289/288: Difference between revisions
m →Temperaments: move semitonic (owing to its simplicity as a subgroup temperament) |
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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[[Subgroup]]: 2.3.17 | [[Subgroup]]: 2.3.17 | ||
{{ | {{Optimal ET sequence|legend=1| 12, 58, 70, 82, 94, 106, 118, 224g }} | ||
{{Mapping|legend=2| 2 2 7 | 0 1 1 }} | |||
[CTE]] generators: ~17/12 = 1\2, ~3/2 = 702.3472 (~17/16 = 102.3472) | |||
=== Semitonismic === | === Semitonismic === | ||
Revision as of 16:56, 16 January 2024
| 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-2 semitonic temperament in the 2.3.17 subgroup, or the rank-6 semitonismic temperament in the full 17-limit. 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
Subgroup: 2.3.17
Optimal ET sequence: 12, 58, 70, 82, 94, 106, 118, 224g
Subgroup-val mapping: [⟨2 2 7], ⟨0 1 1]]
[CTE]] generators: ~17/12 = 1\2, ~3/2 = 702.3472 (~17/16 = 102.3472)
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.