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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]]ism and polychromatics. They are acoustically significant with regards to containing {{w|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'') equal temperament which repeats ''y'' times within that period, its "wireframe" is ''y'' equal temperament. 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 policy and fractional-octave practice ===
Vanilla regular temperament theory does not distinguish periods and generators, so it treats divisions of periods (for example, what [[hemiennealimmal]] is to [[ennealimmal]]) as [[extension]]s of a temperament with a subset period, just like how it treats divisions of generators. However, fractional-octave temperaments and scales are sometimes sought for being able to treat 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]].
 
Besides, on this wiki, temperament collection pages are used to collect temperaments that temper out a common comma. For example, the [[landscape microtemperaments]] list features temperaments which all temper out the [[landscape comma]], as they are all related by that. A fractional-octave temperament user might consider that 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.
 
== Octave-splitting comma ==
An '''octave-splitting comma''' is a rational [[interval]] which induces a fractional-octave temperament. If tempering out the comma leads to splitting the [[octave]] into ''N'' equal parts (''N'' > 1), the comma is an octave-splitting comma which induces an ''N''th-octave temperament. This terminology was developed by [[Xenllium]].
 
=== Property ===
While a comma is given in the ''p''-limit [[monzo]] form {{monzo| ''a''<sub>2</sub> ''a''<sub>3</sub> ''a''<sub>5</sub> … ''a''<sub>''p''</sub> }}, as a rational interval, it is an octave-splitting comma if and only if GCD(''a''<sub>2</sub>, ''a''<sub>3</sub>, ''a''<sub>5</sub>, …, ''a''<sub>''p''</sub>) = 1 and ''N'' = GCD(''a''<sub>3</sub>, ''a''<sub>5</sub>, …, ''a''<sub>''p''</sub>) > 1, and leads to splitting the octave into ''N'' equal parts.
 
=== Examples ===
Below is a list of octave-splitting commas for common temperaments:
 
{| class="wikitable"
|-
! Comma !! Associated <br>temperament !! Harmonic <br>limit !! Splitting <br>order
|-
| [[256/243]] || [[Blackwood family|Blackwood]] || 3 || 5
|-
| [[2187/2048]] || [[Apotome family|Whitewood]] || 3 || 7
|-
| [[531441/524288]] || [[Compton family|Compton]] || 3 || 12
|-
| [[2048/2025]] || [[Diaschismic family|Diaschismic]] || 5 || 2
|-
| [[128/125]] || [[Augmented family|Augmented]] || 5 || 3
|-
| [[648/625]] || [[Diminished family|Diminished]] || 5 || 4
|-
| [[50/49]] || [[Jubilismic clan|Jubilismic]] || 7 || 2
|-
| [[250047/250000]] || [[Landscape microtemperaments|Landscape]] || 7 || 3
|-
| [[9801/9800]] || [[Kalismic temperaments|Kalismic]] || 11 || 2
|-
| [[289/288]] || [[Semitonismic]] || 17 || 2
|}
 
 
== Individual pages of temperaments by subtending equal division ==
=== 2 to 100 ===
{| class="wikitable center-all"
|+
|
| colspan="3" | [[2nd- to 4th-octave temperaments|2–4]]
| [[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]]/[[Compton family|C]]
| [[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|CC]]
| [[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]]
|}
 
* C = compton family
* CC = countercomp family
* M = mercator family equated with 53rd-octave temperaments until otherwise documented, also contains 106th-octave temperaments
 
=== 101 and up ===
[[111th-octave temperaments|111]], [[118th-octave temperaments|118]], [[159th-octave temperaments|159]], [[400th-octave temperaments|400]], [[665th-octave temperaments|665]]
 
