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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.


* [[Ragismic microtemperaments #Chlorine|Chlorine]] (1\17 period)
=== Terminology ===
* [[Ragismic microtemperaments #Enneadecal|Enneadecal]] (1\19 period)
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.
* [[Porwell temperaments #Icositritonic|Icositritonic]] (1\23 period)
* [[Compton family #Hours|Hours]] (1\24 period)
* [[26th-octave temperaments|Bosonic]] (1\26 period)
* [[31st-octave temperaments|Birds]] (1\31 period)
* [[Compton family #Decades|Decades]] (1\36 period)
* [[Counterpyth family|Counterpyth]] (1\41 period)
* [[Mitonismic temperaments #Meridic|Meridic]] (1\43 period)
* [[Mercator family|Mercator]] (1\53 period)
* [[Compton family #Omicronbeta|Omicronbeta]] (1\72 period)
* [[Stearnsmic clan #Garistearn|Garistearn]] (1\94 period)
* [[Tritrizo clan #Undecentic|Undecentic]] (1\99 period)


== 37th-octave temperaments ==
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.
[[37edo|37EDO]] is accurate for harmonics 5, 7, 11, and 13, so various 37th-octave temperaments actually make sense.


=== Rubidium ===
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]].
The name of rubidium temperament comes from Rubidium, the 37th element.


Subgroup: 2.3.5.7
=== 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.  


[[Comma list]]: 3136/3125, 4194304/4117715
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.


[[Mapping]]: [{{val|37 0 86 104}}, {{val|0 1 0 0}}]
== 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]].


[[POTE generator]]: ~3/2 = 703.3903
=== 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.  


{{Val list|legend=1| 37, 74, 111 }}
=== Examples ===
Below is a list of octave-splitting commas for common temperaments:


[[Badness]]: 0.312105
{| class="wikitable"
|-
! Comma !! Associated <br>temperament !! Harmonic <br>limit !! Splitting <br>order
|-
| [[256/243]] || [[Limmic temperaments|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
|}


==== 11-limit ====
Subgroup: 2.3.5.7.11


Comma list: 176/175, 1375/1372, 65536/65219
== Individual pages of temperaments by subtending equal division ==
=== 2 to 100 ===
{| class="wikitable center-all"
|+
|
| [[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]]/[[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]]
|}


Mapping: [{{val|37 0 86 104 128}}, {{val|0 1 0 0 0}}]
=== 101 and up ===
[[111th-octave temperaments|111]], [[118th-octave temperaments|118]], [[159th-octave temperaments|159]], [[400th-octave temperaments|400]], [[665th-octave temperaments|665]]


POTE generator: ~3/2 = 703.0355
* C = compton family
* CC = countercomp family
* M = mercator family equated with 53rd-octave temperaments until otherwise documented, also contains 106th-octave temperaments


Vals: {{Val list| 37, 74, 111 }}
== Temperaments discussed elsewhere ==
Temperaments discussed as a part of a commatic family, or otherwise in temperament lists unrelated to fractional-octave theory include:


Badness: 0.101001
* 1\2 period temperaments
 
** [[Diaschismic family|Diaschismic temperaments]]
==== 13-limit ====
** [[Vishnuzmic family|Vishnuzmic temperaments]]
Subgroup: 2.3.5.7.11.13
** [[Jubilismic clan|Jubilismic temperaments]]
 
** [[Varunismic temperaments]]
[[Comma list]]: 176/175, 640/637, 847/845, 1375/1372
** [[Lokismic temperaments]]
 
** [[Nimona|Nimona temperament]]
[[Mapping]]: [{{val|37 0 86 104 128 137}}, {{val|0 1 0 0 0 0}}]
* 1\3 period temperaments
 
** [[Augmented family|Augmented temperaments]]
[[POTE generator]]: ~3/2 = 703.0520
** [[Misty family|Misty temperaments]]
 
** [[Landscape microtemperaments|Landscape temperaments]]
{{Val list|legend=1| 37, 74, 111 }}
* 1\4 period temperaments
 
** [[Diminished family|Diminished temperaments]]
[[Badness]]: 0.048732
** [[Undim family|Undim temperaments]]
 
* 1\5 period temperaments
== 65th-octave temperaments ==
** [[Quintile family|Quintile temperaments]]
[[65edo|65EDO]] is accurate for harmonics 3. 5, and 11, so various 65th-octave temperaments actually make sense.
** [[Quintosec family|Quintosec temperaments]]
 
** [[Trisedodge family|Trisedodge temperaments]]
=== Terbium ===
** [[Cloudy clan|Cloudy temperaments]]
The name of terbium temperament comes from Terbium, the 65th element.
** [[Limmic temperaments]]
 
* 1\6 period temperaments
Subgroup: 2.3.5.7
** [[Augmented family #Hexe|Hexe]]
 
** [[Landscape microtemperaments #Sextile|Sextile]]
[[Comma list]]: 32805/32768, 78732/78125
** [[Stearnsmic clan #Stearnscape|Stearnscape]]
 
* [[Akjaysma|Akjaysmic temperaments]] (1\7 period)
[[Mapping]]: [{{val|65 103 151 0}}, {{val|0 0 0 1}}]
** [[Ragismic microtemperaments #Brahmagupta|Brahmagupta]]
 
