Biyatismic clan: Difference between revisions

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* ''[[Sonic]]'' (+55/54 or 100/99) → [[Porcupine rank-3 family #Sonic|Porcupine rank-3 family]]

* ''[[Urania]]'' (+81/80) → [[Rastmic rank-3 clan #Urania|Rastmic rank-3 clan]]

* ''[[Big brother]]'' (+99/98) → [[Nuwell family #big Brother|Nuwell 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 ==

The head of this clan is protomere, named by [[Aura]] as part of a series of names ending in ''mere'' (Greek [[wiktionary: %CE%BC%CE%AD%CF%81%CE%BF%CF%82 #Ancient Greek|μέρος]], "part") where a number of [[rastma]]s are equated with a [[syntonic comma]]; in this case, it is a single rastma, hence ''proto-''.

The systematically derived name of the temperament from ''biyatisma'' would be ''biyatic''.

[[Subgroup]]: 2.3.5.11

[[Comma list]]: 121/120

: mapping generators: ~2, ~3, ~11/10

[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1200.~~0000~~{{c}}, ~3/2 = 701.~~4578~~{{c}}, ~11/10 = 157.~~7466~~{{c}}

[[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 }}

[[Badness]] (~~Smith~~): 0.~~0297 × 10-3~~

[[Badness]] (Sintel): 0.245

== Zeus ==

Line 40:

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: Angle (11/10, 11/8) = 87.464 degrees

[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1200.0000{{c}}, ~3/2 = 702.~~1530~~{{c}}, ~11~~/10~~ = 157.~~0881~~{{c}}

[[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]]:

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{{Optimal ET sequence|legend=1| 15, 22, 31, 46, 53, 68, 77, 99, 130e }}

[[Badness]] (~~Smith~~): 0.~~400 × 10-3~~

[[Badness]] (Sintel): 0.480

[[Projection pair]]s: 5 600/121 7 2662/375 11 120/11 to 2.3.11/5

Line 68:

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Comma list: 121/120, 176/175, 351/350

~~Mapping:~~ {{~~mapping~~| 1 0 1 4 2 7 | 0 1 1 -1 1 -2 | 0 0 -2 3 -1 -1 }}

{{Mapping|legend=0| 1 0 1 4 2 7 | 0 1 1 -1 1 -2 | 0 0 -2 3 -1 -1 }}

Mapping to lattice: [{{val| 0 1 -1 2 0 -3 }}, {{val| 0 1 1 -1 1 -2 }}]

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: Angle (11/10, 11/8) = 106.7439 degrees

Optimal ~~tuning (POTE)~~: ~2 = 1200.0000{{c}}, ~3/2 = 701.~~8679~~{{c}}, ~11~~/10~~ = 156.~~9582~~{{c}}

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:

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: unchanged-interval (eigenmonzo) basis: 2.3.7/5

{{Optimal ET sequence|legend=1| 15, 22, 31, 46, 53, 77, 99, 130e }}

{{Optimal ET sequence|legend=0| 15f, 22, 31, 46, 53, 77, 99, 130e }}

Badness (~~Smith~~): 0.~~934 × 10-3~~

Badness (Sintel): 0.873

Projection pairs: 5 600/121 7 2662/375 11 120/11 13 1280/99 to 2.3.11/5

Line 100:

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Comma list: 66/65, 121/120, 176/175

~~Mapping:~~ {{~~mapping~~| 1 0 1 4 2 2 | 0 1 1 -1 1 1 | 0 0 -2 3 -1 -1 }}

{{Mapping|legend=0| 1 0 1 4 2 2 | 0 1 1 -1 1 1 | 0 0 -2 3 -1 -1 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.0000{{c}}, ~3/2 = 699.~~3420~~{{c}}, ~11~~/10~~ = 155.~~3666~~{{c}}

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}}

Badness (~~Smith~~): 0.~~808 × 10-3~~

Badness (Sintel): 0.756

== Artemis ==

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[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1200.0000{{c}}, ~3/2 = ~~699~~.~~8719~~{{c}}, ~11/10 = 158.~~3232~~{{c}}

[[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, ~~82e~~, 84e, 113e, 144ee }}

