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Fractional-octave temperaments: Difference between revisions

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All temperaments on this page have a fractional-octave period, such as 1\26, 1\31, or 1\41.
'''Fractional-octave temperaments''' are [[temperament]]s which have a [[period]] which corresponds to a [[just]] [[interval]] mapped to a fraction of the [[octave]], that is one step of an [[edo]].


Temperaments discussed elsewhere include:
== Theory ==
Fractional-octave temperaments are valuable with regards to [[Polysystemic|polysystemicism]] and polychromatics. They are acoustically significant with regards to containing modes of limited transposition, as well as their ability to expand on the harmony of the equal division they are a superset of. Such temperaments are also a way of introducing less common and harmonically less performing equal divisions into music that prefers consonance and is based on regular temperament theory.
 
=== Terminology ===
The terminology was developed by [[Eliora]]. The equal division containing the mos scale of such a temperament, starting from the tonic, is referred to as a ''wireframe'', and individual notes of that equal division are called ''hinges''. Thus in this context, the wireframe is the tuning consisting of only stacks of the period and no stacks of the generator. Temperament-agnostically, this can be used to refer to any structure embedded in an (x,y)-ET which repeats y times within that period, its "wireframe" is y-ET. If an equal division is a subset of a temperament, it is said to ''subtend'' the temperament, just how hinges on a ferris wheel subtend the structure to make it rotate and function.
 
The most common way to produce a fractional-octave temperament is through an excellent approximation of an interval relative to the size of the wireframe edo. For example, [[compton family]] tempers out the Pythagorean comma and maps 7 steps of 12edo to [[3/2]]. Likewise, a lot of 10th-octave temperaments have a [[13/8]] as 7\10, and 26th-octave temperaments often have a [[7/4]] for 21\26.
 
However, an equal division does not have to be harmonically decent to be a wireframe for a fractional-octave temperament. If an equal division has multiples which are high in consistency or are zeta equal divisions or otherwise harmonically strong, it can produce a lot of such temperaments—notable examples being [[20edo]] or [[32edo]]. Likewise, proximity of a step of equal division to a comma is often a source of these temperaments—for example [[56edo]]'s step being directly close to [[81/80]], and 44edo's step being extremely close to [[64/63]].
 
=== Disagreement between temperament catalog strategy and fractional-octave practice ===
Traditional regular temperament perspective on periods and generators has a shortcoming when it comes to handling fractional-octave temperaments, as it treats divisions of periods (for example, what [[hemiennealimmal]] is to [[ennealimmal]]) as [[extension]]s of a temperament with a subset period. However, fractional-octave temperaments and scales are sought for being able to treat an each equal division as an entity in its own right, so a composer might find hemiennealimmal to be a drastically different system to ennealimmal in line with [[18edo]] being very different from [[9edo]]. This facet is reflected by the distinction of strong and weak extensions.
 
A particularly strong offender of this is the [[landscape microtemperaments]] list, which features temperaments which are all supersets of 3edo, but from a composer's perspective it contains wildly different temperaments due to the fact that edo multiples of 3 themselves are different. For example, magnesium (12), and zinc (30), are both landscape systems due to being multiples of 3, but 30edo is drastically different from 12edo in terms of composition, and therefore such temperaments are not alike at all.
 
