Biyatismic clan: Difference between revisions
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The '''biyatismic clan''' of [[ | The '''biyatismic clan''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the biyatisma, [[121/120]]. | ||
Temperaments discussed elsewhere are: | Temperaments discussed elsewhere are: | ||
* ''[[Sonic]]'' (+55/54 or 100/99) → [[Porcupine rank | * ''[[Sonic]]'' (+55/54 or 100/99) → [[Porcupine rank-3 family #Sonic|Porcupine rank-3 family]] | ||
* ''[[Urania]]'' (+81/80) → [[ | * ''[[Urania]]'' (+81/80) → [[Rastmic rank-3 clan #Urania|Rastmic rank-3 clan]] | ||
* ''[[Bisector]]'' (+245/243) → [[Sensamagic family #Bisector|Sensamagic family]] | * ''[[Bisector]]'' (+245/243) → [[Sensamagic family #Bisector|Sensamagic family]] | ||
Considered below are zeus, artemis, oxpecker, aphrodite, and the no-7 subgroup temperament, protomere. For the rank-4 biyatismic temperament, see [[Rank-4 temperament #Biyatismic (121/120)]]. | Considered below are zeus, artemis, oxpecker, kahoupokane, big brother, aphrodite, and the no-7 subgroup temperament, protomere. For the rank-4 biyatismic temperament, see [[Rank-4 temperament #Biyatismic (121/120)]]. | ||
== Protomere == | == Protomere == | ||
| Line 16: | Line 15: | ||
{{Mapping|legend=2| 1 0 1 2 | 0 1 1 1 | 0 0 -2 -1 }} | {{Mapping|legend=2| 1 0 1 2 | 0 1 1 1 | 0 0 -2 -1 }} | ||
: mapping generators: ~2, ~3, ~11/10 | |||
: | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1200.6628{{c}}, ~3/2 = 701.8452{{c}}, ~11/10 = 157.8337{{c}} | |||
[[ | : [[error map]]: {{val| +0.663 +0.553 +1.190 -5.318 }} | ||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9848{{c}}, ~12/11 = 157.6099{{c}} | |||
: error map: {{val| 0.000 +0.030 +0.451 -6.943 }} | |||
{{Optimal ET sequence|legend=1| 7, 15, 22, 31, 46, 53, 137e, 183ee, 190ee }} | {{Optimal ET sequence|legend=1| 7, 15, 22, 31, 46, 53, 137e, 183ee, 190ee }} | ||
[[Badness]]: 0. | [[Badness]] (Sintel): 0.245 | ||
== Zeus == | == Zeus == | ||
| Line 41: | Line 43: | ||
: Angle (11/10, 11/8) = 87.464 degrees | : Angle (11/10, 11/8) = 87.464 degrees | ||
[[Optimal tuning]] | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1200.1693{{c}}, ~3/2 = 702.2521{{c}}, ~12/11 = 157.1102{{c}} | |||
: [[error map]]: {{val| +0.169 +0.466 +2.057 +0.761 -5.668 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2478{{c}}, ~12/11 = 157.1265{{c}} | |||
: error map: {{val| 0.000 +0.293 +1.681 +0.306 -6.197 }} | |||
[[Minimax tuning]]: | [[Minimax tuning]]: | ||
* [[11-odd-limit]] | * [[11-odd-limit]] | ||
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 11/9 10/9 -1/3 -2/9 0 }}, {{monzo| 22/9 2/9 1/3 -4/9 0 }}, {{monzo| 22/9 2/9 -2/3 5/9 0 }}, {{monzo| 10/3 2/3 0 -1/3 0 }}] | : [{{monzo| 1 0 0 0 0 }}, {{monzo| 11/9 10/9 -1/3 -2/9 0 }}, {{monzo| 22/9 2/9 1/3 -4/9 0 }}, {{monzo| 22/9 2/9 -2/3 5/9 0 }}, {{monzo| 10/3 2/3 0 -1/3 0 }}] | ||
: [[ | : [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.9/7 | ||
{{Optimal ET sequence|legend=1| 15, 22, 31, 46, 53, 68, 77, 99, 130e }} | {{Optimal ET sequence|legend=1| 15, 22, 31, 46, 53, 68, 77, 99, 130e }} | ||
[[Badness]]: 0. | [[Badness]] (Sintel): 0.480 | ||
[[Projection pair]]s: 5 600/121 7 2662/375 11 120/11 to 2.3.11/5 | [[Projection pair]]s: 5 600/121 7 2662/375 11 120/11 to 2.3.11/5 | ||
| Line 77: | Line 83: | ||
: Angle (11/10, 11/8) = 106.7439 degrees | : Angle (11/10, 11/8) = 106.7439 degrees | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.2411{{c}}, ~3/2 = 702.0090{{c}}, ~12/11 = 156.9897{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.8818{{c}}, ~12/11 = 156.9568{{c}} | |||
Minimax tuning: | Minimax tuning: | ||
* 13-odd-limit | * 13-odd-limit | ||