{{Optimal ET sequence|legend=1| 9, 15d, 16d, 20, 22, 31, 53, 60e, 84e, 91e, 113e, 144ee }}

[[Badness]] (Sintel): 0.713

=== 13-limit ===

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Comma list: 105/104, 121/120, 196/195

~~Mapping:~~ {{~~mapping~~| 1 0 1 -3 2 -5 | 0 1 1 4 1 6 | 0 0 -2 -4 -1 -6 }}

{{Mapping|legend=0| 1 0 1 -3 2 -5 | 0 1 1 4 1 6 | 0 0 -2 -4 -1 -6 }}

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 ~~tuning (POTE): ~2~~ = ~~1200.0000{{c}}~~, ~~~3/2 = 698.7090{{c}}~~, ~~~11/10 = 158.7117{{c~~}}

{{Optimal ET sequence|legend=0| 9, 20, 22f, 29, 31, 60e, 129cddee }}

~~{{Optimal ET sequence|legend=0| 9, 20, 22f, 29, 31 }}~~

Badness (Sintel): 1.04

=== Diana ===

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Comma list: 121/120, 225/224, 275/273

~~Mapping:~~ {{~~mapping~~| 1 0 1 -3 2 7 | 0 1 1 4 1 -2 | 0 0 -2 -4 -1 -1 }}

{{Mapping|legend=0| 1 0 1 -3 2 7 | 0 1 1 4 1 -2 | 0 0 -2 -4 -1 -1 }}

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 ~~tuning (POTE): ~2~~ = ~~1200.0000{{c}}~~, ~~~3/2 = 700.9789{{c}}~~, ~~~11/10 = 159.0048{{c~~}}

{{Optimal ET sequence|legend=0| 22, 29, 31, 53, 82e, 84e, 113e }}

~~{{Optimal ET sequence|legend=0| 22, 29, 31, 53, 82e, 84e, 113e, 166ee }}~~

Badness (Sintel): 1.07

== Oxpecker ==

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[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1200.~~000~~{{c}}, ~3/2 = ~~700~~.~~8882~~{{c}}, ~11/10 = 155.~~7756~~{{c}}

[[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 }}

[[Badness]] (~~Smith~~): 0.~~699 × 10-3~~

[[Badness]] (Sintel): 0.840

=== Woodpecker ===

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Comma list: 66/65, 121/120, 126/125

~~Mapping:~~ {{~~mapping~~| 1 0 1 2 2 2 | 0 1 1 1 1 1 | 0 0 -2 -6 -1 1 }}

{{Mapping|legend=0| 1 0 1 2 2 2 | 0 1 1 1 1 1 | 0 0 -2 -6 -1 1 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.0000{{c}}, ~3/2 = ~~701~~.~~5946~~{{c}}, ~11~~/10~~ = 154.~~8652~~{{c}}

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}}

Badness (~~Smith~~): 1.~~093 ×~~ 10~~~~-3</~~sup>~~

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

[[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|legend=0| 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=0| 7, 17c, 24d, 29, 46, 53, 82e, 99, 181eef }}

Badness (Sintel): 1.27

== Big brother ==

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[[Comma list]]: 99/98, 121/120

: mapping generators: ~2, ~3, ~11/7

{{~~Optimal ET sequence~~|~~legend~~=~~1| 8d~~, 9, ~~22, 31, 53~~ }}

[[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 }}

~~[[Badness]]: 0.507 × 10-3~~

{{Optimal ET sequence|legend=1| 8d, 9, 14c, 17c, 22, 31, 53, 84e }}

[[Projection pair]]s: 5 2401/486 11 98/9 to 2.3.7

[[Badness]] (Sintel): 0.609

[[Projection pair]]s: <code>5 2401/486, 11 98/9</code> to 2.3.7

=== 13-limit ===

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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 }}

{{Mapping|legend=0| 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 }}

~~{{Optimal ET sequence|legend=~~0~~| 8d, 9, 22f, 31 }}~~

Badness (Sintel): 0.889

== Aphrodite ==

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: mapping generators: ~2, ~3, ~22/21

Optimal tuning ~~(POTE)~~: ~2 = 1200.0000{{c}}, ~3/2 = 700.~~3200~~{{c}}, ~21~~/20~~ = 78.~~6421~~{{c}}

[[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]] (~~Smith~~): 0.~~583 × 10-3~~

[[Badness]] (Sintel): 0.701

===~~= 13-limit =~~===

=== Eros ===

~~Subgroup: 2~~.3.5.~~7.11.13~~

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.

