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. | '''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. | ||
== Commatic relations == | |||
This comma identifies itself as the difference between the following superparticular pairs: | |||
* [[85/84]] and [[120/119]] | |||
* [[136/135]] and [[256/255]] | |||
* [[273/272]] and [[4914/4913]] | |||
It factors into the following superparticular pairs: | |||
* [[561/560]] and [[595/594]] | |||
* [[442/441]] and [[833/832]] | |||
* [[385/384]] and [[1156/1155]] | |||
* [[325/324]] and [[2601/2600]] | |||
== Temperaments == | == Temperaments == | ||
Tempering it out 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. | Tempering it out 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]]: <br> | |||
[{{val| 2 0 0 0 0 0 5 }}<br> | |||
{{val| 0 1 0 0 0 0 1 }}<br> | |||
{{val| 0 0 1 0 0 0 0 }}<br> | |||
{{val| 0 0 0 1 0 0 0 }}<br> | |||
{{val| 0 0 0 0 1 0 0 }}<br> | |||
{{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 | |||
{{Val list|legend=1| 34d, 38df, 46, 58, 72, 130, 140, 190g, 212g, 270g, 342fg, 402defgg, 460gg, 532dfggg, 672defggg }} | |||
=== Semitonic === | |||
[[Subgroup]]: 2.3.17 | |||
[[Sval]] [[mapping]]: [{{val| 2 0 5 }} {{val| 0 1 1 }}] | |||
Mapping generators: ~17/12, ~3 | |||
[[Optimal tuning]] ([[CTE]]): ~17/12 = 1\2, ~3/2 = 702.3472 (~17/16 = 102.3472) | |||
{{Val list|legend=1| 12, 58, 70, 82, 94, 106, 118, 224g }} | |||
== Etymology == | == Etymology == | ||
Revision as of 11:11, 23 February 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.
Commatic relations
This comma identifies itself as the difference between the following superparticular pairs:
It factors into the following superparticular pairs:
Temperaments
Tempering it out 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
Semitonic
Subgroup: 2.3.17
Sval mapping: [⟨2 0 5] ⟨0 1 1]]
Mapping generators: ~17/12, ~3
Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 702.3472 (~17/16 = 102.3472)
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.