118edo: Difference between revisions

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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}}


== Just approximation ==
== Rank-2 temperaments ==
{{primes in edo|118|prec=2}}
{| 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]]

Revision as of 09:39, 9 June 2021

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.Script error: No such module "primes_in_edo".

Rank-2 temperaments

Table of rank-2 temperaments by generator
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
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