Xenharmonic Wiki

118edo: Difference between revisions

← Older edit
VisualWikitext
ArrowHead294 (talk | contribs)
15,231 edits
mNo edit summary
 
(22 intermediate revisions by 8 users not shown)
Line 1: Line 1:
{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|118}}
{{ED intro}}


== Theory ==
== 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. 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.
118edo is the 17th [[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.


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.
Line 15: Line 15:
=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|118}}
{{Harmonics in equal|118}}
=== 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 ===
=== 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 ==
== Intervals ==
Line 350: Line 353:
| 355.93
| 355.93
|  
|  
| [[27/22]], 16/13 I**
| [[27/22]], [[16/13]] I**
| Minor tridecimal neurtral third, "major-neutral" third
| Minor tridecimal neutral third, "major-neutral" third
| bromine
| bromine
| {{UDnote|step=35}}
| {{UDnote|step=35}}
Line 513: Line 516:
|  
|  
| [[15/11]]
| [[15/11]]
|
| iodine
| {{UDnote|step=53}}
| {{UDnote|step=53}}
|
| iodine
| OG, uGb
| OG, uGb
|-
|-
Line 522: Line 525:
|  
|  
| [[48/35]], [[11/8]]
| [[48/35]], [[11/8]]
|
| xenon
| {{UDnote|step=54}}
| {{UDnote|step=54}}
|
| xenon
| UG, oAb
| UG, oAb
|-
|-
Line 1,120: Line 1,123:
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.   
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.
== Approximation to JI ==
=== Zeta peak index ===
{{ZPI
| zpi = 706
| steps = 117.969513574257
| step size = 10.1721195895637
| tempered height = 9.850823
| pure height = 8.968412
| integral = 1.544280
| gap = 18.861062
| octave = 1200.31011156852
| consistent = 12
| distinct = 12
}}


== Regular temperament properties ==
== Regular temperament properties ==
{{comma basis begin}}
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3
| 2.3
| {{monzo| -187 118 }}
| {{Monzo| -187 118 }}
| {{mapping| 118 187 }}
| {{Mapping| 118 187 }}
| &minus;0.119
| −0.119
| 0.082
| 0.082
| 0.81
| 0.81
Line 1,133: Line 1,160:
| 2.3.5
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| 32805/32768, {{monzo| 8 14 -13 }}
| {{mapping| 118 187 274 }}
| {{Mapping| 118 187 274 }}
| +0.036
| +0.036
| 0.093
| 0.093
Line 1,140: Line 1,167:
| 2.3.5.7
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| 1029/1024, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 }}
| {{Mapping| 118 187 274 331 }}
| +0.270
| +0.270
| 0.412
| 0.412
Line 1,147: Line 1,174:
| 2.3.5.7.11
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| 385/384, 441/440, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 408 }}
| {{Mapping| 118 187 274 331 408 }}
| +0.341
| +0.341
| 0.370
| 0.370
Line 1,154: Line 1,181:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| 196/195, 352/351, 384/384, 625/624, 729/728
| {{mapping| 118 187 274 331 408 437 }} (118)
| {{Mapping| 118 187 274 331 408 437 }} (118)
| +0.125
| +0.125
| 0.604
| 0.604
Line 1,161: Line 1,188:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| {{mapping| 118 187 274 331 408 436 }} (118f)
| {{Mapping| 118 187 274 331 408 436 }} (118f)
| +0.583
| +0.583
| 0.650
| 0.650
Line 1,168: Line 1,195:
| 2.3.5.7.11.17
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| {{mapping| 118 187 274 331 408 482 }}
| {{Mapping| 118 187 274 331 408 482 }}
| +0.417
| +0.417
| 0.399
| 0.399
Line 1,175: Line 1,202:
| 2.3.5.7.11.17.19
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| {{mapping| 118 187 274 331 408 482 501 }}
| {{Mapping| 118 187 274 331 408 482 501 }}
| +0.445
| +0.445
| 0.376
| 0.376
| 3.69
| 3.69
{{comma basis end}}
|}
* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.
* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{{rank-2 begin}}
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>ratio*
! Temperament
|-
|-
| 1
| 1
Line 1,195: Line 1,229:
| 193.22
| 193.22
| 28/25
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
| [[Hemithirds]] / [[lunatic]]
|-
|-
| 1
| 1
Line 1,201: Line 1,235:
| 233.90
| 233.90
| 8/7
| 8/7
| [[Slendric]] / [[guiron]]
| [[Guiron]]
|-
|-
| 1
| 1
Line 1,207: Line 1,241:
| 315.25
| 315.25
| 6/5
| 6/5
| [[Parakleismic]] / paralytic
| [[Parakleismic]] / [[counterpara]]
|-
|-
| 1
| 1
Line 1,219: Line 1,253:
| 498.31
| 498.31
| 4/3
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
| [[Pontiac]] / helenoid / pontic
|-
|-
| 1
| 1
Line 1,231: Line 1,265:
| 20.34
| 20.34
| 81/80
| 81/80
| [[Commatic]]
| [[Bicommatic]]
|-
|-
| 2
| 2
Line 1,243: Line 1,277:
| 71.19
| 71.19
| 25/24
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
| [[Vishnu]] / [[narayana]]
|-
|-
| 2
| 2
Line 1,255: Line 1,289:
| 162.71
| 162.71
| 11/10
| 11/10
| [[Kwazy]] / [[bisupermajor]]
| [[Bisupermajor]]
|-
|-
| 2
| 2
Line 1,270: Line 1,304:
|-
|-
| 2
| 2
| 31\118<br />(28\118)
| 28\118
| 315.25<br />(284.75)
| 284.75
| 6/5<br />(33/28)
| 33/28
| [[Semiparakleismic]]
| [[Semiparakleismic]]
{{rank-2 end}}
|}
{{orf}}
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]


== Instruments ==
== Instruments ==
Retrieved from "https://en.xen.wiki/w/118edo"