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 11/9 10/9 -1/3 -2/9 0 0 }}, {{monzo| 22/9 2/9 1/3 -4/9 0 0 }}, {{monzo| 22/9 2/9 -2/3 5/9 0 0 }}, {{monzo| 10/3 2/3 0 -1/3 0 0 }}, {{monzo| 14/3 -8/3 1 1/3 0 0 }}] | : [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 11/9 10/9 -1/3 -2/9 0 0 }}, {{monzo| 22/9 2/9 1/3 -4/9 0 0 }}, {{monzo| 22/9 2/9 -2/3 5/9 0 0 }}, {{monzo| 10/3 2/3 0 -1/3 0 0 }}, {{monzo| 14/3 -8/3 1 1/3 0 0 }}] | ||
: | : unchanged-interval (eigenmonzo) basis: 2.9/5.9/7 | ||
* 15-odd-limit | * 15-odd-limit | ||
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 0 1 0 0 0 0 }}, {{monzo| 11/5 1/5 2/5 -2/5 0 0 }}, {{monzo| 11/5 1/5 -3/5 3/5 0 0 }}, {{monzo| 13/5 3/5 1/5 -1/5 0 0 }}, {{monzo| 38/5 -12/5 1/5 -1/5 0 0 }}] | : [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 0 1 0 0 0 0 }}, {{monzo| 11/5 1/5 2/5 -2/5 0 0 }}, {{monzo| 11/5 1/5 -3/5 3/5 0 0 }}, {{monzo| 13/5 3/5 1/5 -1/5 0 0 }}, {{monzo| 38/5 -12/5 1/5 -1/5 0 0 }}] | ||
: | : unchanged-interval (eigenmonzo) basis: 2.3.7/5 | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 15f, 22, 31, 46, 53, 77, 99, 130e }} | ||
Badness: 0. | Badness (Sintel): 0.873 | ||
Projection pairs: 5 600/121 7 2662/375 11 120/11 13 1280/99 to 2.3.11/5 | Projection pairs: 5 600/121 7 2662/375 11 120/11 13 1280/99 to 2.3.11/5 | ||
| Line 103: | Line 111: | ||
Mapping: {{mapping| 1 0 1 4 2 2 | 0 1 1 -1 1 1 | 0 0 -2 3 -1 -1 }} | Mapping: {{mapping| 1 0 1 4 2 2 | 0 1 1 -1 1 1 | 0 0 -2 3 -1 -1 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.9251{{c}}, ~3/2 = 699.2984{{c}}, ~12/11 = 155.3569{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.2982{{c}}, ~12/11 = 155.3484{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 7, 9, 15, 22f, 24, 31 }} | ||
Badness: 0. | Badness (Sintel): 0.756 | ||
== Artemis == | == Artemis == | ||
| Line 118: | Line 128: | ||
{{Mapping|legend=1| 1 0 1 -3 2 | 0 1 1 4 1 | 0 0 -2 -4 -1 }} | {{Mapping|legend=1| 1 0 1 -3 2 | 0 1 1 4 1 | 0 0 -2 -4 -1 }} | ||
[[Optimal tuning]] | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1201.2783{{c}}, ~3/2 = 700.6174{{c}}, ~11/10 = 158.4919{{c}} | |||
: [[error map]]: {{val| +1.278 -0.059 -0.123 +0.955 -5.357 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.2777{{c}}, ~11/10 = 158.3100{{c}} | |||
: error map: {{val| 0.000 -1.677 -2.656 -0.955 -9.350 }} | |||
{{Optimal ET sequence|legend=1| 9, 15d, 16d, 20, 22, 31, 53, | {{Optimal ET sequence|legend=1| 9, 15d, 16d, 20, 22, 31, 53, 60e, 84e, 91e, 113e, 144ee }} | ||
[[Badness]] (Sintel): 0.713 | |||
=== 13-limit === | === 13-limit === | ||
| Line 129: | Line 145: | ||
Mapping: {{mapping| 1 0 1 -3 2 -5 | 0 1 1 4 1 6 | 0 0 -2 -4 -1 -6 }} | Mapping: {{mapping| 1 0 1 -3 2 -5 | 0 1 1 4 1 6 | 0 0 -2 -4 -1 -6 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1201.7896{{c}}, ~3/2 = 699.7509{{c}}, ~11/10 = 158.9484{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.1687{{c}}, ~11/10 = 158.7345{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 9, 20, 22f, 29, 31, 60e, 129cddee }} | ||
Badness (Sintel): 1.04 | |||
=== Diana === | === Diana === | ||
| Line 140: | Line 160: | ||
Mapping: {{mapping| 1 0 1 -3 2 7 | 0 1 1 4 1 -2 | 0 0 -2 -4 -1 -1 }} | Mapping: {{mapping| 1 0 1 -3 2 7 | 0 1 1 4 1 -2 | 0 0 -2 -4 -1 -1 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.9110{{c}}, ~3/2 = 701.5110{{c}}, ~11/10 = 159.1256{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.9717{{c}}, ~11/10 = 158.7903{{c}} | |||
{{Optimal ET sequence|legend=0| 22, 29, 31, 53, 82e, 84e, 113e }} | |||
Badness (Sintel): 1.07 | |||
== Oxpecker == | == Oxpecker == | ||
| Line 151: | Line 175: | ||
{{Mapping|legend=1| 1 0 1 2 2 | 0 1 1 1 1 | 0 0 -2 -6 -1 }} | {{Mapping|legend=1| 1 0 1 2 2 | 0 1 1 1 1 | 0 0 -2 -6 -1 }} | ||
[[Optimal tuning]] | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1200.4124{{c}}, ~3/2 = 701.1291{{c}}, ~12/11 = 155.8292{{c}} | |||
