Biyatismic clan: Difference between revisions

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The '''biyatismic clan''' of [[~~Rank~~-3 temperament|rank-3]] [[~~Temperament~~|temperaments]] [[~~Tempering~~ out|tempers out]] the [[~~biyatisma]],~~ 121/120 ~~= {{monzo| -3 -1 -1 0 2 }}~~.

The '''biyatismic clan''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the biyatisma, [[121/120]].

Temperaments discussed elsewhere are:

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

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

* ''[[Urania]]'' (+81/80) → [[~~Didymus~~ rank ~~three family~~ #Urania|~~Didymus~~ 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 ==

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

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

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.6628{{c}}, ~3/2 = 701.8452{{c}}, ~11/10 = 157.8337{{c}}

[[~~Optimal tuning~~]] ([[~~POTE~~]]): ~2 = ~~1\1~~, ~3/2 = 701.~~4578~~, ~11~~/10~~ = 157.~~7466~~

: [[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]]: 0.~~0297 × 10-3~~

[[Badness]] (Sintel): 0.245

== Zeus ==

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

[[Optimal tuning]] ([[~~POTE~~]]): ~2 = ~~1\1~~, ~3/2 = 702.~~1530~~, ~11/10 = 157.~~0881~~

[[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]]:

* [[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 }}]

: [[~~Eigenmonzo~~ basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.9/5.9/7

: [[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 }}

[[Badness]]: 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

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~3/2 = 701.~~8679~~, ~11~~/10~~ = 156.~~9582~~

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:

* 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 }}]

: ~~eigenmonzo (~~unchanged-interval) basis: 2.9/5.9/7

: unchanged-interval (eigenmonzo) basis: 2.9/5.9/7

* 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 }}]

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

: 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: 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

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~3/2 = 699.~~3420~~, ~11/10 = 155.~~3666~~

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: 0.~~808 × 10-3~~

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]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19673.html Yahoo! Tuning Group | ''Artemis and friends'']</ref>, although it is a [[weak extension]]. However, the alternative 13-limit extension called diana is more accurate.

[[Subgroup]]: 2.3.5.7.11

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[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1\1, ~3/2 = ~~699~~.~~8719~~, ~11/10 = 158.~~3232~~

[[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, 60e, 84e, 91e, 113e, 144ee }}

~~{{Optimal ET sequence|legend=1| 9, 15d, 16d, 20, 22, 31, 53, 82e, 84e, 113e, 144ee }}~~

[[Badness]] (Sintel): 0.713

=== 13-limit ===

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~3/2 = ~~698~~.~~7090~~, ~11/10 = 158.~~7117~~

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=0| 9, 20, 22f, 29, 31, 60e, 129cddee }}

~~{{Optimal ET sequence|legend=~~1~~| 9, 20, 22f, 29, 31 }}~~

Badness (Sintel): 1.04

=== Diana ===

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~3/2 = 700.~~9789~~, ~11/10 = ~~159~~.~~0048~~

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=1| 22, 29, 31, 53, 82e, 84e, 113e~~, 166ee~~ }}

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

Badness (Sintel): 1.07

== Oxpecker ==

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[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1\1, ~3/2 = ~~700~~.~~8882~~, ~11~~/10~~ = 155.~~7756~~

[[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]]: 0.~~699 × 10-3~~

[[Badness]] (Sintel): 0.840

=== Woodpecker ===

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~3/2 = ~~701~~.~~5946~~, ~11/10 = 154.~~8652~~

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=~~1~~| 7d, 8d, 15, 23de, 24d, 31 }}~~

Badness (Sintel): 1.02

~~Badness: 1~~.~~093 × 10-3~~

== Kahoupokane ==

Named by [[Tristan Bay]] in 2025, Kahoupokane tempers out [[5120/5103]] and may be described as the {{nowrap| 29 & 46 & 53 }} temperament.

== ~~Aphrodite~~ ==

[[Subgroup]]: 2.3.5.7.11

~~Aphrodite tempers out the squalentine comma, 64827/64000, in the 7~~-~~limit~~. ~~Its generators can be taken to be~~ 2, 3~~, and 21~~/20, ~~and it equates (21~~/~~20)3 with 8/7~~.

