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Fractional-octave temperaments: Difference between revisions

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


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


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val|37 0 86 104}}, {{val|0 1 0 0}}]
[[Mapping]]: [{{val| 37 0 86 104 }}, {{val| 0 1 0 0 }}]
 
Mapping generators: ~50/49, ~3


[[POTE generator]]: ~3/2 = 703.3903
[[Optimal tuning]] ([[POTE]]): ~3/2 = 703.3903


{{Val list|legend=1| 37, 74, 111 }}
{{Val list|legend=1| 37, 74, 111 }}
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Comma list: 176/175, 1375/1372, 65536/65219
Comma list: 176/175, 1375/1372, 65536/65219


Mapping: [{{val|37 0 86 104 128}}, {{val|0 1 0 0 0}}]
Mapping: [{{val| 37 0 86 104 128 }}, {{val| 0 1 0 0 0 }}]


POTE generator: ~3/2 = 703.0355
Optimal tuning (POTE): ~3/2 = 703.0355


Vals: {{Val list| 37, 74, 111 }}
Optimal GPV sequence: {{Val list| 37, 74, 111 }}


Badness: 0.101001
Badness: 0.101001
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Comma list: 176/175, 640/637, 847/845, 1375/1372
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}}]
Mapping: [{{val| 37 0 86 104 128 137 }}, {{val| 0 1 0 0 0 0 }}]


POTE generator: ~3/2 = 703.0520
Optimal tuning (POTE): ~3/2 = 703.0520


Vals: {{Val list| 37, 74, 111 }}
Optimal GPV sequence: {{Val list| 37, 74, 111 }}


Badness: 0.048732
Badness: 0.048732
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[[Comma list]]: 244140625/242121642, 283115520/282475249
[[Comma list]]: 244140625/242121642, 283115520/282475249


[[Mapping]]: [{{val|37 23 74 92}}, {{val|0 3 1 1}}]
[[Mapping]]: [{{val| 37 2 67 85 }}, {{val| 0 3 1 1 }}]
 
Mapping generator: ~50/49, ~24000/16807


[[POTE generator]]: ~5/4 = 385.3041
[[Optimal tuning]] ([[CTE]]): ~24000/16807 = 612.4003


{{Val list|legend=1| 37, 259b, 296, 629, 925c }}
{{Val list|legend=1| 37, 222b, 259b, 296, 629 }}


[[Badness]]: 0.784746
[[Badness]]: 0.784746
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Comma list: 1375/1372, 4000/3993, 226492416/226474325
Comma list: 1375/1372, 4000/3993, 226492416/226474325


Mapping: [{{val|37 23 74 92 128}}, {{val|0 3 1 1 0}}]
Mapping: [{{val| 37 2 67 85 128 }}, {{val| 0 3 1 1 0 }}]


POTE generator: ~5/4 = 385.3281
Optimal tuning (CTE): ~768/359 = 612.4003


Vals: {{Val list| 37, 259b, 296, 629 }}
Optimal GPV sequence: {{Val list| 37, 259b, 296, 629 }}


Badness: 0.167327
Badness: 0.167327
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Comma list: 625/624, 1375/1372, 4000/3993, 15379/15360
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}}]
Mapping: [{{val| 37 2 67 85 128 118 }}, {{val| 0 3 1 1 0 1 }}]


POTE generator: ~5/4 = 385.3067
Optimal tuning (CTE): ~462/325 = 612.4206


Vals: {{Val list| 37, 259b, 296, 629f, 925cf }}
Optimal GPV sequence: {{Val list| 37, 259b, 296, 629f }}


Badness: 0.076183
Badness: 0.076183
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Comma list: 625/624, 715/714, 1225/1224, 4000/3993, 11271/11264
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}}]
Mapping: [{{val| 37 2 67 85 128 118 189 }}, {{val| 0 3 1 1 0 1 -2 }}]


POTE generator: ~5/4 = 385.3427
Optimal tuning (CTE): ~121/85 = 612.4187


Vals: {{Val list| 37, 259b, 296, 629f }}
Optimal GPV sequence: {{Val list| 37, 259b, 296, 629f }}


Badness: 0.052475
Badness: 0.052475


== 65th-octave temperaments ==
== 65th-octave temperaments ==
[[65edo|65EDO]] is accurate for harmonics 3, 5, and 11, so various 65th-octave temperaments actually make sense.
[[65edo]] is accurate for harmonics 3, 5, and 11, so various 65th-octave temperaments actually make sense.