== 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
** [[Diaschismic family|Diaschismic temperaments]]
** [[Vishnuzmic family|Vishnuzmic temperaments]]
** [[Jubilismic clan|Jubilismic temperaments]]
** [[Varunismic temperaments]]
** [[Lokismic temperaments]]
* 1\3 period temperaments
** [[Augmented family|Augmented temperaments]]
** [[Misty family|Misty temperaments]]
** [[Landscape microtemperaments|Landscape temperaments]]
* 1\4 period temperaments
** [[Dimipent family|Diminished 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 comma #Temperaments|Cloudy temperaments]]
** [[Limmic temperaments]]
** [[Blackwood family]]
* 1\6 period temperaments
** [[Augmented family #Hexe|Hexe]]
** [[Landscape microtemperaments #Sextile|Sextile]]
** [[Stearnsmic clan #Stearnscape|Stearnscape]]
* [[Akjaysma|Akjaysmic temperaments]] (1\7 period)
** [[Ragismic microtemperaments #Brahmagupta|Brahmagupta]]
** [[Schismatic family #Septant|Septant]]
** [[Apotome family #Whitewood|Whitewood]]
** [[Keemic temperaments #Sevond|Sevond]]
** [[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]] ([[20th-octave temperaments|1\20 period]])
* [[Hemimage temperaments #Degrees|Degrees]] (1\20 period)
* [[Akjayland]] ([[21st-octave temperaments|1\21 period]])
* [[Akjayland]] (1\21 period)
* [[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)
* [[Landscape microtemperaments #Chromium|Chromium]] (1\24 period)
* [[Septiennealimmal clan #Ennealimmal|Trinealimmal]], [[Tritrizo clan #Cobalt|cobalt]] (1\27 period)
* [[26th-octave temperaments|Bosonic]] ([[26th-octave temperaments|1\26 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)
* [[Windrose]] ([[32nd-octave temperaments|1\32 period]])
* [[Compton family #Gamelstearn|Gamelstearn]] (1\36 period)
* [[Compton family #Decades|Decades]] (1\36 period)
* [[Ragismic microtemperaments #Enneadecal|Hemienneadecal]], [[semihemienneadecal]] (1\38 period)
* [[Ragismic microtemperaments #Enneadecal|Hemienneadecal]] (1\38 period)
* [[Countercomp family|Countercomp temperaments]], [[niobium]] (1\41 period)
* [[Counterpyth family|Counterpyth temperaments]] (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)
* [[Parkleiness temperaments #Octogintic|Octogintic]] ([[80th-octave temperaments|1\80 period]])
* [[The Flashmob#Iridium|Iridium]] (1\77 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 simple 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. When tuned in 742edo, it is generated by a [[53edo]] fifth intermingled with [[14edo]] periods.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 14348907/14336000, 56358560858112/56296884765625
 
[[Mapping]]: [{{val| 14 0 -145 239 }}, {{val| 0 1 8 -9 }}]
 
Mapping generators: ~6125/5832, ~3
 
[[Optimal tuning]] ([[CTE]]): ~3/2 = 701.870
 
{{Val list|legend=1| 224, 518, 742, 966, 1708, 2674 }}
 
[[Badness]]: 0.196
 
==== 11-limit ====
Subgroup: 2.3.5.7.11
 
Comma list: 9801/9800, 1240029/1239040, 2359296/2358125
 
Mapping: [{{val| 14 0 -145 239 115 }}, {{val| 0 1 8 -9 -3 }}]
 
Optimal tuning (CTE): ~3/2 = 701.872
 
Optimal GPV sequence: {{val list| 224, 518, 742, 966, 1708, 4382e }}
 
Badness: 0.0450
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 4096/4095, 6656/6655, 9801/9800, 24192/24167
 
Mapping: [{{val| 14 0 -145 239 115 74 }}, {{val| 0 1 8 -9 -3 -1 }}]
 
Optimal tuning (CTE): ~3/2 = 701.8733
 
Optimal GPV sequence: {{val list| 224, 518, 742, 966, 1708f }}
 
Badness: 0.0269
 
== 37th-octave temperaments ==
[[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 }}]
 
Mapping generators: ~50/49, ~3
 
[[Optimal tuning]] ([[POTE]]): ~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 }}]
 
Optimal tuning (POTE): ~3/2 = 703.0355
 
Optimal GPV sequence: {{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 }}]
 
Optimal tuning (POTE): ~3/2 = 703.0520
 
Optimal GPV sequence: {{Val list| 37, 74, 111 }}
 
Badness: 0.048732
 
=== Triacontaheptoid ===
Subgroup: 2.3.5.7
 
[[Comma list]]: 244140625/242121642, 283115520/282475249
 
[[Mapping]]: [{{val| 37 2 67 85 }}, {{val| 0 3 1 1 }}]
 
Mapping generator: ~50/49, ~24000/16807
 
[[Optimal tuning]] ([[CTE]]): ~24000/16807 = 612.4003
 
{{Val list|legend=1| 37, 222b, 259b, 296, 629 }}
 
[[Badness]]: 0.784746
 
==== 11-limit ====
Subgroup: 2.3.5.7.11
 
Comma list: 1375/1372, 4000/3993, 226492416/226474325
 
Mapping: [{{val| 37 2 67 85 128 }}, {{val| 0 3 1 1 0 }}]
 