[[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 }}

=~~== 7-limit (squalentine) ===~~

{{Optimal ET sequence|legend=1| 7, 17c, 24d, 29, 46, 53, 82e, 99 }}

~~[[Subgroup]]: 2.3.5.~~7

[[~~Comma list~~]]: ~~64827/64000~~

[[Badness]] (Sintel): 2.73

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

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

: ~~Mapping generators: ~2~~, ~3, ~~~21~~/20

Comma list: 121/120, 169/168, 352/351

~~[[Optimal tuning]] ([[POTE]])~~: ~2 = 1\1~~, ~3/~~2 ~~= 700.2144, ~21/20 = 78.5694~~

Mapping: {{mapping| 1 0 1 11 2 7 | 0 1 1 -5 1 -2 | 0 0 -2 -2 -1 -1 }}

{{~~Optimal ET sequence|legend~~=~~1| 14c~~, 15, 29, ~~31, 46, 60, 77, 91, 122, 137d, 168d~~ }}

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}}

~~[[Badness]]:~~ 0~~.943 × 10-3~~

{{Optimal ET sequence|legend=0| 7, 17c, 24d, 29, 46, 53, 82e, 99, 181eef }}

[[~~Projection pair~~]]~~s: 5 320000/64827 7 64000/9261 to 2.3~~.~~7/5~~

Badness (Sintel): 1.27

== Big brother ==

: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Nuwell]].''

~~=== 11-limit ===~~

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 121/120~~, 441/440~~

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

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

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

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

[[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 ~~tuning (POTE): ~2~~ = 1\1, ~~~3/2 = 700.3200~~, ~~~21/20 = 78.6421~~

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

~~{{Optimal ET sequence|legend=1| 14c, 15, 29, 31, 46, 60e, 77, 91e, 137de, 168dee }}~~

[[Badness]] (Sintel): 0.609

[[~~Badness~~]]: ~~0.583 × 10~~<~~sup~~>-3</~~sup~~>

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

==== 13-limit ====

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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 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 }}

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

{{Optimal ET sequence|legend=1| 14c, 15, 29, 31, 46, 60e, 77, 91e, 137de, 168dee }}

~~Optimal tuning~~ (~~POTE~~): ~~~2 = 1\1, ~3/2 = 700~~.~~1158, ~21/20 = 78.5211~~

[[Badness]] (Sintel): 0.701

~~{{Optimal ET sequence|legend~~=~~1| 14cf, 31, 45ef, 46, 77~~, ~~122ee~~, ~~137def~~, ~~168deef }}~~

=== 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.

~~Badness:~~ 1.~~456 × 10-3~~

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.

~~==== Eros ====~~

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.

~~Eros fairs impressively into~~ the 23-limit ~~as a rank 3 temperament; not only is it fairly simple (considering this 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 224:

Line 304:

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~3/2 = 701.~~5014~~, ~21~~/20~~ = 78.~~6143~~

Optimal tunings:

* WE: ~2 = 1200.6419{{c}}, ~3/2 = 701.8766{{c}}, ~22/21 = 78.6564{{c}}

{{~~Optimal ET sequence|legend~~=~~1| 17c~~, ~~29, 31, 46, 60e, 77, 106de, 183dee~~ }}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.5612{{c}}, ~22/21 = 78.4778{{c}}

~~Badness: 1.150 × 10-3~~

{{Optimal ET sequence|legend=0| 17c, 29, 31, 46, 60e, 77, 106de, 183dee }}

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

Badness (Sintel): 1.08

~~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

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Optimal tunings:

* ~~CTE~~: ~2 = ~~1\1~~, ~3/2 = ~~701~~.~~9299~~, ~22/21 = 78.~~2539~~

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

* CWE: ~2 = ~~1\1~~, ~3/2 = 701.7925, ~22/21 = 78.6203

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7925{{c}}, ~22/21 = 78.6203{{c}}