** [[Schismatic family #Septant|Septant]]
[[POTE generator]]: ~8/7 = 230.8641
** [[Whitewood family #Whitewood|Whitewood temperaments]]
 
** [[Keemic temperaments #Sevond|Sevond]]
{{Val list|legend=1| 65, 130 }}
** [[Mistismic temperaments #Neutron|Neutron]]
 
* [[Ragismic microtemperaments #Octoid|Octoid]], [[Schismatic family #Octant|octant]] (1\8 period)
[[Badness]]: 0.169778
* [[Septiennealimmal clan|Septiennealimmal temperaments]] (1\9 period)
 
** [[Ragismic microtemperaments #Ennealimmal|Ennealimmal]]
==== 11-limit ====
** [[Augmented family #Niner|Niner]]
Subgroup: 2.3.5.7.11
** [[Marvel temperaments #Enneaportent|Enneaportent]]
 
** [[Kleismic family #Novemkleismic|Novemkleismic]]
Comma list: 243/242, 4000/3993, 5632/5625
* [[Linus]] temperaments (1\10 period)
 
** [[Breedsmic temperaments #Decoid|Decoid]]
Mapping: [{{val|65 103 151 0 225}}, {{val|0 0 0 1 0}}]
** [[Ragismic microtemperaments #Deca|Deca]]
 
** [[Cloudy clan #Decic|Decic]]
POTE generator: ~8/7 = 230.4285
** [[Stearnsmic clan #Decistearn|Decistearn]]
 
** [[Quintile family #Decile|Decile]]
Vals: {{Val list| 65d, 130 }}
** [[Vishnuzmic family #Decavish|Decavish]]
 
** [[Metric microtemperaments #Decimetra|Decimetra]]
Badness: 0.059966
* [[Porwell temperaments #Hendecatonic|Hendecatonic]] (1\11 period)
 
* [[Compton family|Compton]], [[Very high accuracy temperaments #Atomic|atomic]] (1\12 period)
==== 13-limit ====
* [[Orwellismic temperaments #Triskaidekic|Triskaidekic]], [[Octagar temperaments #Tridecatonic|tridecatonic]], [[Ragismic microtemperaments #Trideci|trideci]], [[aluminium]] (1\13 period)
Subgroup: 2.3.5.7.11.13
* [[Silicon]] (1\14 period)
 
* [[Cloudy clan #Pentadecal|Pentadecal]], [[Trienstonic clan #Quindecic|quindecic]] (1\15 period)
Comma list: 243/242, 351/350, 2080/2079, 3584/3575
* [[Ragismic microtemperaments #Octoid|Hexadecoid]], [[Jubilismic clan #Sedecic|sedecic]] (1\16 period)
 
* [[Ragismic microtemperaments #Chlorine|Chlorine]] (1\17 period)
Mapping: [{{val|65 103 151 0 225 58}}, {{val|0 0 0 1 0 1}}]
* [[Septiennealimmal clan #Ennealimmal|Hemiennealimmal]] (1\18 period)
 
* [[Ragismic microtemperaments #Enneadecal|Enneadecal]], [[Meantone family #Meanmag|meanmag]] (1\19 period)
POTE generator: ~8/7 = 230.0388
* [[Hemimage temperaments #Degrees|Degrees]] (1\20 period)
 
* [[Akjayland]] (1\21 period)
Vals: {{Val list| 65d, 130 }}
* [[Porwell temperaments #Hendecatonic|Icosidillic]] (1\22 period)
* [[Porwell temperaments #Icositritonic|Icositritonic]] (1\23 period)
* [[Compton family #Hours|Hours]], [[chromium]] (1\24 period)
* [[Septiennealimmal clan #Ennealimmal|Trinealimmal]], [[Tritrizo clan #Cobalt|cobalt]] (1\27 period)
* [[Horwell temperaments #Oquatonic|Oquatonic]] (1\28 period)
* [[Hemifamity temperaments #Mystery|Mystery]], [[Copper comma|copper]] (1\29 period)
* [[31st-octave temperaments|Birds]] (1\31 period)
* [[Compton family #Gamelstearn|Gamelstearn]] (1\36 period)
* [[Ragismic microtemperaments #Enneadecal|Hemienneadecal]], [[semihemienneadecal]] (1\38 period)
* [[Countercomp family|Countercomp temperaments]], [[niobium]] (1\41 period)
* [[Mitonismic temperaments #Meridic|Meridic]] (1\43 period)
* [[Ragismic microtemperaments #Palladium|Palladium]] (1\46 period)
* [[Compton family #Omicronbeta|Omicronbeta]] (1\72 period)
* [[The Flashmob#Iridium|Iridium]] (1\77 period)
* [[Parkleiness temperaments #Octogintic|Octogintic]] (1\80 period)
* [[Stearnsmic clan #Garistearn|Garistearn]] (1\94 period)
* [[Septiennealimmal clan #Undecentic|Undecentic]] (1\99 period)
* [[Septiennealimmal clan #Schisennealimmal|Schisennealimmal]] (1\171 period)
* [[Septiennealimmal clan #Lunennealimmal|Lunennealimmal]] (1\441 period)


Badness: 0.036267
== See also ==
* [[Map of rank-2 temperaments]]: Visual map of many of the temperaments listed here.


[[Category:Regular temperament theory]]
[[Category:Temperament collections]]
[[Category:Temperament collection]]
[[Category:Rank 2]]
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