 968.3544{{c}}, ~11/8 = 550.7369{{c}}, ~13/8 = 839.9061{{c}} | |||
{{Optimal ET sequence|legend=1| 34d, 38df, 46, 58, 72, 130, 140, 190g, 212g, 270g, 342fg, 402defgg, 460gg, 532dfggg, 672defggg }} * | |||
[[ | <nowiki>*</nowiki> [[optimal patent val]]: [[280edo|280]] | ||
[[Badness]] (Sintel): 3.70 | |||
== Etymology == | == Etymology == | ||
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== See also == | == See also == | ||
* [[List of superparticular intervals]] | * [[List of superparticular intervals]] | ||
[[Category:Semitonismic]] | [[Category:Semitonismic]] | ||
[[Category:Commas named for the intervals they stack]] | |||