== Individual pages of temperaments by subtending equal division ==
 
=== 2 to 100 ===
Many pages are yet to be created.
{| class="wikitable"
|+
|
| [[2nd-octave temperaments|2]]
| [[3rd-octave temperaments|3]]
| [[4th-octave temperaments|4]]
| [[5th-octave temperaments|5]]
| [[6th-octave temperaments|6]]
| [[7th-octave temperaments|7]]
| [[8th-octave temperaments|8]]
| [[9th-octave temperaments|9]]
| [[10th-octave temperaments|10]]
|-
| [[11th-octave temperaments|11]]
| [[12th-octave temperaments|12]]
| [[13th-octave temperaments|13]]
| [[14th-octave temperaments|14]]
| [[15th-octave temperaments|15]]
| [[16th-octave temperaments|16]]
| [[17th-octave temperaments|17]]
| [[18th-octave temperaments|18]]
| [[19th-octave temperaments|19]]
| [[20th-octave temperaments|20]]
|-
| [[21st-octave temperaments|21]]
| [[22nd-octave temperaments|22]]
| [[23rd-octave temperaments|23]]
| [[24th-octave temperaments|24]]
| [[25th-octave temperaments|25]]
| [[26th-octave temperaments|26]]
| [[27th-octave temperaments|27]]
| [[28th-octave temperaments|28]]
| [[29th-octave temperaments|29]]
| [[30th-octave temperaments|30]]
|-
| [[31st-octave temperaments|31]]
| [[32nd-octave temperaments|32]]
| [[33rd-octave temperaments|33]]
| [[34th-octave temperaments|34]]
| [[35th-octave temperaments|35]]
| [[36th-octave temperaments|36]]
| [[37th-octave temperaments|37]]
| [[38th-octave temperaments|38]]
| [[39th-octave temperaments|39]]
| [[40th-octave temperaments|40]]
|-
| [[41st-octave temperaments|41]] / [[Countercomp family|C]]
| [[42nd-octave temperaments|42]]
| [[43rd-octave temperaments|43]]
| [[44th-octave temperaments|44]]
| [[45th-octave temperaments|45]]
| [[46th-octave temperaments|46]]
| [[47th-octave temperaments|47]]
| [[48th-octave temperaments|48]]
| [[49th-octave temperaments|49]]
| [[50th-octave temperaments|50]]
|-
| [[51st-octave temperaments|51]]
| [[52nd-octave temperaments|52]]
| [[53rd-octave temperaments|53]] / [[Mercator family|M]]
| [[54th-octave temperaments|54]]
| [[55th-octave temperaments|55]]
| [[56th-octave temperaments|56]]
| [[57th-octave temperaments|57]]
| [[58th-octave temperaments|58]]
| [[59th-octave temperaments|59]]
| [[60th-octave temperaments|60]]
|-
| [[61st-octave temperaments|61]]
| [[62nd-octave temperaments|62]]
| [[63rd-octave temperaments|63]]
| [[64th-octave temperaments|64]]
| [[65th-octave temperaments|65]]
| [[66th-octave temperaments|66]]
| [[67th-octave temperaments|67]]
| [[68th-octave temperaments|68]]
| [[69th-octave temperaments|69]]
| [[70th-octave temperaments|70]]
|-
| [[71st-octave temperaments|71]]
| [[72nd-octave temperaments|72]]
| [[73rd-octave temperaments|73]]
| [[74th-octave temperaments|74]]
| [[75th-octave temperaments|75]]
| [[76th-octave temperaments|76]]
| [[77th-octave temperaments|77]]
| [[78th-octave temperaments|78]]
| [[79th-octave temperaments|79]]
| [[80th-octave temperaments|80]]
|-
| [[81st-octave temperaments|81]]
| [[82nd-octave temperaments|82]]
| [[83rd-octave temperaments|83]]
| [[84th-octave temperaments|84]]
| [[85th-octave temperaments|85]]
| [[86th-octave temperaments|86]]
| [[87th-octave temperaments|87]]
| [[88th-octave temperaments|88]]
| [[89th-octave temperaments|89]]
| [[90th-octave temperaments|90]]
|-
| [[91st-octave temperaments|91]]
| [[92nd-octave temperaments|92]]
| [[93rd-octave temperaments|93]]
| [[94th-octave temperaments|94]]
| [[95th-octave temperaments|95]]
| [[96th-octave temperaments|96]]
| [[97th-octave temperaments|97]]
| [[98th-octave temperaments|98]]
| [[99th-octave temperaments|99]]
| [[100th-octave temperaments|100]]
|}
 