: [[error map]]: {{val| +0.412 -0.414 +3.982 -1.435 -4.781 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.2359{{c}}, ~12/11 = 155.7399{{c}} | |||
: error map: {{val| 0.000 -0.719 +3.442 -2.029 -5.822 }} | |||
{{Optimal ET sequence|legend=1| 7d, 8d, 15, 23de, 24d, 31, 46, 77 }} | {{Optimal ET sequence|legend=1| 7d, 8d, 15, 23de, 24d, 31, 46, 77 }} | ||
[[Badness]]: 0. | [[Badness]] (Sintel): 0.840 | ||
=== Woodpecker === | === Woodpecker === | ||
| Line 164: | Line 192: | ||
Mapping: {{mapping| 1 0 1 2 2 2 | 0 1 1 1 1 1 | 0 0 -2 -6 -1 1 }} | Mapping: {{mapping| 1 0 1 2 2 2 | 0 1 1 1 1 1 | 0 0 -2 -6 -1 1 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1198.9113{{c}}, ~3/2 = 700.9581{{c}}, ~12/11 = 154.7247{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.6409{{c}}, ~12/11 = 154.9115{{c}} | |||
{{Optimal ET sequence|legend=0| 7d, 8d, 15, 23de, 24d, 31 }} | |||
Badness (Sintel): 1.02 | |||
== Kahoupokane == | |||
Named by [[Tristan Bay]] in 2025, Kahoupokane tempers out [[5120/5103]] and may be described as the {{nowrap| 29 & 46 & 53 }} temperament. | |||
[[Subgroup]]: 2.3.5.7.11 | |||
[[Comma list]]: 121/120, 5120/5103 | |||
{{ | {{Mapping|legend=1| 1 0 1 11 2 | 0 1 1 -5 1 | 0 0 -2 -2 -1 }} | ||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1200.1911{{c}}, ~3/2 = 703.1412{{c}}, ~11/10 = 158.1068{{c}} | |||
: [[error map]]: {{val| +0.191 +1.377 +0.996 +0.401 -5.710 }} | |||
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 703.0417{{c}}, ~11/10 = 157.9917{{c}} | |||
: error map: {{val| 0.000 +1.087 +0.744 -0.018 -6.268 }} | |||
= | {{Optimal ET sequence|legend=1| 7, 17c, 24d, 29, 46, 53, 82e, 99 }} | ||
[[Badness]] (Sintel): 2.73 | |||
[[ | |||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 121/120, 169/168, 352/351 | |||
Mapping: {{mapping| 1 0 1 11 2 7 | 0 1 1 -5 1 -2 | 0 0 -2 -2 -1 -1 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1200.4435{{c}}, ~3/2 = 703.1443{{c}}, ~11/10 = 158.4176{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.9013{{c}}, ~11/10 = 158.1657{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 7, 17c, 24d, 29, 46, 53, 82e, 99, 181eef }} | ||
Badness (Sintel): 1.27 | |||
[[ | == Big brother == | ||
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Nuwell]].'' | |||
[[Subgroup]]: 2.3.5.7.11 | [[Subgroup]]: 2.3.5.7.11 | ||
[[Comma list]]: 121/120 | [[Comma list]]: 99/98, 121/120 | ||
{{Mapping|legend=1| 1 0 1 | {{Mapping|legend=1| 1 0 -5 -1 -1 | 0 1 3 2 2 | 0 0 4 1 2 }} | ||
: mapping generators: ~2, ~3, ~11/7 | |||
: | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1200.6559{{c}}, ~3/2 = 700.2627{{c}}, ~11/7 = 771.8821{{c}} | |||
: [[error map]]: {{val| +0.656 -1.036 +0.691 +4.237 -6.372 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.4019{{c}}, ~11/7 = 771.2671{{c}} | |||
: error map: {{val| 0.000 -1.553 -0.039 +3.245 -7.980 }} | |||
Optimal | {{Optimal ET sequence|legend=1| 8d, 9, 14c, 17c, 22, 31, 53, 84e }} | ||
[[Badness]] (Sintel): 0.609 | |||
[[ | [[Projection pair]]s: <code>5 2401/486, 11 98/9</code> to 2.3.7 | ||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: 121/120, | Comma list: 66/65, 99/98, 121/120 | ||
Mapping: {{mapping| 1 0 3 1 3 1 | 0 1 3 2 2 0 | 0 0 -4 -1 -2 2 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.0121{{c}}, ~3/2 = 699.9867{{c}}, ~11/7 = 771.9817{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.7360{{c}}, ~11/7 = 773.0154{{c}} | |||
{{Optimal ET sequence|legend=0| 8d, 9, 14c, 17c, 22f, 31, 79cf }} | |||
Badness (Sintel): 0.889 | |||
== Aphrodite == | |||
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Squalentine]].'' | |||
[[Subgroup]]: 2.3.5.7.11 | |||
[[Comma list]]: 121/120, 441/440 | |||
{{Mapping|legend=1| 1 0 1 3 2 | 0 1 1 0 1 | 0 0 -4 -3 -2 }} | |||
