289/288
Ratio | 289/288 |
Subgroup monzo | 2.3.17 [-5 -2 2⟩ |
Size in cents | 6.0008173¢ |
Name | semitonisma |
Color name | 17oo2, soso 2nd, Soso comma |
FJS name | [math]\text{d2}^{17,17}[/math] |
Special properties | square superparticular, reduced |
Tenney height (log_{2} nd) | 16.3449 |
Weil height (log_{2} max(n, d)) | 16.3499 |
Wilson height (sopfr (nd)) | 50 |
Harmonic entropy (Shannon, [math]\sqrt{n\cdot d}[/math]) |
~2.59027 bits |
Comma size | small |
S-expression | S17 |
open this interval in xen-calc |
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
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: 10, 12, 58, 70, 82, 94, 106, 118, 224g
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.