~~Comma list:~~ 121/120, ~~351~~/~~350~~, ~~441~~/~~440~~

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.

~~Mapping: {{mapping| 1 0 1 3 2 6 | 0 1 1 0 1 -1 | 0 0 -4 -3 -2 -11 }}~~

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.

~~Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 700.1158{{c}}, ~21/20 = 78~~.~~5211{{c}}~~

~~{{Optimal ET sequence|legend=0| 14cf, 31, 45ef, 46, 77, 122ee, 137def, 168deef }}~~

~~Badness (Smith): 1.456 × 10-3~~

~~==== 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 all the generators are positive (or only 1 into the negatives in the case of the fifth) meaning it's even simpler than it might appear and~~ has the ~~pleasing property~~ of ~~all harmonics and subharmonics being "on the same side"; specifically: -3 to 1 fifths ([[2L 3s]]) and -5 to 0 ~[[23~~/~~22]]'s will get you every prime, up to octave equivalence; you can think of this as a 5 by 6 grid if you like~~ and ~~is a recommendable place to start looking at its structure. Tempering the less accurate comma [[121~~/120|S11]] can be seen as a consequence of tempering {[[441/440|S21]], [[484/483|S22]], [[529/528|S23]]} so is very natural and given its properties certainly excusable. Therefore characteristic of any good tuning is the ~11 being the most flat prime, with other primes having strictly less than 5{{cent}} of error. This temperament was first logged on x31eq by [[Scott Dakota]].

Subgroup: 2.3.5.7.11.13

Line 229:

Line 306:

Comma list: 121/120, 196/195, 352/351

~~Mapping:~~ {{~~mapping~~| 1 0 1 3 2 7 | 0 1 1 0 1 -2 | 0 0 -4 -3 -2 -2 }}

{{Mapping|legend=0| 1 0 1 3 2 7 | 0 1 1 0 1 -2 | 0 0 -4 -3 -2 -2 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.0000{{c}}, ~3/2 = 701.~~5014~~{{c}}, ~21~~/20~~ = 78.~~6143~~{{c}}

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=0| 17c, 29, 31, 46, 60e, 77, 106de, 183dee }}

Badness (~~Smith~~): 1.~~150 × 10-3~~

Badness (Sintel): 1.08

~~===== 17-limit =====~~

~~Note that this extension requires the 29g val for 29edo, which has the sizes of 17/16 and 18/17 swapped.~~

==== 17-limit ====

Subgroup: 2.3.5.7.11.13.17

Comma list: 121/120, 154/153, 196/195, 352/351

~~Mapping:~~ {{~~mapping~~| 1 0 1 3 2 7 6 | 0 1 1 0 1 -2 -1 | 0 0 -4 -3 -2 -2 -5 }}

{{Mapping|legend=0| 1 0 1 3 2 7 6 | 0 1 1 0 1 -2 -1 | 0 0 -4 -3 -2 -2 -5 }}

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~0000~~{{c}}, ~3/2 = ~~701~~.~~9299~~{{c}}, ~22/21 = 78.~~2539~~{{c}}

* WE: ~2 = 1200.6172{{c}}, ~3/2 = 702.1026{{c}}, ~22/21 = 78.7963{{c}}

* 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

* Smith: 0.979 × 10-3

* Sintel: 0.931

===== 19-limit =====

==== 19-limit ====

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 121/120, 154/153, 196/195, 286/285, 352/351

~~Mapping:~~ {{~~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 }}

{{Mapping|legend=0| 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:

* ~~CTE~~: ~2 = 1200.~~0000~~{{c}}, ~3/2 = ~~701~~.~~5642~~{{c}}, ~22/21 = 78.~~2353~~{{c}}

* WE: ~2 = 1200.6224{{c}}, ~3/2 = 702.0959{{c}}, ~22/21 = 78.8004{{c}}

* 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

* Smith: 1.13 × 10-3

* Sintel: 1.~~159~~

===== 23-limit =====

==== 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:~~ {{~~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 }}