: mapping generators: ~2, ~3, ~22/21 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1201.0691{{c}}, ~3/2 = 700.9439{{c}}, ~22/21 = 78.7122{{c}} | |||
: [[error map]]: {{val| +1.069 +0.058 +1.920 -1.755 -4.591 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.8990{{c}}, ~22/21 = 78.4412{{c}} | |||
: error map: {{val| 0.000 -1.056 +0.820 -4.150 -7.301 }} | |||
{{Optimal ET sequence|legend=1| 14c, 15, 29, 31, 46, 60e, 77, 91e, 137de, 168dee }} | |||
[[Badness]] (Sintel): 0.701 | |||
=== Eros === | |||
Eros fairs impressively into the 23-limit as a rank-3 temperament; not only is it fairly simple (considering this is a subgroup as complex as the full 23-limit, with many challenges) but has the pleasing property that all the harmonics are on the negative side of the last generator. Specifically, -3 to 2 fifths and -5 to 0 [[~]][[23/22]]'s will provide odd harmonics 1–23 up to octave equivalence; you can think of this as a 6×6 grid, which is a recommendable place to start looking at its structure. | |||
Tempering out the less accurate comma 121/120 can be seen as an implication of tempering out [[441/440]] ({{S|21}}), [[484/483]] ({{S|22}}), and [[529/528]] ({{S|23}}). Therefore, characteristic of any good tuning is the prime [[11/1|11]] being the flattest prime, with other primes having strictly less than 5{{cent}} of error. | |||
This temperament was discovered by [[Scott Dakota]]. Note that the 17-limit extension requires the 29g val for 29edo, which has the sizes of 17/16 and 18/17 swapped. | |||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 227: | Line 304: | ||
Mapping: {{mapping| 1 0 1 3 2 7 | 0 1 1 0 1 -2 | 0 0 -4 -3 -2 -2 }} | Mapping: {{mapping| 1 0 1 3 2 7 | 0 1 1 0 1 -2 | 0 0 -4 -3 -2 -2 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.6419{{c}}, ~3/2 = 701.8766{{c}}, ~22/21 = 78.6564{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.5612{{c}}, ~22/21 = 78.4778{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 17c, 29, 31, 46, 60e, 77, 106de, 183dee }} | ||
Badness: 1. | Badness (Sintel): 1.08 | ||
==== 17-limit ==== | |||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
| Line 243: | Line 320: | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.6172{{c}}, ~3/2 = 702.1026{{c}}, ~22/21 = 78.7963{{c}} | ||
* CWE: ~2 = | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7925{{c}}, ~22/21 = 78.6203{{c}} | ||
{{Optimal ET sequence|legend=0| 17cg, 29g, 31, 46, 60e, 77, 106de }} | |||
Badness | Badness (Sintel): 0.931 | ||
==== 19-limit ==== | |||
Subgroup: 2.3.5.7.11.13.17.19 | Subgroup: 2.3.5.7.11.13.17.19 | ||
| Line 260: | Line 335: | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.6224{{c}}, ~3/2 = 702.0959{{c}}, ~22/21 = 78.8004{{c}} | ||
* CWE: ~2 = | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.6963{{c}}, ~22/21 = 78.6479{{c}} | ||
{{Optimal ET sequence|legend=0| 17cg, 29g, 31, 46, 60e, 75dfgh, 77, 106de }} | |||
Badness | Badness (Sintel): 1.16 | ||
==== 23-limit ==== | |||
Subgroup: 2.3.5.7.11.13.17.19.23 | Subgroup: 2.3.5.7.11.13.17.19.23 | ||
| Line 277: | Line 350: | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.7268{{c}}, ~3/2 = 702.2463{{c}}, ~22/21 = 78.8824{{c}} | ||
* CWE: ~2 = | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.8010{{c}}, ~23/22 = 78.7188{{c}} | ||
{{Optimal ET sequence|legend=0| 17cg, 29g, 31, 46, 60e, 75dfgh, 77, 106de }} | |||
Badness: | Badness (Sintel): 1.08 | ||
=== Astarte === | |||
This extension is catalogued as tridecimal aphrodite in [[Graham Breed]]'s temperament finder. | |||
==== Inanna | Subgroup: 2.3.5.7.11.13 | ||
Comma list: 121/120, 351/350, 441/440 | |||
Mapping: {{mapping| 1 0 1 3 2 6 | 0 1 1 0 1 -1 | 0 0 -4 -3 -2 -11 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1201.0656{{c}}, ~3/2 = 700.7374{{c}}, ~22/21 = 78.5908{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.7005{{c}}, ~22/21 = 78.3253{{c}} | |||