Optimal tuning (CTE): ~768/359 = 612.4003
 
Optimal GPV sequence: {{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 2 67 85 128 118 }}, {{val| 0 3 1 1 0 1 }}]
 
Optimal tuning (CTE): ~462/325 = 612.4206
 
Optimal GPV sequence: {{Val list| 37, 259b, 296, 629f }}
 
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 2 67 85 128 118 189 }}, {{val| 0 3 1 1 0 1 -2 }}]
 
Optimal tuning (CTE): ~121/85 = 612.4187
 
Optimal GPV sequence: {{Val list| 37, 259b, 296, 629f }}
 
Badness: 0.052475
 
== 44th-octave temperaments ==
One step of 44edo is very close to the septimal comma, [[64/63]]. The relationship is preserved even up thousands of edos.
 
=== Ruthenium ===
Ruthenium is named after the 44th element, and can be expressed as the 1848 & 2684 temperament.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: {{monzo| -8  23 -5 -6 }}, {{monzo| 51 -13 -1 -10 }}
 
[[Mapping]]: [{{val| 44 0 -386 263 }}, {{val| 0 1 7 -2 }}]
 
Mapping generators: ~64/63, ~3
 
[[Optimal tuning]] ([[CTE]]): ~3/2 = 701.9420
 
{{Val list|legend=1| 176, 660, 836, 1848, 2684, 4532, 19976, 24508, 29040, 33572 }}
 
[[Badness]]: 0.111
 
==== 11-limit ====
Subgroup: 2.3.5.7.11
 
Comma list: 9801/9800, 1771561/1771470, 67110351/67108864
 
Mapping: [{{val| 44 0 -386 263 -57 }}, {{val| 0 1 7 -2 3 }}]
 
Optimal tuning (CTE): ~3/2 = 701.9429
 
Optiml GPV sequence: {{val list| 176, 660, 836, 1848, 2684, 4532, 15444, 19976e }}
 
Badness: 0.0209
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 9801/9800, 196625/196608, 823680/823543, 1771561/1771470
 
Mapping: [{{val| 44 0 -386 263 -57 1976 }}, {{val| 0 1 7 -2 3 -26 }}]
 
Optimal tuning (CTE): ~3/2 = 701.939
 
Optiml GPV sequence: {{val list| 836, 1848, 2684, 7216, 9900, 12584 }}
 
Badness: 0.0396
 
== 56th-octave temperaments ==
=== Barium ===
One step of 56edo is close to a syntonic comma. Named after the 56th element, barium tempers out the {{monzo| -225 224 -56 }} comma, which sets 56 syntonic commas equal to the octave. It can be expressed as the 224 & 2072 temperament.
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: {{monzo| -225 24 -56 }}
 
[[Mapping]]: [{{val| 56 0 -225 }}, {{val| 0 1 4 }}]
 
Mapping generators: ~81/80, ~3
 
[[Optimal tuning]] ([[CTE]]): ~3/2 = 701.9379
 
{{Val list|legend=1| 224, 1176, 1400, 1624, 1848, 2072, 5992, 8064, 26264, 34328b, 42392b }}
 
[[Badness]]: 4.70
 
==== 7-limit ====
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: {{monzo| -12 29 -11 -3 }}, {{monzo| 47 -7 -7 -7 }}
 
[[Mapping]]: [{{val| 56 0 -225 601 }}, {{val| 0 1 4 -5 }}]
 
[[Optimal tuning]] ([[CTE]]): ~3/2 = 701.9433
 
{{Val list|legend=1| 224, 1176, 1400, 1624, 1848, 2072, 5768, 7616, 17080, 24696cd }}
 
[[Badness]]: 0.227
 
==== 11-limit ====
Subgroup: 2.3.5.7.11
 
Comma list: 9801/9800, 1019215872/1019046875, 14765025303/14763950080
 
Mapping: [{{val| 56 0 -225 601 460 }}, {{val| 0 1 4 5 -3 }}]
 
Optimal tuning (CTE): ~3/2 = 701.9431
 
Optimal GPV sequence: {{Val list| 224, 1176, 1400, 1624, 1848, 3920, 5768, 7616, 21000cd, 28616cd }}
 
