289/288: Difference between revisions
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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 [[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. | '''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 == | ||
| Line 20: | Line 20: | ||
== Temperaments == | == Temperaments == | ||
Tempering | 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 === | |||
See [[No-fives subgroup temperaments #Semitonic]]. | |||
=== Semitonismic === | === Semitonismic === | ||
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[[Mapping]]: <br> | [[Mapping]]: <br> | ||
[ | {| class="right-all" | ||
|- | |||
| [⟨ || 2 || 0 || 0 || 0 || 0 || 0 || 5 || ], | |||
|- | |||
| ⟨ || 0 || 1 || 0 || 0 || 0 || 0 || 1 || ], | |||
|- | |||
| ⟨ || 0 || 0 || 1 || 0 || 0 || 0 || 0 || ], | |||
|- | |||
| ⟨ || 0 || 0 || 0 || 1 || 0 || 0 || 0 || ], | |||
|- | |||
| ⟨ || 0 || 0 || 0 || 0 || 1 || 0 || 0 || ], | |||
|- | |||
| ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]] | |||
|} | |||
: mapping generators: ~17/12, ~3, ~5, ~7, ~11, ~13 | |||
[[Optimal tuning]] ([[CTE]]): ~17/12 = 1\2, ~3/2 = 702.3472, 5/4, 7/4, 11/8, 13/8 | [[Optimal tuning]] ([[CTE]]): ~17/12 = 1\2, ~3/2 = 702.3472, 5/4, 7/4, 11/8, 13/8 | ||
{{ | {{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]] | |||
== Etymology == | == Etymology == | ||
| Line 58: | Line 60: | ||
[[Category:Semitonismic]] | [[Category:Semitonismic]] | ||
[[Category:Commas named for the intervals they stack]] | |||
Latest revision as of 00:39, 16 November 2024
| Interval information |
septendecimal semitones comma,
septendecimal 6-cent comma
Soso comma
reduced
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 cents. 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
This comma identifies itself as the difference between the following superparticular pairs:
It factors into the following superparticular pairs:
Temperaments
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
See No-fives subgroup temperaments #Semitonic.
Semitonismic
Subgroup: 2.3.5.7.11.13.17
| [⟨ | 2 | 0 | 0 | 0 | 0 | 0 | 5 | ], |
| ⟨ | 0 | 1 | 0 | 0 | 0 | 0 | 1 | ], |
| ⟨ | 0 | 0 | 1 | 0 | 0 | 0 | 0 | ], |
| ⟨ | 0 | 0 | 0 | 1 | 0 | 0 | 0 | ], |
| ⟨ | 0 | 0 | 0 | 0 | 1 | 0 | 0 | ], |
| ⟨ | 0 | 0 | 0 | 0 | 0 | 1 | 0 | ]] |
- mapping generators: ~17/12, ~3, ~5, ~7, ~11, ~13
Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 702.3472, 5/4, 7/4, 11/8, 13/8
Optimal ET sequence: 34d, 38df, 46, 58, 72, 130, 140, 190g, 212g, 270g, 342fg, 402defgg, 460gg, 532dfggg, 672defggg*
Etymology
The semitonisma was named by Flora Canou in 2023. It is a contraction of septendecimal semitones comma into a single word consisting of Latin semi ("half") and tonus ("tone"). However, septendecimal semitones comma and septendecimal 6-cent comma were attested much earlier.