{{Optimal ET sequence|legend=0| 14cf, 29ff, 31, 45ef, 46, 77, 122ee, 137def, 168deef }} | |||
Badness (Sintel): 1.36 | |||
=== Inanna === | |||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 293: | Line 381: | ||
Mapping: {{mapping| 1 0 1 3 2 1 | 0 1 1 0 1 2 | 0 0 -4 -3 -2 -7 }} | Mapping: {{mapping| 1 0 1 3 2 1 | 0 1 1 0 1 2 | 0 0 -4 -3 -2 -7 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1201.7881{{c}}, ~3/2 = 699.8166{{c}}, ~22/21 = 79.7282{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.5926{{c}}, ~22/21 = 79.3822{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 14cf, 15, 29, 31, 45ef, 60e }} | ||
Badness: 1. | Badness (Sintel): 1.01 | ||
=== Ishtar === | |||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 306: | Line 396: | ||
Mapping: {{mapping| 1 0 1 3 2 -1 | 0 1 1 0 1 3 | 0 0 -4 -3 -2 -1 }} | Mapping: {{mapping| 1 0 1 3 2 -1 | 0 1 1 0 1 3 | 0 0 -4 -3 -2 -1 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.7875{{c}}, ~3/2 = 703.8568{{c}}, ~22/21 = 79.0096{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.7640{{c}}, ~22/21 = 78.8025{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 14cf, 15, 17c, 29, 31f, 46, 106deff, 121def }} | ||
Badness: 1. | Badness (Sintel): 1.08 | ||
== | == References == | ||
[[Category:Temperament clans]] | [[Category:Temperament clans]] | ||
[[Category:Biyatismic clan| ]] <!-- main article --> | [[Category:Biyatismic clan| ]] <!-- main article --> | ||
[[Category:Rank 3]] | [[Category:Rank 3]] | ||
Latest revision as of 03:43, 25 April 2026
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
The biyatismic clan of rank-3 temperaments tempers out the biyatisma, 121/120.
Temperaments discussed elsewhere are:
- Sonic (+55/54 or 100/99) → Porcupine rank-3 family
- Urania (+81/80) → Rastmic rank-3 clan
- Bisector (+245/243) → Sensamagic family
Considered below are zeus, artemis, oxpecker, kahoupokane, big brother, aphrodite, and the no-7 subgroup temperament, protomere. For the rank-4 biyatismic temperament, see Rank-4 temperament #Biyatismic (121/120).
Protomere
Subgroup: 2.3.5.11
Comma list: 121/120
Subgroup-val mapping: [⟨1 0 1 2], ⟨0 1 1 1], ⟨0 0 -2 -1]]
- mapping generators: ~2, ~3, ~11/10
- WE: ~2 = 1200.6628 ¢, ~3/2 = 701.8452 ¢, ~11/10 = 157.8337 ¢
- error map: ⟨+0.663 +0.553 +1.190 -5.318]
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9848 ¢, ~12/11 = 157.6099 ¢
- error map: ⟨0.000 +0.030 +0.451 -6.943]
Optimal ET sequence: 7, 15, 22, 31, 46, 53, 137e, 183ee, 190ee
Badness (Sintel): 0.245
Zeus
Subgroup: 2.3.5.7.11
Comma list: 121/120, 176/175
Mapping: [⟨1 0 1 4 2], ⟨0 1 1 -1 1], ⟨0 0 -2 3 1]]
Mapping to lattice: [⟨0 1 -1 2 0], ⟨0 1 1 -1 1]]
Lattice basis:
- 11/10, 11/8
- Angle (11/10, 11/8) = 87.464 degrees
- WE: ~2 = 1200.1693 ¢, ~3/2 = 702.2521 ¢, ~12/11 = 157.1102 ¢
- error map: ⟨+0.169 +0.466 +2.057 +0.761 -5.668]
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2478 ¢, ~12/11 = 157.1265 ¢
- error map: ⟨0.000 +0.293 +1.681 +0.306 -6.197]
- [[1 0 0 0 0⟩, [11/9 10/9 -1/3 -2/9 0⟩, [22/9 2/9 1/3 -4/9 0⟩, [22/9 2/9 -2/3 5/9 0⟩, [10/3 2/3 0 -1/3 0⟩]
- unchanged-interval (eigenmonzo) basis: 2.9/5.9/7
Optimal ET sequence: 15, 22, 31, 46, 53, 68, 77, 99, 130e
Badness (Sintel): 0.480
Projection pairs: 5 600/121 7 2662/375 11 120/11 to 2.3.11/5
Zeus11[22] hobbit transversal
- 33/32, 16/15, 11/10, 8/7, 64/55, 77/64, 5/4, 14/11, 4/3,
- 11/8, 45/32, 16/11, 3/2, 11/7, 8/5, 5/3, 55/32, 7/4,
- 11/6, 15/8, 64/33, 2
Zeus11[24] hobbit transversal
- 33/32, 16/15, 11/10, 9/8, 8/7, 77/64, 11/9, 5/4, 21/16, 4/3,
- 11/8, 45/32, 16/11, 3/2, 32/21, 8/5, 18/11, 5/3, 7/4, 16/9,
- 11/6, 15/8, 64/33, 2
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 121/120, 176/175, 351/350
Mapping: [⟨1 0 1 4 2 7], ⟨0 1 1 -1 1 -2], ⟨0 0 -2 3 -1 -1]]
