118edo: Difference between revisions
→Theory: rework on the opening paragraph. -shruti (5-limit accuracy isn't a focus of it) |
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" |
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== Theory == | == Theory == | ||
118edo is the first [[5-limit]] equal division which clearly gives [[microtemperament|microtempering]], with [[error]]s well under half a cent. It 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 comma]], {{monzo| -53 10 16 }}. | 118edo is the first [[5-limit]] equal division which clearly gives [[microtemperament|microtempering]], with [[error]]s well under half a cent. It 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 comma]], {{monzo| -53 10 16 }}. | ||
118edo is the 17th [[The Riemann zeta function and tuning|zeta peak edo]], and it has decent approximations to harmonics [[7/1|7]], [[11/1|11]], [[17/1|17]], and [[19/1|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that. | 118edo is the 17th [[The Riemann zeta function and tuning|zeta peak edo]], and it has decent approximations to harmonics [[7/1|7]], [[11/1|11]], [[17/1|17]], and [[19/1|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that. | ||
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| 355.93 | | 355.93 | ||
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| [[27/22]], 16/13 I** | | [[27/22]], [[16/13]] I** | ||
| Minor tridecimal neurtral third, "major-neutral" third | | Minor tridecimal neurtral third, "major-neutral" third | ||
| bromine | | bromine | ||
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| [[15/11]] | | [[15/11]] | ||
| | |||
| iodine | |||
| {{UDnote|step=53}} | | {{UDnote|step=53}} | ||
| OG, uGb | | OG, uGb | ||
|- | |- | ||
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| | | | ||
| [[48/35]], [[11/8]] | | [[48/35]], [[11/8]] | ||
| | |||
| xenon | |||
| {{UDnote|step=54}} | | {{UDnote|step=54}} | ||
| UG, oAb | | UG, oAb | ||
|- | |- | ||
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| 498.31 | | 498.31 | ||
| 4/3 | | 4/3 | ||
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic | | [[Helmholtz (temperament)|Helmholtz]] / [[pontiac]] / helenoid / pontic | ||
|- | |- | ||
| 1 | | 1 | ||
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| [[Semiparakleismic]] | | [[Semiparakleismic]] | ||
|} | |} | ||
<nowiki/>* [[Normal | <nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | ||
== Instruments == | == Instruments == | ||