{{Mapping|legend=0| 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:

* ~~CTE~~: ~2 = 1200.~~0000~~{{c}}, ~3/2 = ~~701~~.~~7115~~{{c}}, ~23/22 = 78.~~2054~~{{c}}

* WE: ~2 = 1200.7268{{c}}, ~3/2 = 702.2463{{c}}, ~22/21 = 78.8824{{c}}

* 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

* Smith: 0.939 × 10-3

* Sintel: 1.~~084~~

=== 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|legend=0| 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

Comma list: 105/104, 121/120, 275/273

~~Mapping:~~ {{~~mapping~~| 1 0 1 3 2 1 | 0 1 1 0 1 2 | 0 0 -4 -3 -2 -7 }}

{{Mapping|legend=0| 1 0 1 3 2 1 | 0 1 1 0 1 2 | 0 0 -4 -3 -2 -7 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.0000{{c}}, ~3/2 = ~~698~~.~~7754~~{{c}}, ~21~~/20~~ = 79.~~6096~~{{c}}

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}}

Badness (Sintel): 1.~~077 × 10-3~~

Badness (Sintel): 1.01

==== Ishtar ====

=== Ishtar ===

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 121/120, 441/440

~~Mapping:~~ {{~~mapping~~| 1 0 1 3 2 -1 | 0 1 1 0 1 3 | 0 0 -4 -3 -2 -1 }}

{{Mapping|legend=0| 1 0 1 3 2 -1 | 0 1 1 0 1 3 | 0 0 -4 -3 -2 -1 }}

Optimal ~~tuning (POTE)~~: ~2 = 1200.0000{{c}}, ~3/2 = 703.~~3952~~{{c}}, ~21~~/20~~ = 78.~~9578~~{{c}}

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=0| 14cf, 15, 17c, 29, 31f, 46, 106deff, 121def }}

Badness (Sintel): 1.~~151 × 10-3~~

Badness (Sintel): 1.08

== ~~Notes~~ ==

== References ==

~~[[Category:Temperament clans]]~~

[[Category:Biyatismic clan| ]]

[[Category:Biyatismic| ]]

[[Category:~~Rank~~ 3]]

[[Category:Temperament clans]]

[[Category:Catalogs of rank-3 temperaments]]