Mapping to lattice: [⟨0 1 -1 2 0 -3], ⟨0 1 1 -1 1 -2]]
Lattice basis:
- 11/10 length = 0.7898, 11/8 length = 1.002
- Angle (11/10, 11/8) = 106.7439 degrees
Optimal tunings:
- WE: ~2 = 1200.2411 ¢, ~3/2 = 702.0090 ¢, ~12/11 = 156.9897 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.8818 ¢, ~12/11 = 156.9568 ¢
Minimax tuning:
- 13-odd-limit
- [[1 0 0 0 0 0⟩, [11/9 10/9 -1/3 -2/9 0 0⟩, [22/9 2/9 1/3 -4/9 0 0⟩, [22/9 2/9 -2/3 5/9 0 0⟩, [10/3 2/3 0 -1/3 0 0⟩, [14/3 -8/3 1 1/3 0 0⟩]
- unchanged-interval (eigenmonzo) basis: 2.9/5.9/7
- 15-odd-limit
- [[1 0 0 0 0 0⟩, [0 1 0 0 0 0⟩, [11/5 1/5 2/5 -2/5 0 0⟩, [11/5 1/5 -3/5 3/5 0 0⟩, [13/5 3/5 1/5 -1/5 0 0⟩, [38/5 -12/5 1/5 -1/5 0 0⟩]
- unchanged-interval (eigenmonzo) basis: 2.3.7/5
Optimal ET sequence: 15f, 22, 31, 46, 53, 77, 99, 130e
Badness (Sintel): 0.873
Projection pairs: 5 600/121 7 2662/375 11 120/11 13 1280/99 to 2.3.11/5
Zeus13[22] hobbit transversal
- 260/243, 88/81, 11/10, 44/39, 162/143, 11/9, 16/13, 320/243, 4/3, 1040/729, 13/9, 729/520, 3/2, 99/65, 44/27, 18/11, 1280/729, 16/9, 11/6, 24/13, 243/130, 2
Tinia
Subgroup: 2.3.5.7.11.13
Comma list: 66/65, 121/120, 176/175
Mapping: [⟨1 0 1 4 2 2], ⟨0 1 1 -1 1 1], ⟨0 0 -2 3 -1 -1]]
Optimal tunings:
- WE: ~2 = 1199.9251 ¢, ~3/2 = 699.2984 ¢, ~12/11 = 155.3569 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 699.2982 ¢, ~12/11 = 155.3484 ¢
Optimal ET sequence: 7, 9, 15, 22f, 24, 31
Badness (Sintel): 0.756
Artemis
Named by Graham Breed in 2011, artemis was found to be locally efficient in the higher limits among rank-3 extensions of marvel[1], although it is a weak extension. However, the alternative 13-limit extension called diana is more accurate.
Subgroup: 2.3.5.7.11
Comma list: 121/120, 225/224
Mapping: [⟨1 0 1 -3 2], ⟨0 1 1 4 1], ⟨0 0 -2 -4 -1]]
- WE: ~2 = 1201.2783 ¢, ~3/2 = 700.6174 ¢, ~11/10 = 158.4919 ¢
- error map: ⟨+1.278 -0.059 -0.123 +0.955 -5.357]
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.2777 ¢, ~11/10 = 158.3100 ¢
- error map: ⟨0.000 -1.677 -2.656 -0.955 -9.350]
Optimal ET sequence: 9, 15d, 16d, 20, 22, 31, 53, 60e, 84e, 91e, 113e, 144ee
Badness (Sintel): 0.713
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 105/104, 121/120, 196/195
Mapping: [⟨1 0 1 -3 2 -5], ⟨0 1 1 4 1 6], ⟨0 0 -2 -4 -1 -6]]
Optimal tunings:
- WE: ~2 = 1201.7896 ¢, ~3/2 = 699.7509 ¢, ~11/10 = 158.9484 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 699.1687 ¢, ~11/10 = 158.7345 ¢
Optimal ET sequence: 9, 20, 22f, 29, 31, 60e, 129cddee
Badness (Sintel): 1.04
Diana
Subgroup: 2.3.5.7.11.13
Comma list: 121/120, 225/224, 275/273
Mapping: [⟨1 0 1 -3 2 7], ⟨0 1 1 4 1 -2], ⟨0 0 -2 -4 -1 -1]]
Optimal tunings:
- WE: ~2 = 1200.9110 ¢, ~3/2 = 701.5110 ¢, ~11/10 = 159.1256 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.9717 ¢, ~11/10 = 158.7903 ¢
Optimal ET sequence: 22, 29, 31, 53, 82e, 84e, 113e
Badness (Sintel): 1.07
Oxpecker
Subgroup: 2.3.5.7.11
Comma list: 121/120, 126/125
Mapping: [⟨1 0 1 2 2], ⟨0 1 1 1 1], ⟨0 0 -2 -6 -1]]
- WE: ~2 = 1200.4124 ¢, ~3/2 = 701.1291 ¢, ~12/11 = 155.8292 ¢
- error map: ⟨+0.412 -0.414 +3.982 -1.435 -4.781]
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.2359 ¢, ~12/11 = 155.7399 ¢
- error map: ⟨0.000 -0.719 +3.442 -2.029 -5.822]
Optimal ET sequence: 7d, 8d, 15, 23de, 24d, 31, 46, 77
Badness (Sintel): 0.840
Woodpecker
Subgroup: 2.3.5.7.11.13
Comma list: 66/65, 121/120, 126/125
Mapping: [⟨1 0 1 2 2 2], ⟨0 1 1 1 1 1], ⟨0 0 -2 -6 -1 1]]
Optimal tunings:
- WE: ~2 = 1198.9113 ¢, ~3/2 = 700.9581 ¢, ~12/11 = 154.7247 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.6409 ¢, ~12/11 = 154.9115 ¢
Optimal ET sequence: 7d, 8d, 15, 23de, 24d, 31
Badness (Sintel): 1.02
Kahoupokane
Named by Tristan Bay in 2025, Kahoupokane tempers out 5120/5103 and may be described as the 29 & 46 & 53 temperament.
