289/288: Difference between revisions
I noticed that this was a super common comma in 12ed2 when you consider the strength of its 17 limit properties. |
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== Temperaments == | == Temperaments == | ||
Tempering | Tempering out this comma 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 so is the whole tone, each representing 17/16~18/17. Odd edos cannot temper it out. | ||
=== Semitonismic === | === Semitonismic === | ||
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{{val| 0 0 0 0 0 1 0 }}] | {{val| 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 | ||
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[[Subgroup]]: 2.3.17 | [[Subgroup]]: 2.3.17 | ||
{{Mapping|legend=2| 2 0 5 | 0 1 1 }} | |||
: sval mapping generators: ~17/12, ~3 | |||
[[Optimal tuning]] ([[CTE]]): ~17/12 = 1\2, ~3/2 = 702.3472 (~17/16 = 102.3472) | [[Optimal tuning]] ([[CTE]]): ~17/12 = 1\2, ~3/2 = 702.3472 (~17/16 = 102.3472) | ||
Revision as of 12:05, 19 July 2023
| Interval information |
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 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 so is the whole tone, each representing 17/16~18/17. Odd edos cannot temper it out.
Semitonismic
Subgroup: 2.3.5.7.11.13.17
Mapping:
[⟨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
Semitonic
Subgroup: 2.3.17
Sval mapping: [⟨2 0 5], ⟨0 1 1]]
- sval mapping generators: ~17/12, ~3
Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 702.3472 (~17/16 = 102.3472)
Optimal ET sequence: 12, 58, 70, 82, 94, 106, 118, 224g
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.