=== 101 and up ===
[[111th-octave temperaments|111]], [[118th-octave temperaments|118]], [[159th-octave temperaments|159]], [[400th-octave temperaments|400]], [[665th-octave temperaments|665]]
 
 
C = Countercomp family
 
M = Mercator family equated with 53rd-octave temperaments until otherwise discovered, also contains 106th-octave temperaments
 
== Temperaments discussed elsewhere ==
Temperaments discussed as a part of a commatic family, or otherwise in temperament lists unrelated to fractional-octave theory include:


* 1\2 period temperaments
* 1\2 period temperaments
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** [[Varunismic temperaments]]
** [[Varunismic temperaments]]
** [[Lokismic temperaments]]
** [[Lokismic temperaments]]
** [[Nimona|Nimona temperament]]
* 1\3 period temperaments
* 1\3 period temperaments
** [[Augmented family|Augmented temperaments]]
** [[Augmented family|Augmented temperaments]]
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** [[Landscape microtemperaments|Landscape temperaments]]
** [[Landscape microtemperaments|Landscape temperaments]]
* 1\4 period temperaments
* 1\4 period temperaments
** [[Dimipent family|Diminished temperaments]]
** [[Diminished family|Diminished temperaments]]
** [[Undim family|Undim temperaments]]
** [[Undim family|Undim temperaments]]
* 1\5 period temperaments
* 1\5 period temperaments
** [[Pental family|Pental temperaments]]
** [[Quintile family|Quintile temperaments]]
** [[Qintosec family|Qintosec temperaments]]
** [[Quintosec family|Quintosec temperaments]]
** [[Trisedodge family|Trisedodge temperaments]]
** [[Trisedodge family|Trisedodge temperaments]]
** [[Cloudy clan|Cloudy temperaments]]
** [[Cloudy clan|Cloudy temperaments]]
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** [[Ragismic microtemperaments #Brahmagupta|Brahmagupta]]
** [[Ragismic microtemperaments #Brahmagupta|Brahmagupta]]
** [[Schismatic family #Septant|Septant]]
** [[Schismatic family #Septant|Septant]]
** [[Apotome family #Whitewood|Whitewood]]
** [[Whitewood family #Whitewood|Whitewood temperaments]]
** [[Keemic temperaments #Sevond|Sevond]]
** [[Keemic temperaments #Sevond|Sevond]]
** [[Mistismic temperaments #Neutron|Neutron]]
** [[Mistismic temperaments #Neutron|Neutron]]
* [[Ragismic microtemperaments #Octoid|Octoid]], [[Schismatic family #Octant|octant]] (1\8 period)
* [[Ragismic microtemperaments #Octoid|Octoid]], [[Schismatic family #Octant|octant]] (1\8 period)
* [[Tritrizo clan|Tritrizo temperaments]] (1\9 period)
* [[Septiennealimmal clan|Septiennealimmal temperaments]] (1\9 period)
** [[Ragismic microtemperaments #Ennealimmal|Ennealimmal]]
** [[Ragismic microtemperaments #Ennealimmal|Ennealimmal]]
** [[Augmented family #Niner|Niner]]
** [[Augmented family #Niner|Niner]]
** [[Marvel temperaments #Enneaportent|Enneaportent]]
** [[Marvel temperaments #Enneaportent|Enneaportent]]
** [[Kleismic family #Novemkleismic|Novemkleismic]]
** [[Kleismic family #Novemkleismic|Novemkleismic]]
* [[15/14 equal-step tuning|Linus temperaments]] (1\10 period)
* [[Linus]] temperaments (1\10 period)
** [[Breedsmic temperaments #Decoid|Decoid]]
** [[Breedsmic temperaments #Decoid|Decoid]]
** [[Ragismic microtemperaments #Deca|Deca]]
** [[Ragismic microtemperaments #Deca|Deca]]
** [[Cloudy clan #Decic|Decic]]
** [[Cloudy clan #Decic|Decic]]
** [[Stearnsmic clan #Decistearn|Decistearn]]
** [[Stearnsmic clan #Decistearn|Decistearn]]
** [[Pental family #Decal|Decal]]
** [[Quintile family #Decile|Decile]]
** [[Vishnuzmic family #Decavish|Decavish]]
** [[Vishnuzmic family #Decavish|Decavish]]
** [[Metric microtemperaments #Decimetra|Decimetra]]
** [[Metric microtemperaments #Decimetra|Decimetra]]
* [[Porwell temperaments #Hendecatonic|Hendecatonic]], [[Keemic temperaments #Undeka|undeka]] (1\11 period)
* [[Porwell temperaments #Hendecatonic|Hendecatonic]] (1\11 period)
* [[Compton family|Compton]], [[Very high accuracy temperaments #Atomic|atomic]] (1\12 period)
* [[Compton family|Compton]], [[Very high accuracy temperaments #Atomic|atomic]] (1\12 period)
