118edo: Difference between revisions
prime intervals via template, monzo formatting |
+table of rank-2 temperaments |
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'''118edo''' is the [[equal division of the octave]] into 118 parts of 10.1695 cents each. | '''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]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. | 118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. | ||
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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. | 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 [[The Riemann Zeta Function and Tuning|zeta peak edo]]. | 118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]].{{primes in edo|118|prec=2}} | ||
== | == Rank-2 temperaments == | ||
{ | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |||
! Periods<br>per octave | |||
! Generator<br>(reduced) | |||
! Cents<br>(reduced) | |||
! Associated<br>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]]<br>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<br>(28\118) | |||
| 315.25<br>(284.75) | |||
| 6/5<br>(33/28) | |||
| [[Semiparakleismic]] | |||
|} | |||
[[Category:Equal divisions of the octave]] | [[Category:Equal divisions of the octave]] | ||