~~Optimal ET sequence:~~ {{Optimal ET sequence| 17cg, 29g, 31, 46, 60e, 77, 106de }}

{{Optimal ET sequence|legend=0| 17cg, 29g, 31, 46, 60e, 77, 106de }}

Badness:

Badness (Sintel): 0.931

* Smith: 0.979 × 10-3

* Dirichlet: 0.931

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

==== 19-limit ====

Subgroup: 2.3.5.7.11.13.17.19

Line 257:

Line 335:

Optimal tunings:

* ~~CTE~~: ~2 = ~~1\1~~, ~3/2 = ~~701~~.~~5642~~, ~22/21 = 78.~~2353~~

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

* CWE: ~2 = ~~1\1~~, ~3/2 = 701.6963, ~22/21 = 78.6479

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.6963{{c}}, ~22/21 = 78.6479{{c}}

~~Optimal ET sequence:~~ {{Optimal ET sequence| 17cg, 29g, 31, 46, 60e, 75dfgh, 77, 106de }}

{{Optimal ET sequence|legend=0| 17cg, 29g, 31, 46, 60e, 75dfgh, 77, 106de }}

Badness:

Badness (Sintel): 1.16

* Smith: 1.13 × 10-3

* Dirichlet: 1.~~159~~

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

==== 23-limit ====

Subgroup: 2.3.5.7.11.13.17.19.23

Line 274:

Line 350:

Optimal tunings:

* ~~CTE~~: ~2 = ~~1\1~~, ~3 = ~~1901~~.~~7115~~, ~23/22 = 78.~~2054~~

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

* CWE: ~2 = ~~1\1~~, ~3 = ~~1901~~.8010, ~23/22 = 78.7188

* 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 (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: {{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:~~ {{Optimal ET sequence| ~~17cg~~, ~~29g~~, 31, 46, ~~60e~~, ~~75dfgh~~, 77, ~~106de~~ }}

{{Optimal ET sequence|legend=0| 14cf, 29ff, 31, 45ef, 46, 77, 122ee, 137def, 168deef }}

Badness:

Badness (Sintel): 1.36

* Smith: 0.939 × 10-3

* Dirichlet: 1.~~084~~

==== Inanna ====

=== Inanna ===

Subgroup: 2.3.5.7.11.13

Line 290:

Line 381:

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~3/2 = ~~698~~.~~7754~~, ~21/20 = 79.~~6096~~

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: 1.~~077 × 10-3~~

Badness (Sintel): 1.01

==== Ishtar ====

=== Ishtar ===

Subgroup: 2.3.5.7.11.13

Line 303:

Line 396:

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~3/2 = 703.~~3952~~, ~21/20 = 78.~~9578~~

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 }}

~~{{Optimal ET sequence|legend=~~1~~| 14cf, 15, 17c, 29, 31f, 46, 106deff, 121def }}~~

Badness (Sintel): 1.08

~~Badness: 1.151 × 10-3~~

== References ==

[[Category:Temperament clans]]

[[Category:Biyatismic clan| ]]

~~[[Category:Biyatismic| ]] ~~

[[Category:Rank 3]]