Subgroup: 2.3.5.7.11
Comma list: 121/120, 5120/5103
Mapping: [⟨1 0 1 11 2], ⟨0 1 1 -5 1], ⟨0 0 -2 -2 -1]]
- WE: ~2 = 1200.1911 ¢, ~3/2 = 703.1412 ¢, ~11/10 = 158.1068 ¢
- error map: ⟨+0.191 +1.377 +0.996 +0.401 -5.710]
- CWE: ~2 = 1200.000 ¢, ~3/2 = 703.0417 ¢, ~11/10 = 157.9917 ¢
- error map: ⟨0.000 +1.087 +0.744 -0.018 -6.268]
Optimal ET sequence: 7, 17c, 24d, 29, 46, 53, 82e, 99
Badness (Sintel): 2.73
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 121/120, 169/168, 352/351
Mapping: [⟨1 0 1 11 2 7], ⟨0 1 1 -5 1 -2], ⟨0 0 -2 -2 -1 -1]]
Optimal tunings:
- WE: ~2 = 1200.4435 ¢, ~3/2 = 703.1443 ¢, ~11/10 = 158.4176 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.9013 ¢, ~11/10 = 158.1657 ¢
Optimal ET sequence: 7, 17c, 24d, 29, 46, 53, 82e, 99, 181eef
Badness (Sintel): 1.27
Big brother
- For the 7-limit version, see Miscellaneous 7-limit temperaments #Nuwell.
Subgroup: 2.3.5.7.11
Comma list: 99/98, 121/120
Mapping: [⟨1 0 -5 -1 -1], ⟨0 1 3 2 2], ⟨0 0 4 1 2]]
- mapping generators: ~2, ~3, ~11/7
- WE: ~2 = 1200.6559 ¢, ~3/2 = 700.2627 ¢, ~11/7 = 771.8821 ¢
- error map: ⟨+0.656 -1.036 +0.691 +4.237 -6.372]
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.4019 ¢, ~11/7 = 771.2671 ¢
- error map: ⟨0.000 -1.553 -0.039 +3.245 -7.980]
Optimal ET sequence: 8d, 9, 14c, 17c, 22, 31, 53, 84e
Badness (Sintel): 0.609
Projection pairs: 5 2401/486, 11 98/9 to 2.3.7
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 66/65, 99/98, 121/120
Mapping: [⟨1 0 3 1 3 1], ⟨0 1 3 2 2 0], ⟨0 0 -4 -1 -2 2]]
Optimal tunings:
- WE: ~2 = 1199.0121 ¢, ~3/2 = 699.9867 ¢, ~11/7 = 771.9817 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 699.7360 ¢, ~11/7 = 773.0154 ¢
Optimal ET sequence: 8d, 9, 14c, 17c, 22f, 31, 79cf
Badness (Sintel): 0.889
Aphrodite
- For the 7-limit version, see Miscellaneous 7-limit temperaments #Squalentine.
Subgroup: 2.3.5.7.11
Comma list: 121/120, 441/440
Mapping: [⟨1 0 1 3 2], ⟨0 1 1 0 1], ⟨0 0 -4 -3 -2]]
- mapping generators: ~2, ~3, ~22/21
- WE: ~2 = 1201.0691 ¢, ~3/2 = 700.9439 ¢, ~22/21 = 78.7122 ¢
- error map: ⟨+1.069 +0.058 +1.920 -1.755 -4.591]
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8990 ¢, ~22/21 = 78.4412 ¢
- error map: ⟨0.000 -1.056 +0.820 -4.150 -7.301]
Optimal ET sequence: 14c, 15, 29, 31, 46, 60e, 77, 91e, 137de, 168dee
Badness (Sintel): 0.701
Eros
Eros fairs impressively into the 23-limit as a rank-3 temperament; not only is it fairly simple (considering this is a subgroup as complex as the full 23-limit, with many challenges) but has the pleasing property that all the harmonics are on the negative side of the last generator. Specifically, -3 to 2 fifths and -5 to 0 ~23/22's will provide odd harmonics 1–23 up to octave equivalence; you can think of this as a 6×6 grid, which is a recommendable place to start looking at its structure.