=== Terbium ===
=== Terbium ===
The name of terbium temperament comes from Terbium, the 65th element.
The name of terbium temperament comes from Terbium, the 65th element.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 32805/32768, 78732/78125
[[Comma list]]: 32805/32768, 78732/78125


[[Mapping]]: [{{val|65 103 151 0}}, {{val|0 0 0 1}}]
[[Mapping]]: [{{val| 65 103 151 0 }}, {{val| 0 0 0 1 }}]


[[POTE generator]]: ~8/7 = 230.8641
Mapping generators: ~81/80, ~7
 
[[Optimal tuning]] ([[POTE]]): ~7/4 = 969.1359


{{Val list|legend=1| 65, 130 }}
{{Val list|legend=1| 65, 130 }}
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Comma list: 243/242, 4000/3993, 5632/5625
Comma list: 243/242, 4000/3993, 5632/5625


Mapping: [{{val|65 103 151 0 225}}, {{val|0 0 0 1 0}}]
Mapping: [{{val| 65 103 151 0 225 }}, {{val| 0 0 0 1 0 }}]


POTE generator: ~8/7 = 230.4285
Optimal tuning (POTE): ~7/4 = 969.5715


Vals: {{Val list| 65d, 130 }}
Optimal GPV sequence: {{Val list| 65d, 130 }}


Badness: 0.059966
Badness: 0.059966
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Comma list: 243/242, 351/350, 2080/2079, 3584/3575
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}}]
Mapping: [{{val| 65 103 151 0 225 58 }}, {{val| 0 0 0 1 0 1 }}]


POTE generator: ~8/7 = 230.0388
Optimal tuning (POTE): ~7/4 = 969.9612


Vals: {{Val list| 65d, 130 }}
Optimal GPV sequence: {{Val list| 65d, 130 }}


Badness: 0.036267
Badness: 0.036267


== 91st-octave temperaments ==
== 91st-octave temperaments ==
=== Protactinium ===
=== Protactinium ===
Defined as the 364 & 1547 temperament and named after the 91st element.
Protactinium is described as the 364 & 1547 temperament and named after the 91st element.


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 4096/4095, 91125/91091, 2912000/2910897, 369754/369603
Comma list: 4096/4095, 91125/91091, 369754/369603, 2912000/2910897


Mapping: [{{val|91 91 371 149 581 390}}, {{val|0 1 -3 -2 -5 -1}}]
Mapping: [{{val| 91 0 644 -33 1036 481 }}, {{val| 0 1 -3 -2 -5 -1 }}]


Mapping generators: ~1728/1715, ~3/2
Mapping generators: ~1728/1715, ~3


Optimal tuning (CTE): ~3/2 = 702.020c
Optimal tuning (CTE): ~3/2 = 702.0195


Optimal GPV sequence: {{EDOs|364, 1183, 1547}}
Optimal GPV sequence: {{Val list| 364, 819e, 1183, 1547 }}
====17-limit====
 
Badness: 0.0777
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 4096/4095, 14400/14399, 42500/42471, 75735/75712, 2100875/2100384
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: [{{val| 91 0 644 -33 1036 481 -205 }}, {{val| 0 1 -3 -2 -5 -1 4 }}]


Mapping generators: ~3773/3744, ~3/2
Optimal tuning (CTE): ~3/2 = 702.0269


Optimal tuning (CTE): ~3/2 = 702.027c
Optimal GPV sequence: {{Val list| 364, 1183, 1547, 1911 }}


Optimal GPV sequence: {{EDOs|364, 1183, 1547}}
Badness: 0.0582


== 118th-octave temperaments ==
== 118th-octave temperaments ==
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Optimal tuning (POTE): ~7/4 = 968.5117
Optimal tuning (POTE): ~7/4 = 968.5117


Vals: {{Val list| 118, 354, 472 }}
Optimal GPV sequence: {{Val list| 118, 354, 472 }}


Badness: 0.049316
Badness: 0.049316
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