289/288: Difference between revisions
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'''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]]. | '''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]]. | ||
== Commatic relations == | == Commatic relations == | ||
| Line 58: | Line 58: | ||
== See also == | == See also == | ||
* [[List of superparticular intervals]] | * [[List of superparticular intervals]] | ||
[[Category:Semitonismic]] | [[Category:Semitonismic]] | ||
[[Category:Commas named for the intervals they stack]] | [[Category:Commas named for the intervals they stack]] | ||
Latest revision as of 14:41, 21 March 2026
| 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 small 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
- WE: ~17/12 = 600.1471 ¢, ~3/2 = 701.9563 ¢, ~5/4 = 385.7246 ¢, ~7/4 = 968.2366 ¢, ~11/8 = 550.4343 ¢, ~13/8 = 839.6439 ¢
- CWE: ~17/12 = 600.0000 ¢, ~3/2 = 702.0260 ¢, ~5/4 = 385.9237 ¢, ~7/4 = 968.3544 ¢, ~11/8 = 550.7369 ¢, ~13/8 = 839.9061 ¢
Optimal ET sequence: 34d, 38df, 46, 58, 72, 130, 140, 190g, 212g, 270g, 342fg, 402defgg, 460gg, 532dfggg, 672defggg *
Badness (Sintel): 3.70
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.