Tempering out the less accurate comma 121/120 can be seen as an implication of tempering out 441/440 (S21), 484/483 (S22), and 529/528 (S23). Therefore, characteristic of any good tuning is the prime 11 being the flattest prime, with other primes having strictly less than 5 ¢ of error.
This temperament was discovered by Scott Dakota. Note that the 17-limit extension requires the 29g val for 29edo, which has the sizes of 17/16 and 18/17 swapped.
Subgroup: 2.3.5.7.11.13
Comma list: 121/120, 196/195, 352/351
Mapping: [⟨1 0 1 3 2 7], ⟨0 1 1 0 1 -2], ⟨0 0 -4 -3 -2 -2]]
Optimal tunings:
- WE: ~2 = 1200.6419 ¢, ~3/2 = 701.8766 ¢, ~22/21 = 78.6564 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.5612 ¢, ~22/21 = 78.4778 ¢
Optimal ET sequence: 17c, 29, 31, 46, 60e, 77, 106de, 183dee
Badness (Sintel): 1.08
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 121/120, 154/153, 196/195, 352/351
Mapping: [⟨1 0 1 3 2 7 6], ⟨0 1 1 0 1 -2 -1], ⟨0 0 -4 -3 -2 -2 -5]]
Optimal tunings:
- WE: ~2 = 1200.6172 ¢, ~3/2 = 702.1026 ¢, ~22/21 = 78.7963 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7925 ¢, ~22/21 = 78.6203 ¢
Optimal ET sequence: 17cg, 29g, 31, 46, 60e, 77, 106de
Badness (Sintel): 0.931
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 121/120, 154/153, 196/195, 286/285, 352/351
Mapping: [⟨1 0 1 3 2 7 6 9], ⟨0 1 1 0 1 -2 -1 -3], ⟨0 0 -4 -3 -2 -2 -5 0]]
Optimal tunings:
- WE: ~2 = 1200.6224 ¢, ~3/2 = 702.0959 ¢, ~22/21 = 78.8004 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.6963 ¢, ~22/21 = 78.6479 ¢
Optimal ET sequence: 17cg, 29g, 31, 46, 60e, 75dfgh, 77, 106de
Badness (Sintel): 1.16
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 121/120, 154/153, 161/160, 196/195, 286/285, 352/351
Mapping: [⟨1 0 1 3 2 7 6 9 3], ⟨0 1 1 0 1 -2 -1 -3 1], ⟨0 0 -4 -3 -2 -2 -5 0 -1]]
Optimal tunings:
- WE: ~2 = 1200.7268 ¢, ~3/2 = 702.2463 ¢, ~22/21 = 78.8824 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.8010 ¢, ~23/22 = 78.7188 ¢
Optimal ET sequence: 17cg, 29g, 31, 46, 60e, 75dfgh, 77, 106de
Badness (Sintel): 1.08
Astarte
This extension is catalogued as tridecimal aphrodite in Graham Breed's temperament finder.
Subgroup: 2.3.5.7.11.13
Comma list: 121/120, 351/350, 441/440
Mapping: [⟨1 0 1 3 2 6], ⟨0 1 1 0 1 -1], ⟨0 0 -4 -3 -2 -11]]
Optimal tunings:
- WE: ~2 = 1201.0656 ¢, ~3/2 = 700.7374 ¢, ~22/21 = 78.5908 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.7005 ¢, ~22/21 = 78.3253 ¢
Optimal ET sequence: 14cf, 29ff, 31, 45ef, 46, 77, 122ee, 137def, 168deef
Badness (Sintel): 1.36
Inanna
Subgroup: 2.3.5.7.11.13
Comma list: 105/104, 121/120, 275/273
Mapping: [⟨1 0 1 3 2 1], ⟨0 1 1 0 1 2], ⟨0 0 -4 -3 -2 -7]]
Optimal tunings:
- WE: ~2 = 1201.7881 ¢, ~3/2 = 699.8166 ¢, ~22/21 = 79.7282 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 699.5926 ¢, ~22/21 = 79.3822 ¢
Optimal ET sequence: 14cf, 15, 29, 31, 45ef, 60e
Badness (Sintel): 1.01
Ishtar
Subgroup: 2.3.5.7.11.13
Comma list: 91/90, 121/120, 441/440
Mapping: [⟨1 0 1 3 2 -1], ⟨0 1 1 0 1 3], ⟨0 0 -4 -3 -2 -1]]
Optimal tunings:
- WE: ~2 = 1200.7875 ¢, ~3/2 = 703.8568 ¢, ~22/21 = 79.0096 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.7640 ¢, ~22/21 = 78.8025 ¢
Optimal ET sequence: 14cf, 15, 17c, 29, 31f, 46, 106deff, 121def
Badness (Sintel): 1.08