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Fractional-octave temperaments: Difference between revisions

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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]]
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