@@ Line 5: / Line 5: @@
 * ''[[Sonic]]'' (+55/54 or 100/99) → [[Porcupine rank-3 family #Sonic|Porcupine rank-3 family]]
 * ''[[Urania]]'' (+81/80) → [[Rastmic rank-3 clan #Urania|Rastmic rank-3 clan]]
-* ''[[Big brother]]'' (+99/98) → [[Nuwell family #big Brother|Nuwell 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 ==
+The head of this clan is protomere, named by [[Aura]] as part of a series of names ending in ''mere'' (Greek [[wiktionary: %CE%BC%CE%AD%CF%81%CE%BF%CF%82 #Ancient Greek|μέρος]], "part") where a number of [[rastma]]s are equated with a [[syntonic comma]]; in this case, it is a single rastma, hence ''proto-''.
+The systematically derived name of the temperament from ''biyatisma'' would be ''biyatic''.
 [[Subgroup]]: 2.3.5.11
 [[Comma list]]: 121/120
-{{Mapping|legend=2| 1 0 1 2 | 0 1 1 1 | 0 0 -2 -1 }}
+{{Mapping|legend=3| 1 0 1 2 | 0 1 1 1 | 0 0 -2 -1 }}
 : mapping generators: ~2, ~3, ~11/10
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 701.4578{{c}}, ~11/10 = 157.7466{{c}}
+[[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 }}
-[[Badness]] (Smith): 0.0297 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 0.245
 == Zeus ==
@@ Line 40: / Line 47: @@
 : Angle (11/10, 11/8) = 87.464 degrees
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 702.1530{{c}}, ~11/10 = 157.0881{{c}}
+[[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]]:
@@ Line 49: / Line 60: @@
 {{Optimal ET sequence|legend=1| 15, 22, 31, 46, 53, 68, 77, 99, 130e }}
-[[Badness]] (Smith): 0.400 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 0.480
 [[Projection pair]]s: 5 600/121 7 2662/375 11 120/11 to 2.3.11/5
@@ Line 68: / Line 79: @@
 Comma list: 121/120, 176/175, 351/350
-Mapping: {{mapping| 1 0 1 4 2 7 | 0 1 1 -1 1 -2 | 0 0 -2 3 -1 -1 }}
+{{Mapping|legend=0| 1 0 1 4 2 7 | 0 1 1 -1 1 -2 | 0 0 -2 3 -1 -1 }}
 Mapping to lattice: [{{val| 0 1 -1 2 0 -3 }}, {{val| 0 1 1 -1 1 -2 }}]
@@ Line 76: / Line 87: @@
 : Angle (11/10, 11/8) = 106.7439 degrees
-Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 701.8679{{c}}, ~11/10 = 156.9582{{c}}
+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:
@@ Line 86: / Line 99: @@
 : unchanged-interval (eigenmonzo) basis: 2.3.7/5
-{{Optimal ET sequence|legend=1| 15, 22, 31, 46, 53, 77, 99, 130e }}
+{{Optimal ET sequence|legend=0| 15f, 22, 31, 46, 53, 77, 99, 130e }}
-Badness (Smith): 0.934 × 10<sup>-3</sup>
+Badness (Sintel): 0.873
 Projection pairs: 5 600/121 7 2662/375 11 120/11 13 1280/99 to 2.3.11/5
@@ Line 100: / Line 113: @@
 Comma list: 66/65, 121/120, 176/175
-Mapping: {{mapping| 1 0 1 4 2 2 | 0 1 1 -1 1 1 | 0 0 -2 3 -1 -1 }}
+{{Mapping|legend=0| 1 0 1 4 2 2 | 0 1 1 -1 1 1 | 0 0 -2 3 -1 -1 }}
-Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 699.3420{{c}}, ~11/10 = 155.3666{{c}}
+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=0| 7, 9, 15, 22f, 24, 31 }}
-Badness (Smith): 0.808 × 10<sup>-3</sup>
+Badness (Sintel): 0.756
 == Artemis ==
@@ Line 117: / Line 132: @@
 {{Mapping|legend=1| 1 0 1 -3 2 | 0 1 1 4 1 | 0 0 -2 -4 -1 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 699.8719{{c}}, ~11/10 = 158.3232{{c}}
+[[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, 82e, 84e, 113e, 144ee }}
+{{Optimal ET sequence|legend=1| 9, 15d, 16d, 20, 22, 31, 53, 60e, 84e, 91e, 113e, 144ee }}
+[[Badness]] (Sintel): 0.713
 === 13-limit ===
@@ Line 126: / Line 147: @@
 Comma list: 105/104, 121/120, 196/195
-Mapping: {{mapping| 1 0 1 -3 2 -5 | 0 1 1 4 1 6 | 0 0 -2 -4 -1 -6 }}
+{{Mapping|legend=0| 1 0 1 -3 2 -5 | 0 1 1 4 1 6 | 0 0 -2 -4 -1 -6 }}
+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 tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 698.7090{{c}}, ~11/10 = 158.7117{{c}}
+{{Optimal ET sequence|legend=0| 9, 20, 22f, 29, 31, 60e, 129cddee }}
-{{Optimal ET sequence|legend=0| 9, 20, 22f, 29, 31 }}
