118edo: Difference between revisions

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== 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 comma]], {{monzo| -53 10 16 }}~~. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. As a result, 118edo also excellently approximates the 22 Shruti scale~~.

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.

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=== Prime harmonics ===

=== Octave stretch ===

118edo's approximated harmonics 7, 11, 17 and 19 can be improved by employing a moderate [[stretched and compressed tuning|octave stretch]], using tunings such as [[69edf]] or [[187edt]], only at the cost of a little less accurate 5-limit part.

=== Subsets and supersets ===

118edo contains [[2edo]] and [[59edo]] as ~~subsets~~. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].

Since 118 factors into primes as {{nowrap| 2 × 59 }}, 118edo contains [[2edo]] and [[59edo]] as subset edos. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].

== Intervals ==

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| 355.93

|

| [[27/22]], 16/13 I**

| [[27/22]], [[16/13]] I**

| Minor tridecimal neurtral third, "major-neutral" third

| bromine

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| 498.31

| 4/3

| [[Helmholtz]] / [[pontiac]] / helenoid / pontic

| [[Helmholtz (temperament)|Helmholtz]] / [[pontiac]] / helenoid / pontic

|-

| 1

@@ Line 3: / Line 3: @@
 == Theory ==
-edo 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 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. As a result, 118edo also excellently approximates the 22 Shruti scale.
+edo 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 }}.
 edo 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.
@@ Line 15: / Line 15: @@
 === Prime harmonics ===
 {{Harmonics in equal|118}}
+=== Octave stretch ===
+edo's approximated harmonics 7, 11, 17 and 19 can be improved by employing a moderate [[stretched and compressed tuning|octave stretch]], using tunings such as [[69edf]] or [[187edt]], only at the cost of a little less accurate 5-limit part.
 === Subsets and supersets ===
-edo contains [[2edo]] and [[59edo]] as subsets. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].
+Since 118 factors into primes as {{nowrap| 2 × 59 }}, 118edo contains [[2edo]] and [[59edo]] as subset edos. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].
 == Intervals ==
@@ Line 350: / Line 353: @@
 | 355.93
 |
-| [[27/22]], 16/13 I**
+| [[27/22]], [[16/13]] I**
 | Minor tridecimal neurtral third, "major-neutral" third
 | bromine
@@ Line 1,250: / Line 1,253: @@
 | 498.31
 | 4/3
-| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
+| [[Helmholtz (temperament)|Helmholtz]] / [[pontiac]] / helenoid / pontic
 |-
 | 1