Badness: 0.0345
 
== 65th-octave temperaments ==
[[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 }}]
 
Mapping generators: ~81/80, ~7
 
[[Optimal tuning]] ([[POTE]]): ~7/4 = 969.1359
 
{{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 }}]
 
Optimal tuning (POTE): ~7/4 = 969.5715
 
Optimal GPV sequence: {{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 }}]
 
Optimal tuning (POTE): ~7/4 = 969.9612
 
Optimal GPV sequence: {{Val list| 65d, 130 }}
 
Badness: 0.036267
== 91st-octave temperaments ==
=== Protactinium ===
Protactinium is described as the 364 & 1547 temperament and named after the 91st element.
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 4096/4095, 91125/91091, 369754/369603, 2912000/2910897
 
Mapping: [{{val| 91 0 644 -33 1036 481 }}, {{val| 0 1 -3 -2 -5 -1 }}]
 
Mapping generators: ~1728/1715, ~3
 
Optimal tuning (CTE): ~3/2 = 702.0195
 
Optimal GPV sequence: {{Val list| 364, 819e, 1183, 1547 }}
 
Badness: 0.0777
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 4096/4095, 14400/14399, 42500/42471, 75735/75712, 2100875/2100384
 
Mapping: [{{val| 91 0 644 -33 1036 481 -205 }}, {{val| 0 1 -3 -2 -5 -1 4 }}]
 
Optimal tuning (CTE): ~3/2 = 702.0269
 
Optimal GPV sequence: {{Val list| 364, 1183, 1547, 1911 }}
 
Badness: 0.0582
 
== 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
 
Optimal GPV sequence: {{Val list| 118, 354, 472 }}
 
Badness: 0.049316
 
==== Centenniamajor ====
Named after the fact that 18 is the age of majority in most countries, and 100 (centennial) + 18 (major) = 118.
 
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
 
=== Oganesson ===
Named after the 118th element. In the 13-limit, the period corresponds to [[169/168]], and in the 17-limit, it corresponds also to [[170/169]], meaning that 28561/28560 is tempered out. As opposed to being an extension of parakleischis, this has the generator that splits the third harmonic into three equal parts.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: {{monzo| 30 10 -27 6 }}, {{monzo| 77 -20 -5 -12 }}
 
[[Mapping]]: [{{val| 118 0 274 643 }}, {{val| 0 3 0 -5 }}]
 
Mapping generators: ~2097152/2083725, ~1953125/1354752
 
[[Optimal tuning]] ([[CTE]]): ~1953125/1354752 = 634.0068
 
{{Val list|legend=1| 354, 2360, 2714, 3068, 3442, 3776, 7198cd, 10974bccdd }}
 
[[Badness]]: 2.66
 
==== 11-limit ====
Subgroup: 2.3.5.7.11
 
Comma list: 9801/9800, {{monzo| 13 -1 4 -16 7 }}, {{monzo| 55 -7 -15 -2 -1 }}
 
Mapping: [{{val| 118 0 274 643 1094 }}, {{val| 0 3 0 -5 -11 }}]
 
Optimal tuning (CTE): ~1953125/1354752 = 634.0085
 
Optimal GPV sequence: {{Val list| 354, 3068e, 3442, 3776, 11682ccdde }}
 
Badness: 0.568
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 4096/4095, 9801/9800, 537403776/537109375, 453874312332/453857421875
 
Mapping: [{{val| 118 0 274 643 1094 499}}, {{val| 0 3 0 -5 -11 -1}}]
 
Mapping generators: ~169/168, ~1124864/779625
 
Optimal tuning (CTE): ~1124864/779625 = 634.0087
 
Optimal GPV sequence: {{Val list| 354, 3068e, 3422, 3776 }}
 
Badness: 0.172
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 4096/4095, 9801/9800, 34391/34375, 361250/361179, 562432/562275
 
Mapping:  [{{val| 118 0 274 643 1094 499 607 }}, {{val| 0 3 0 -5 -11 -1 2 }}]
 
Mapping generators: ~170/169, ~238/165
 
Optimal tuning (CTE): ~238/165 = 634.0080
 
Optimal GPV sequence: {{Val list| 354, 3068e, 3422, 3776 }}


Badness: 0.105
== 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]]
Retrieved from "https://en.xen.wiki/w/Fractional-octave_temperaments"