@@ Line 1: / Line 1: @@
-The '''biyatismic clan''' of [[Rank-3 temperament|rank-3]] [[Temperament|temperaments]] [[Tempering out|tempers out]] the [[biyatisma]], 121/120 = {{monzo| -3 -1 -1 0 2 }}.
+{{Technical data page}}
+The '''biyatismic clan''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the biyatisma, [[121/120]].
 Temperaments discussed elsewhere are:
-* ''[[Sonic]]'' (+55/54 or 100/99) → [[Porcupine rank three family #Sonic|Porcupine rank-3 family]]
+* ''[[Sonic]]'' (+55/54 or 100/99) → [[Porcupine rank-3 family #Sonic|Porcupine rank-3 family]]
-* ''[[Urania]]'' (+81/80) → [[Didymus rank three family #Urania|Didymus 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 ==
@@ Line 15: / Line 15: @@
 {{Mapping|legend=2| 1 0 1 2 | 0 1 1 1 | 0 0 -2 -1 }}
+: mapping generators: ~2, ~3, ~11/10
-: Mapping generators: ~2, ~3, ~11/10
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.6628{{c}}, ~3/2 = 701.8452{{c}}, ~11/10 = 157.8337{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.4578, ~11/10 = 157.7466
+: [[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]]: 0.0297 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 0.245
 == Zeus ==
@@ Line 40: / Line 43: @@
 : Angle (11/10, 11/8) = 87.464 degrees
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.1530, ~11/10 = 157.0881
+[[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]]:
 * [[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 }}]
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/5.9/7
+: [[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 }}
-[[Badness]]: 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 76: / Line 83: @@
 : Angle (11/10, 11/8) = 106.7439 degrees
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.8679, ~11/10 = 156.9582
+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:
 * 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 }}]
-: eigenmonzo (unchanged-interval) basis: 2.9/5.9/7
+: unchanged-interval (eigenmonzo) basis: 2.9/5.9/7
 * 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 }}]
-: eigenmonzo (unchanged-interval) basis: 2.3.7/5
+: 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: 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 102: / Line 111: @@
 Mapping: {{mapping| 1 0 1 4 2 2 | 0 1 1 -1 1 1 | 0 0 -2 3 -1 -1 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 699.3420, ~11/10 = 155.3666
+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=1| 7, 9, 15, 22f, 24, 31 }}
+{{Optimal ET sequence|legend=0| 7, 9, 15, 22f, 24, 31 }}
-Badness: 0.808 × 10<sup>-3</sup>
+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]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19673.html Yahoo! Tuning Group | ''Artemis and friends'']</ref>, although it is a [[weak extension]]. However, the alternative 13-limit extension called diana is more accurate.
 [[Subgroup]]: 2.3.5.7.11
@@ Line 115: / Line 128: @@
 {{Mapping|legend=1| 1 0 1 -3 2 | 0 1 1 4 1 | 0 0 -2 -4 -1 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 699.8719, ~11/10 = 158.3232
+[[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, 60e, 84e, 91e, 113e, 144ee }}
-{{Optimal ET sequence|legend=1| 9, 15d, 16d, 20, 22, 31, 53, 82e, 84e, 113e, 144ee }}
+[[Badness]] (Sintel): 0.713
 === 13-limit ===
@@ Line 126: / Line 145: @@
 Mapping: {{mapping| 1 0 1 -3 2 -5 | 0 1 1 4 1 6 | 0 0 -2 -4 -1 -6 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 698.7090, ~11/10 = 158.7117
+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=0| 9, 20, 22f, 29, 31, 60e, 129cddee }}
-{{Optimal ET sequence|legend=1| 9, 20, 22f, 29, 31 }}
+Badness (Sintel): 1.04
 === Diana ===
@@ Line 137: / Line 160: @@