* [[Orwellismic temperaments #Triskaidekic|Triskaidekic]], [[Octagar temperaments #Tridecatonic|tridecatonic]], [[Ragismic microtemperaments #Trideci|trideci]] (1\13 period)
* [[Orwellismic temperaments #Triskaidekic|Triskaidekic]], [[Octagar temperaments #Tridecatonic|tridecatonic]], [[Ragismic microtemperaments #Trideci|trideci]], [[aluminium]] (1\13 period)
* [[Silicon]] (1\14 period)
* [[Cloudy clan #Pentadecal|Pentadecal]], [[Trienstonic clan #Quindecic|quindecic]] (1\15 period)
* [[Cloudy clan #Pentadecal|Pentadecal]], [[Trienstonic clan #Quindecic|quindecic]] (1\15 period)
* [[Ragismic microtemperaments #Octoid|Hexadecoid]], [[Jubilismic clan #Sedecic|sedecic]] (1\16 period)
* [[Ragismic microtemperaments #Octoid|Hexadecoid]], [[Jubilismic clan #Sedecic|sedecic]] (1\16 period)
* [[Ragismic microtemperaments #Chlorine|Chlorine]] (1\17 period)
* [[Ragismic microtemperaments #Chlorine|Chlorine]] (1\17 period)
* [[Ragismic microtemperaments #Ennealimmal|Hemiennealimmal]] (1\18 period)
* [[Septiennealimmal clan #Ennealimmal|Hemiennealimmal]] (1\18 period)
* [[Ragismic microtemperaments #Enneadecal|Enneadecal]], [[Meantone family #Meanmag|meanmag]] (1\19 period)
* [[Ragismic microtemperaments #Enneadecal|Enneadecal]], [[Meantone family #Meanmag|meanmag]] (1\19 period)
* [[Hemimage temperaments #Degrees|Degrees]] (1\20 period)
* [[Hemimage temperaments #Degrees|Degrees]] (1\20 period)
Line 58: Line 201:
* [[Porwell temperaments #Hendecatonic|Icosidillic]] (1\22 period)
* [[Porwell temperaments #Hendecatonic|Icosidillic]] (1\22 period)
* [[Porwell temperaments #Icositritonic|Icositritonic]] (1\23 period)
* [[Porwell temperaments #Icositritonic|Icositritonic]] (1\23 period)
* [[Compton family #Hours|Hours]] (1\24 period)
* [[Compton family #Hours|Hours]], [[chromium]] (1\24 period)
* [[26th-octave temperaments|Bosonic]] (1\26 period)
* [[Septiennealimmal clan #Ennealimmal|Trinealimmal]], [[Tritrizo clan #Cobalt|cobalt]] (1\27 period)
* [[Ragismic microtemperaments #Ennealimmal|Trinealimmal]], [[Tritrizo clan #Cobalt|cobalt]] (1\27 period)
* [[Horwell temperaments #Oquatonic|Oquatonic]] (1\28 period)
* [[Horwell temperaments #Oquatonic|Oquatonic]] (1\28 period)
* [[Hemifamity temperaments #Mystery|Mystery]] (1\29 period)
* [[Hemifamity temperaments #Mystery|Mystery]], [[Copper comma|copper]] (1\29 period)
* [[31st-octave temperaments|Birds]] (1\31 period)
* [[31st-octave temperaments|Birds]] (1\31 period)
* [[Compton family #Decades|Decades]] (1\36 period)
* [[Compton family #Gamelstearn|Gamelstearn]] (1\36 period)
* [[Ragismic microtemperaments #Enneadecal|Hemienneadecal]] (1\38 period)
* [[Ragismic microtemperaments #Enneadecal|Hemienneadecal]], [[semihemienneadecal]] (1\38 period)
* [[Counterpyth family|Counterpyth temperaments]] (1\41 period)
* [[Countercomp family|Countercomp temperaments]], [[niobium]] (1\41 period)
* [[Mitonismic temperaments #Meridic|Meridic]] (1\43 period)
* [[Mitonismic temperaments #Meridic|Meridic]] (1\43 period)
* [[Ragismic microtemperaments #Palladium|Palladium]] (1\46 period)
* [[Ragismic microtemperaments #Palladium|Palladium]] (1\46 period)
* [[Mercator family|Mercator temperaments]] (1\53 period)
* [[Compton family #Omicronbeta|Omicronbeta]] (1\72 period)
* [[Compton family #Omicronbeta|Omicronbeta]] (1\72 period)
* [[The Flashmob#Iridium|Iridium]] (1\77 period)
* [[Parkleiness temperaments #Octogintic|Octogintic]] (1\80 period)
* [[Parkleiness temperaments #Octogintic|Octogintic]] (1\80 period)
* [[Stearnsmic clan #Garistearn|Garistearn]] (1\94 period)
* [[Stearnsmic clan #Garistearn|Garistearn]] (1\94 period)
* [[Tritrizo clan #Undecentic|Undecentic]] (1\99 period)
* [[Septiennealimmal clan #Undecentic|Undecentic]] (1\99 period)
* [[Tritrizo clan #Schisennealimmal|Schisennealimmal]] (1\171 period)
* [[Septiennealimmal clan #Schisennealimmal|Schisennealimmal]] (1\171 period)
* [[Tritrizo clan #Lunennealimmal|Lunennealimmal]] (1\441 period)
* [[Septiennealimmal clan #Lunennealimmal|Lunennealimmal]] (1\441 period)
 