+Badness (Sintel): 1.04
 === Diana ===
@@ Line 137: / Line 162: @@
 Comma list: 121/120, 225/224, 275/273
-Mapping: {{mapping| 1 0 1 -3 2 7 | 0 1 1 4 1 -2 | 0 0 -2 -4 -1 -1 }}
+{{Mapping|legend=0| 1 0 1 -3 2 7 | 0 1 1 4 1 -2 | 0 0 -2 -4 -1 -1 }}
+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 tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 700.9789{{c}}, ~11/10 = 159.0048{{c}}
+{{Optimal ET sequence|legend=0| 22, 29, 31, 53, 82e, 84e, 113e }}
-{{Optimal ET sequence|legend=0| 22, 29, 31, 53, 82e, 84e, 113e, 166ee }}
+Badness (Sintel): 1.07
 == Oxpecker ==
@@ Line 150: / Line 179: @@
 {{Mapping|legend=1| 1 0 1 2 2 | 0 1 1 1 1 | 0 0 -2 -6 -1 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 700.8882{{c}}, ~11/10 = 155.7756{{c}}
+[[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 }}
-[[Badness]] (Smith): 0.699 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 0.840
 === Woodpecker ===
@@ Line 161: / Line 194: @@
 Comma list: 66/65, 121/120, 126/125
-Mapping: {{mapping| 1 0 1 2 2 2 | 0 1 1 1 1 1 | 0 0 -2 -6 -1 1 }}
+{{Mapping|legend=0| 1 0 1 2 2 2 | 0 1 1 1 1 1 | 0 0 -2 -6 -1 1 }}
-Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 701.5946{{c}}, ~11/10 = 154.8652{{c}}
+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 (Smith): 1.093 × 10<sup>-3</sup>
+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|legend=0| 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=0| 7, 17c, 24d, 29, 46, 53, 82e, 99, 181eef }}
+Badness (Sintel): 1.27
 == Big brother ==
@@ Line 176: / Line 245: @@
 [[Comma list]]: 99/98, 121/120
-{{Mapping|legend=1| 1 0 3 1 3 | 0 1 3 2 2 | 0 0 -4 -1 -2 }}
+{{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 ET sequence|legend=1| 8d, 9, 22, 31, 53 }}
+[[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 }}
-[[Badness]]: 0.507 × 10<sup>-3</sup>
+{{Optimal ET sequence|legend=1| 8d, 9, 14c, 17c, 22, 31, 53, 84e }}
-[[Projection pair]]s: 5 2401/486 11 98/9 to 2.3.7
+[[Badness]] (Sintel): 0.609
+[[Projection pair]]s: <code>5 2401/486, 11 98/9</code> to 2.3.7
 === 13-limit ===
@@ Line 189: / Line 265: @@
 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 }}
+{{Mapping|legend=0| 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 }}
-{{Optimal ET sequence|legend=0| 8d, 9, 22f, 31 }}
+Badness (Sintel): 0.889
 == Aphrodite ==
@@ Line 203: / Line 285: @@
 : mapping generators: ~2, ~3, ~22/21
-Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 700.3200{{c}}, ~21/20 = 78.6421{{c}}
+[[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]] (Smith): 0.583 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 0.701
-==== 13-limit ====
+=== Eros ===
-Subgroup: 2.3.5.7.11.13
+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.
-Comma list: 121/120, 351/350, 441/440
+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.
-Mapping: {{mapping| 1 0 1 3 2 6 | 0 1 1 0 1 -1 | 0 0 -4 -3 -2 -11 }}
+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.
-Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 700.1158{{c}}, ~21/20 = 78.5211{{c}}
-{{Optimal ET sequence|legend=0| 14cf, 31, 45ef, 46, 77, 122ee, 137def, 168deef }}
-Badness (Smith): 1.456 × 10<sup>-3</sup>
-==== 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 all the generators are positive (or only 1 into the negatives in the case of the fifth) meaning it's even simpler than it might appear and has the pleasing property of all harmonics and subharmonics being "on the same side"; specifically: -3 to 1 fifths ([[2L 3s]]) and -5 to 0 ~[[23/22]]'s will get you every prime, up to octave equivalence; you can think of this as a 5 by 6 grid if you like and is a recommendable place to start looking at its structure. Tempering the less accurate comma [[121/120|S11]] can be seen as a consequence of tempering {[[441/440|S21]], [[484/483|S22]], [[529/528|S23]]} so is very natural and given its properties certainly excusable. Therefore characteristic of any good tuning is the ~11 being the most flat prime, with other primes having strictly less than 5{{cent}} of error. This temperament was first logged on x31eq by [[Scott Dakota]].