 Mapping: {{mapping| 1 0 1 -3 2 7 | 0 1 1 4 1 -2 | 0 0 -2 -4 -1 -1 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.9789, ~11/10 = 159.0048
+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=1| 22, 29, 31, 53, 82e, 84e, 113e, 166ee }}
+{{Optimal ET sequence|legend=0| 22, 29, 31, 53, 82e, 84e, 113e }}
+Badness (Sintel): 1.07
 == Oxpecker ==
@@ Line 148: / Line 175: @@
 {{Mapping|legend=1| 1 0 1 2 2 | 0 1 1 1 1 | 0 0 -2 -6 -1 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 700.8882, ~11/10 = 155.7756
+[[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]]: 0.699 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 0.840
 === Woodpecker ===
@@ Line 161: / Line 192: @@
 Mapping: {{mapping| 1 0 1 2 2 2 | 0 1 1 1 1 1 | 0 0 -2 -6 -1 1 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.5946, ~11/10 = 154.8652
+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 }}
-{{Optimal ET sequence|legend=1| 7d, 8d, 15, 23de, 24d, 31 }}
+Badness (Sintel): 1.02
-Badness: 1.093 × 10<sup>-3</sup>
+== Kahoupokane ==
+Named by [[Tristan Bay]] in 2025, Kahoupokane tempers out [[5120/5103]] and may be described as the {{nowrap| 29 & 46 & 53 }} temperament.
-== Aphrodite ==
+[[Subgroup]]: 2.3.5.7.11
-Aphrodite tempers out the squalentine comma, 64827/64000, in the 7-limit. Its generators can be taken to be 2, 3, and 21/20, and it equates (21/20)<sup>3</sup> with 8/7.
+[[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 }}
-=== 7-limit (squalentine) ===
+{{Optimal ET sequence|legend=1| 7, 17c, 24d, 29, 46, 53, 82e, 99 }}
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 64827/64000
+[[Badness]] (Sintel): 2.73
-{{Mapping|legend=1| 1 0 1 3 | 0 1 1 0 | 0 0 -4 -3 }}
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-: Mapping generators: ~2, ~3, ~21/20
+Comma list: 121/120, 169/168, 352/351
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 700.2144, ~21/20 = 78.5694
+Mapping: {{mapping| 1 0 1 11 2 7 | 0 1 1 -5 1 -2 | 0 0 -2 -2 -1 -1 }}
-{{Optimal ET sequence|legend=1| 14c, 15, 29, 31, 46, 60, 77, 91, 122, 137d, 168d }}
+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}}
-[[Badness]]: 0.943 × 10<sup>-3</sup>
+{{Optimal ET sequence|legend=0| 7, 17c, 24d, 29, 46, 53, 82e, 99, 181eef }}
-[[Projection pair]]s: 5 320000/64827 7 64000/9261 to 2.3.7/5
+Badness (Sintel): 1.27
+== Big brother ==
+: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Nuwell]].''
-=== 11-limit ===
 [[Subgroup]]: 2.3.5.7.11
-[[Comma list]]: 121/120, 441/440
+[[Comma list]]: 99/98, 121/120
-{{Mapping|legend=1| 1 0 1 3 2 | 0 1 1 0 1 | 0 0 -4 -3 -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
-: Mapping generators: ~2, ~3, ~22/21
+[[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 tuning (POTE): ~2 = 1\1, ~3/2 = 700.3200, ~21/20 = 78.6421
+{{Optimal ET sequence|legend=1| 8d, 9, 14c, 17c, 22, 31, 53, 84e }}
-{{Optimal ET sequence|legend=1| 14c, 15, 29, 31, 46, 60e, 77, 91e, 137de, 168dee }}
+[[Badness]] (Sintel): 0.609
-[[Badness]]: 0.583 × 10<sup>-3</sup>
+[[Projection pair]]s: <code>5 2401/486, 11 98/9</code> to 2.3.7
-==== 13-limit ====
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 121/120, 351/350, 441/440
+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 }}
-Mapping: {{mapping| 1 0 1 3 2 6 | 0 1 1 0 1 -1 | 0 0 -4 -3 -2 -11 }}
+{{Optimal ET sequence|legend=1| 14c, 15, 29, 31, 46, 60e, 77, 91e, 137de, 168dee }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.1158, ~21/20 = 78.5211
+[[Badness]] (Sintel): 0.701
-{{Optimal ET sequence|legend=1| 14cf, 31, 45ef, 46, 77, 122ee, 137def, 168deef }}
+=== 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.
-Badness: 1.456 × 10<sup>-3</sup>
+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.
-==== Eros ====
+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.