== 14th-octave temperaments ==
While 14edo is poor in LCJI harmonics, some of its multiples (such as [[224edo]] and [[742edo]]) are members of zeta edo list.
 
=== Silicon ===
The name of silicon temperament comes from the 14th element. Defined upwards to the 13-limit. In 742edo, what's also unique is that it is generated by a [[53edo]] fifth intermingled with [[14edo]] periods.
 
==== 5-limit ====
Subgroup: 2.3.5
 
Comma list: {{Monzo|-145 112 -14}}
 
Mapping generators: ~282429536481/268435456000, ~3/2
 
Mapping: [{{val|14 14 -33}}, {{val|0 1 8}}]
 
Optimal tuning (CTE): ~3/2 = 701.864
 
==== 7-limit ====
Subgroup: 2.3.5.7
 
Comma list: 14348907/14336000, 56358560858112/56296884765625
 
Mapping generators: ~6125/5832, ~3/2
 
Mapping: [{{val|14 14 -33 113}}, {{val|0 1 8 -9}}]
 
Optimal tuning (CTE): ~3/2 = 701.870
 
==== 11-limit ====
Subgroup: 2.3.5.7.11
 
Comma list: 9801/9800, 1240029/1239040, 2359296/2358125
 
Mapping generators: ~605/576, ~3/2
 
Mapping: [{{val|14 14 -33 113 73}}, {{val|0 1 8 -9 -3}}]
 
