118edo
118edo is the equal division of the octave into 118 parts of 10.1695 cents each.
Theory
118edo represents the intersection of the 5-limit schismatic and parakleismic temperaments, tempering out both the schisma, [-15 8 1⟩ and the parakleisma, [8 14 -13⟩, as well as the vishnuzma, [23 6 -14⟩, the hemithirds comma, [38 -2 -15⟩, and the kwazy, [-53 10 16⟩. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.
In the 7-limit, it is particularly notable for tempering out the gamelisma, 1029/1024, and is an excellent tuning for the rank three gamelan temperament, and for guiron, the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but 99edo does better with that.
In the 11-limit, it tempers out 385/384 and 441/440, and is an excellent tuning for portent, the temperament tempering out both, and for the 11-limit version of guiron, which does also.
118edo is the 17th zeta peak edo.
Prime harmonics
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Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-187 118⟩ | [⟨118 187]] | -0.119 | 0.082 | 0.81 |
| 2.3.5 | 32805/32768, 1224440064/1220703125 | [⟨118 187 274]] | +0.036 | 0.093 | 0.91 |
| 2.3.5.7 | 1029/1024, 3136/3125, 4375/4374 | [⟨118 187 274 331]] | +0.270 | 0.412 | 4.05 |
| 2.3.5.7.11 | 385/384, 441/440, 3136/3125, 4375/4374 | [⟨118 187 274 331 408]] | +0.341 | 0.370 | 3.89 |
Rank-2 temperaments
| Periods per octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 11\118 | 111.86 | 16/15 | Vavoom |
| 1 | 19\118 | 193.22 | 28/25 | Luna / hemithirds / lunatic |
| 1 | 23\118 | 233.90 | 8/7 | Slendric / guiron |
| 1 | 31\118 | 315.25 | 6/5 | Parakleismic / paralytic |
| 1 | 49\118 | 498.31 | 4/3 | Helmholtz / pontiac / helenoid / pontic |
| 1 | 55\118 | 559.32 | 242/175 | Tritriple |
| 2 | 2\118 | 20.34 | 81/80 | Commatic |
| 2 | 5\118 | 50.85 | 33/32~36/35 | Kleischismic |
| 2 | 7\118 | 71.19 | 25/24 | Vishnuzmic / vishnu / ananta (118) / acyuta (118f) |
| 2 | 10\118 | 101.69 | 35/33 | Bischismic / Bipont (118) / counterbipont (118f) |
| 2 | 16\118 | 162.71 | 11/10 | Kwazy / bisupermajor |
| 2 | 18\118 | 183.05 | 10/9 | Unidec / ekadash (118) / hendec (118f) |
| 2 | 19\118 | 193.22 | 121/108 | Semiluna |
| 2 | 31\118 (28\118) |
315.25 (284.75) |
6/5 (33/28) |
Semiparakleismic |