 Subgroup: 2.3.5.7.11.13
@@ Line 229: / Line 306: @@
 Comma list: 121/120, 196/195, 352/351
-Mapping: {{mapping| 1 0 1 3 2 7 | 0 1 1 0 1 -2 | 0 0 -4 -3 -2 -2 }}
+{{Mapping|legend=0| 1 0 1 3 2 7 | 0 1 1 0 1 -2 | 0 0 -4 -3 -2 -2 }}
-Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 701.5014{{c}}, ~21/20 = 78.6143{{c}}
+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=0| 17c, 29, 31, 46, 60e, 77, 106de, 183dee }}
-Badness (Smith): 1.150 × 10<sup>-3</sup>
+Badness (Sintel): 1.08
-===== 17-limit =====
-Note that this extension requires the 29g val for 29edo, which has the sizes of 17/16 and 18/17 swapped.
+==== 17-limit ====
 Subgroup: 2.3.5.7.11.13.17
 Comma list: 121/120, 154/153, 196/195, 352/351
-Mapping: {{mapping| 1 0 1 3 2 7 6 | 0 1 1 0 1 -2 -1 | 0 0 -4 -3 -2 -2 -5 }}
+{{Mapping|legend=0| 1 0 1 3 2 7 6 | 0 1 1 0 1 -2 -1 | 0 0 -4 -3 -2 -2 -5 }}
 Optimal tunings:
-* CTE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9299{{c}}, ~22/21 = 78.2539{{c}}
+* WE: ~2 = 1200.6172{{c}}, ~3/2 = 702.1026{{c}}, ~22/21 = 78.7963{{c}}
 * 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
-* Smith: 0.979 × 10<sup>-3</sup>
-* Sintel: 0.931
-===== 19-limit =====
+==== 19-limit ====
 Subgroup: 2.3.5.7.11.13.17.19
 Comma list: 121/120, 154/153, 196/195, 286/285, 352/351
-Mapping: {{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 }}
+{{Mapping|legend=0| 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:
-* CTE: ~2 = 1200.0000{{c}}, ~3/2 = 701.5642{{c}}, ~22/21 = 78.2353{{c}}
+* WE: ~2 = 1200.6224{{c}}, ~3/2 = 702.0959{{c}}, ~22/21 = 78.8004{{c}}
 * 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
-* Smith: 1.13 × 10<sup>-3</sup>
-* Sintel: 1.159
-===== 23-limit =====
+==== 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: {{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 }}
+{{Mapping|legend=0| 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:
-* CTE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7115{{c}}, ~23/22 = 78.2054{{c}}
+* WE: ~2 = 1200.7268{{c}}, ~3/2 = 702.2463{{c}}, ~22/21 = 78.8824{{c}}
 * 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
-* Smith: 0.939 × 10<sup>-3</sup>
-* Sintel: 1.084
+=== 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|legend=0| 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
 Comma list: 105/104, 121/120, 275/273
-Mapping: {{mapping| 1 0 1 3 2 1 | 0 1 1 0 1 2 | 0 0 -4 -3 -2 -7 }}
+{{Mapping|legend=0| 1 0 1 3 2 1 | 0 1 1 0 1 2 | 0 0 -4 -3 -2 -7 }}
-Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 698.7754{{c}}, ~21/20 = 79.6096{{c}}
+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=0| 14cf, 15, 29, 31, 45ef, 60e }}
-Badness (Sintel): 1.077 × 10<sup>-3</sup>
+Badness (Sintel): 1.01
-==== Ishtar ====
+=== Ishtar ===
 Subgroup: 2.3.5.7.11.13
 Comma list: 91/90, 121/120, 441/440
-Mapping: {{mapping| 1 0 1 3 2 -1 | 0 1 1 0 1 3 | 0 0 -4 -3 -2 -1 }}
+{{Mapping|legend=0| 1 0 1 3 2 -1 | 0 1 1 0 1 3 | 0 0 -4 -3 -2 -1 }}
-Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 703.3952{{c}}, ~21/20 = 78.9578{{c}}
+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=0| 14cf, 15, 17c, 29, 31f, 46, 106deff, 121def }}
-Badness (Sintel): 1.151 × 10<sup>-3</sup>
+Badness (Sintel): 1.08
-== Notes ==
+== References ==
-[[Category:Temperament clans]]
 [[Category:Biyatismic clan| ]] <!-- main article -->
 [[Category:Biyatismic| ]] <!-- key article -->
-[[Category:Rank 3]]
+[[Category:Temperament clans]]
+[[Category:Catalogs of rank-3 temperaments]]

Biyatismic clan: Difference between revisions

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