-Eros fairs impressively into the 23-limit as a rank 3 temperament; not only is it fairly simple (considering this 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 224: / Line 304: @@
 Mapping: {{mapping| 1 0 1 3 2 7 | 0 1 1 0 1 -2 | 0 0 -4 -3 -2 -2 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.5014, ~21/20 = 78.6143
+Optimal tunings:
+* WE: ~2 = 1200.6419{{c}}, ~3/2 = 701.8766{{c}}, ~22/21 = 78.6564{{c}}
-{{Optimal ET sequence|legend=1| 17c, 29, 31, 46, 60e, 77, 106de, 183dee }}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.5612{{c}}, ~22/21 = 78.4778{{c}}
-Badness: 1.150 × 10<sup>-3</sup>
+{{Optimal ET sequence|legend=0| 17c, 29, 31, 46, 60e, 77, 106de, 183dee }}
-===== 17-limit =====
+Badness (Sintel): 1.08
-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
@@ Line 240: / Line 320: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~3/2 = 701.9299, ~22/21 = 78.2539
+* WE: ~2 = 1200.6172{{c}}, ~3/2 = 702.1026{{c}}, ~22/21 = 78.7963{{c}}
-* CWE: ~2 = 1\1, ~3/2 = 701.7925, ~22/21 = 78.6203
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7925{{c}}, ~22/21 = 78.6203{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 17cg, 29g, 31, 46, 60e, 77, 106de }}
+{{Optimal ET sequence|legend=0| 17cg, 29g, 31, 46, 60e, 77, 106de }}
-Badness:
+Badness (Sintel): 0.931
-* Smith: 0.979 × 10<sup>-3</sup>
-* Dirichlet: 0.931
-===== 19-limit =====
+==== 19-limit ====
 Subgroup: 2.3.5.7.11.13.17.19
@@ Line 257: / Line 335: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~3/2 = 701.5642, ~22/21 = 78.2353
+* WE: ~2 = 1200.6224{{c}}, ~3/2 = 702.0959{{c}}, ~22/21 = 78.8004{{c}}
-* CWE: ~2 = 1\1, ~3/2 = 701.6963, ~22/21 = 78.6479
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.6963{{c}}, ~22/21 = 78.6479{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 17cg, 29g, 31, 46, 60e, 75dfgh, 77, 106de }}
+{{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>
-* Dirichlet: 1.159
-===== 23-limit =====
+==== 23-limit ====
 Subgroup: 2.3.5.7.11.13.17.19.23
@@ Line 274: / Line 350: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~3 = 1901.7115, ~23/22 = 78.2054
+* WE: ~2 = 1200.7268{{c}}, ~3/2 = 702.2463{{c}}, ~22/21 = 78.8824{{c}}
-* CWE: ~2 = 1\1, ~3 = 1901.8010, ~23/22 = 78.7188
+* 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 (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: {{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: {{Optimal ET sequence| 17cg, 29g, 31, 46, 60e, 75dfgh, 77, 106de }}
+{{Optimal ET sequence|legend=0| 14cf, 29ff, 31, 45ef, 46, 77, 122ee, 137def, 168deef }}
-Badness:
+Badness (Sintel): 1.36
-* Smith: 0.939 × 10<sup>-3</sup>
-* Dirichlet: 1.084
-==== Inanna ====
+=== Inanna ===
 Subgroup: 2.3.5.7.11.13
@@ Line 290: / Line 381: @@
 Mapping: {{mapping| 1 0 1 3 2 1 | 0 1 1 0 1 2 | 0 0 -4 -3 -2 -7 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 698.7754, ~21/20 = 79.6096
+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=1| 14cf, 15, 29, 31, 45ef, 60e }}
+{{Optimal ET sequence|legend=0| 14cf, 15, 29, 31, 45ef, 60e }}
-Badness: 1.077 × 10<sup>-3</sup>
+Badness (Sintel): 1.01
-==== Ishtar ====
+=== Ishtar ===
 Subgroup: 2.3.5.7.11.13
@@ Line 303: / Line 396: @@
 Mapping: {{mapping| 1 0 1 3 2 -1 | 0 1 1 0 1 3 | 0 0 -4 -3 -2 -1 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.3952, ~21/20 = 78.9578
+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 }}
-{{Optimal ET sequence|legend=1| 14cf, 15, 17c, 29, 31f, 46, 106deff, 121def }}
+Badness (Sintel): 1.08
-Badness: 1.151 × 10<sup>-3</sup>
+== References ==
 [[Category:Temperament clans]]
 [[Category:Biyatismic clan| ]] <!-- main article -->
-[[Category:Biyatismic| ]] <!-- key article -->
 [[Category:Rank 3]]