Fractional-octave temperaments: Difference between revisions

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→Theory: remove those links because I'll be adding them to the table
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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.
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.


== Temperament collections ==
== Individual pages of temperaments by equal division ==
 
=== 2 to 40 ===
Many pages are yet to be created.
{| class="wikitable"
|+
|
|2
|3
|4
|5
|6
|7
|8
|9
|10
|-
|11
|12
|13
|14
|15
|16
|17
|18
|19
|20
|-
|21
|22
|23
|24
|25
|26
|27
|28
|29
|30
|-
|31
|32
|33
|34
|35
|36
|37
|38
|39
|40
|}
 
=== 40 and up ===
41, 44, 53*, 56, 60, 61, 65, 80, 91, 111, 118
 
== Temperaments discussed elsewhere ==
Temperaments discussed as a part of a commatic family, or otherwise in temperament lists unrelated to fractional-octave theory include:
Temperaments discussed as a part of a commatic family, or otherwise in temperament lists unrelated to fractional-octave theory include:


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** [[Landscape microtemperaments #Sextile|Sextile]]
** [[Landscape microtemperaments #Sextile|Sextile]]
** [[Stearnsmic clan #Stearnscape|Stearnscape]]
** [[Stearnsmic clan #Stearnscape|Stearnscape]]
* [[Akjaysma|Akjaysmic temperaments]] ([[7th-octave temperaments|1\7 period]])
* [[Akjaysma|Akjaysmic temperaments]] (1\7 period)
** [[Ragismic microtemperaments #Brahmagupta|Brahmagupta]]
**[[Ragismic microtemperaments #Brahmagupta|Brahmagupta]]
** [[Schismatic family #Septant|Septant]]
** [[Schismatic family #Septant|Septant]]
** [[Apotome family #Whitewood|Whitewood]]
** [[Apotome family #Whitewood|Whitewood]]
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* [[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]] ([[17th-octave temperaments|1\17 period]])
* [[Ragismic microtemperaments #Chlorine|Chlorine]] (1\17 period)
* [[Ragismic microtemperaments #Ennealimmal|Hemiennealimmal]] (1\18 period)
* [[Ragismic microtemperaments #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]] ([[22nd-octave temperaments|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]], [[chromium]] (1\24 period)
* [[Compton family #Hours|Hours]], [[chromium]] (1\24 period)
* [[26th-octave temperaments|Bosonic]] ([[26th-octave temperaments|1\26 period]])
* [[Ragismic microtemperaments #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]], [[Copper comma|copper]] ([[29th-octave temperaments|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]], [[bezique]] ([[32nd-octave temperaments|1\32 period]])
* Bromine, tritonopod ([[35th-octave temperaments|1\35 period]])
* [[Compton family #Decades|Decades]] (1\36 period)
* [[Compton family #Decades|Decades]] (1\36 period)
* Rubidium, dzelic ([[37th-octave temperaments|1\37 period]])
* Rubidium, dzelic (1\37 period)
* [[Ragismic microtemperaments #Enneadecal|Hemienneadecal]], [[semihemienneadecal]] (1\38 period)
* [[Ragismic microtemperaments #Enneadecal|Hemienneadecal]], [[semihemienneadecal]] (1\38 period)
* [[Counterpyth family|Counterpyth temperaments]], [[niobium]] (1\41 period)
* [[Counterpyth family|Counterpyth temperaments]], [[niobium]] (1\41 period)
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* [[Ragismic microtemperaments #Palladium|Palladium]] (1\46 period)
* [[Ragismic microtemperaments #Palladium|Palladium]] (1\46 period)
* [[Mercator family|Mercator temperaments]] (1\53 period)
* [[Mercator family|Mercator temperaments]] (1\53 period)
* [[60th-octave temperaments|Minutes, magnetic temperaments]] (1\60 period)
* [[Compton family #Omicronbeta|Omicronbeta]], [[The Flashmob#Hafnium|hafnium]] (1\72 period)
* [[Compton family #Omicronbeta|Omicronbeta]], [[The Flashmob#Hafnium|hafnium]] (1\72 period)
* [[The Flashmob#Iridium|Iridium]] (1\77 period)
* [[The Flashmob#Iridium|Iridium]] (1\77 period)
* [[Parkleiness temperaments #Octogintic|Octogintic]], mercury, tetraicosic ([[80th-octave temperaments|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)
* [[Tritrizo clan #Undecentic|Undecentic]] (1\99 period)
* Parakleischis, oganesson ([[118th-octave temperaments|1\118 period]])
* [[Tritrizo clan #Schisennealimmal|Schisennealimmal]] (1\171 period)
* [[Tritrizo clan #Schisennealimmal|Schisennealimmal]] (1\171 period)
* [[Tritrizo clan #Lunennealimmal|Lunennealimmal]] (1\441 period)
* [[Tritrizo clan #Lunennealimmal|Lunennealimmal]] (1\441 period)
== 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
{{Optimal ET sequence|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: {{Optimal ET sequence| 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: {{Optimal ET sequence| 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| -225 224 -56 }}