Optimal tuning (CTE): ~3/2 = 701.872
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 4096/4095, 9801/9800, 67392/67375, 59535/59488
 
Mapping generators: ~104/99, ~3/2
 
Mapping: [{{val|14 14 -33 113 73 60}}, {{val|0 1 8 -9 -3 -1}}]
 
Optimal tuning (CTE): ~3/2 = 701.873
 
Vals: {{EDOs|70d, 224, 294, 448, 518, 672, 742, 966, 1190, 1260}}
 
== 37th-octave temperaments ==
[[37edo|37EDO]] is accurate for harmonics 5, 7, 11, and 13, so various 37th-octave temperaments actually make sense.
 
=== Rubidium ===
The name of rubidium temperament comes from Rubidium, the 37th element.
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 3136/3125, 4194304/4117715
 
[[Mapping]]: [{{val|37 0 86 104}}, {{val|0 1 0 0}}]
 
[[POTE generator]]: ~3/2 = 703.3903
 
{{Val list|legend=1| 37, 74, 111 }}
 
[[Badness]]: 0.312105
 
==== 11-limit ====
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 1375/1372, 65536/65219
 
Mapping: [{{val|37 0 86 104 128}}, {{val|0 1 0 0 0}}]
 
POTE generator: ~3/2 = 703.0355
 
Vals: {{Val list| 37, 74, 111 }}
 
Badness: 0.101001
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 176/175, 640/637, 847/845, 1375/1372
 
Mapping: [{{val|37 0 86 104 128 137}}, {{val|0 1 0 0 0 0}}]
 
POTE generator: ~3/2 = 703.0520
 
Vals: {{Val list| 37, 74, 111 }}
 
Badness: 0.048732
 
=== Triacontaheptoid ===
Subgroup: 2.3.5.7
 
[[Comma list]]: 244140625/242121642, 283115520/282475249
 
[[Mapping]]: [{{val|37 23 74 92}}, {{val|0 3 1 1}}]
 
[[POTE generator]]: ~5/4 = 385.3041
 
{{Val list|legend=1| 37, 259b, 296, 629, 925c }}
 
[[Badness]]: 0.784746
 
==== 11-limit ====
Subgroup: 2.3.5.7.11
 
Comma list: 1375/1372, 4000/3993, 226492416/226474325
 
Mapping: [{{val|37 23 74 92 128}}, {{val|0 3 1 1 0}}]
 
POTE generator: ~5/4 = 385.3281
 
Vals: {{Val list| 37, 259b, 296, 629 }}
 
Badness: 0.167327
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 625/624, 1375/1372, 4000/3993, 15379/15360
 
Mapping: [{{val|37 23 74 92 128 125}}, {{val|0 3 1 1 0 1}}]
 
POTE generator: ~5/4 = 385.3067
 
Vals: {{Val list| 37, 259b, 296, 629f, 925cf }}
 
Badness: 0.076183
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 625/624, 715/714, 1225/1224, 4000/3993, 11271/11264
 
Mapping: [{{val|37 23 74 92 128 125 175}}, {{val|0 3 1 1 0 1 -2}}]
 
POTE generator: ~5/4 = 385.3427
 
Vals: {{Val list| 37, 259b, 296, 629f }}
 
Badness: 0.052475
 
== 65th-octave temperaments ==
[[65edo|65EDO]] is accurate for harmonics 3, 5, and 11, so various 65th-octave temperaments actually make sense.
 