[[Mapping]]: [{{val| 56 0 -225 }}, {{val| 0 1 4 }}]
Mapping generators: ~81/80, ~3
[[Optimal tuning]] ([[CTE]]): ~3/2 = 701.9379
{{Optimal ET sequence|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
{{Optimal ET sequence|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 ET sequence|legend=1| 224, 1176, 1400, 1624, 1848, 3920, 5768, 7616, 21000cd, 28616cd }}
Badness: 0.0345
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 4225/4224, 9801/9800, 67392/67375, 26802913280/26795786661
Mapping: [{{val| 56 0 -225 601 460 651}}, {{val| 0 1 4 -5 -3 -5}}]
Optimal tuning (CTE): ~3/2 = 701.9431
{{Optimal ET sequence|legend=1| 224, 1848, 2072}}, ...
== 61st-octave temperaments ==
=== Promethium ===
Promethium tempers out the [[dipromethia]] and can be described as the 183 & 2684 temperament. By tempering out 4100625/4100096 promethium identifies the diaschisma with [[2025/2002]] in the 13-limit and also in the 17-limit.
Subgroup: 2.3.5.7.11.13
Comma list: 10648/10647, 196625/196608, 4100625/4100096, 204800000/204788493
Mapping: [{{val|61 0 335 703 66 -161}}, {{val|0 2 -4 -11 3 8}}]
Mapping generators: ~2025/2002 = 1\61, ~6875/3969 = 950.970
Optimal tuning (CTE): ~6875/3969 = 950.970
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Comma list: 14400/14399, 37180/37179, 121875/121856, 140800/140777, 3536379/3536000
Mapping: [{{val|61 0 335 703 66 -161 201}}, {{val|0 2 -4 -11 3 8 1}}]
Mapping generators: ~2025/2002 = 1\61, ~11907/6875 = 950.970
Optimal tuning (CTE): ~11907/6875 = 950.970
{{Optimal ET sequence|legend=1|183, 2684}}, ...
== 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
{{Optimal ET sequence|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 ET sequence|legend=1| 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 ET sequence|legend=1| 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
Comma list: {{monzo|47 -7 -7 -7}}, {{monzo|-2 -25 1 14}}
Mapping: [{{val| 91 0 644 -33 1036}}, {{val| 0 1 -3 -2 -5}}]
: mapping generators: ~1728/1715, ~3
Optimal tuning (CTE): ~3/2 = 701.991
[[Support]]ing [[ET]]s: {{EDOs|364, 819, 1183, 1547, 1911, 2730, 3094, 3913, 4277}}
==== 11-limit ====
Subgroup: 2.3.5.7.11
Comma list: 234375/234256, 26214400/26198073, 514714375/514434888
Mapping: [{{val| 91 0 644 -33 1036}}, {{val| 0 1 -3 -2 -5}}]
: mapping generators: ~1728/1715, ~3
Optimal tuning (CTE): ~3/2 = 702.015
[[Support]]ing [[ET]]s: {{EDOs|364, 819e, 1183, 1547, 1911, 2275, 2730e, 3458}}
==== 13-limit ====
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 ET sequence|legend=1| 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 ET sequence|legend=1| 364, 1183, 1547, 1911 }}
Badness: 0.0582
== 111th-octave temperaments ==
=== Roentgenium ===
Roentgenium is defined as 4884 & 8103 in the 19-limit and is named after the 111th element. 111 is 37 x 3, and what's particularly remarkable about this temperament is that it still preserves the relationship of 11/8 to 37edo in EDOs the size of thousands. Developed for a musical composition in [[8103edo]] by Eliora.
Subgroup: 2.3.5.7.11
Comma list: {{monzo|-25 -12 -3 12  5}}, {{monzo|-27  27  0  3 -7}}, {{monzo|26  -8 -2  8 -9}}
Mapping: [{{val|111 111 2855 896 384}}, {{val|0 1 -40 -9 0}}]
Optimal tuning (CTE): ~3/2 = 701.964
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 31213/31212, 486400/486387, 633556/633555, 653429/653400, 1037232/1037153, 9714446/9713275, 24764600/24762387
Mapping: [{{val|111 111 2855 896 384 410 452 472}}, {{val|0 1 -40 -9 0 -11 -25 7}}]
Optimal tuning (CTE): ~3/2 = 701.9...
{{Optimal ET sequence|legend=1|3219c, 4884, 8103, 12987}}, ...
[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Rank 2]]
[[Category:Lists of temperaments]]
[[Category:Lists of temperaments]]

Revision as of 14:02, 15 June 2023

Fractional-octave temperaments, when viewed from a regular temperament theory perspective, are temperaments which have a period which corresponds to a just interval mapped to a fraction of the octave, that is one step of an EDO.

Theory

Fractional-octave temperaments are valuable with regards to 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.

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.

Individual pages of temperaments by equal division

2 to 40

Many pages are yet to be created.

2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40

40 and up

41, 44, 53*, 56, 60, 61, 65, 80, 91, 111, 118

Temperaments discussed elsewhere

Temperaments discussed as a part of a commatic family, or otherwise in temperament lists unrelated to fractional-octave theory include:

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