=== Terbium ===
The name of terbium temperament comes from Terbium, the 65th element.
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 32805/32768, 78732/78125
 
[[Mapping]]: [{{val|65 103 151 0}}, {{val|0 0 0 1}}]
 
[[POTE generator]]: ~8/7 = 230.8641
 
{{Val list|legend=1| 65, 130 }}
 
[[Badness]]: 0.169778
 
==== 11-limit ====
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 4000/3993, 5632/5625
 
Mapping: [{{val|65 103 151 0 225}}, {{val|0 0 0 1 0}}]
 
POTE generator: ~8/7 = 230.4285
 
Vals: {{Val list| 65d, 130 }}
 
Badness: 0.059966
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 243/242, 351/350, 2080/2079, 3584/3575
 
Mapping: [{{val|65 103 151 0 225 58}}, {{val|0 0 0 1 0 1}}]
 
POTE generator: ~8/7 = 230.0388
 
Vals: {{Val list| 65d, 130 }}
 
Badness: 0.036267
 
== 91st-octave temperaments ==
 
=== Protactinium ===
Defined as the 364 & 1547 temperament and named after the 91st element.
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 4096/4095, 91125/91091, 2912000/2910897, 369754/369603
 
Mapping: [{{val|91 91 371 149 581 390}}, {{val|0 1 -3 -2 -5 -1}}]
 
Mapping generators: ~1728/1715, ~3/2
 
Optimal tuning (CTE): ~3/2 = 702.020c
 
Optimal GPV sequence: {{EDOs|364, 1183, 1547}}
====17-limit====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 4096/4095, 14400/14399, 42500/42471, 75735/75712, 2100875/2100384
 
Mapping: [<nowiki/>{{91 91 371 149 581 390 159}}, {{val|0 1 -3 -2 -5 -1 4}}]
 
Mapping generators: ~3773/3744, ~3/2
 
Optimal tuning (CTE): ~3/2 = 702.027c
 
Optimal GPV sequence: {{EDOs|364, 1183, 1547}}
 
== 118th-octave temperaments ==
[[118edo]] is accurate for harmonics 3 and 5, so various 118th-octave temperaments actually make sense.
 
=== Parakleischis ===
118edo and its multiples are members of both [[parakleismic]] and [[Schismatic family|schismic]], and from this it derives its name.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 32805/32768, 1224440064/1220703125
 
[[Mapping]]: [{{val| 118 187 274 0 }}, {{val| 0 0 0 1 }}]
 
Mapping generators: ~15625/15552, ~7
 
[[Optimal tuning]] ([[POTE]]): ~7/4 = 968.7235
 
{{Val list|legend=1| 118, 236, 354, 472, 2242, 2714b, 3186b, 3658b }}
 
[[Badness]]: 0.145166
 
==== 11-limit ====
Subgroup: 2.3.5.7.11
 
Comma list: 9801/9800, 32805/32768, 137781/137500
 
Mapping: [{{val| 118 187 274 0 77 }}, {{val| 0 0 0 1 1 }}]
 
Optimal tuning (POTE): ~7/4 = 968.5117
 
Vals: {{Val list| 118, 354, 472 }}
 
Badness: 0.049316
 
==== Oganesson ====
Named after the 118th element, since a simpler temperament was already named. 82 periods plus a generator correspond to [[13/8]].
 
Subgroup: 2.3.5.7.11
 
Comma list: 32805/32768, 151263/151250, 1224440064/1220703125
 
Mapping: [{{val| 118 187 274 0 -420 }}, {{val| 0 0 0 2 5 }}]
 
Mapping generators: ~15625/15552, ~405504/153125
 
Optimal tuning (CTE): ~202752/153125 = 484.4837
 
Optimal GPV sequence: {{val list| 354, 944e, 1298 }}
 
Badness: 0.357
 
===== 13-limit =====
Subgroup: 2.3.5.7.11.13
 
Comma list: 1716/1715, 32805/32768, 34398/34375, 384912/384475
 
Mapping: [{{val| 118 187 274 0 -420 271 }}, {{val| 0 0 0 2 5 1 }}]
 
Optimal tuning (CTE): ~8125/6144 = 484.4867
 
Optimal GPV sequence: {{val list| 354, 944e, 1298 }}


Badness: 0.122
== See also ==
* [[Map of rank-2 temperaments]]: Visual map of many of the temperaments listed here.


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Rank 2]]
[[Category:Pages with mostly numerical content]]
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