Biyatismic clan: Difference between revisions

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* ''[[Bisector]]'' (+245/243) → [[Sensamagic family #Bisector|Sensamagic family]]

Considered below are zeus, artemis, oxpecker, big brother, 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

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* [[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}}, ~11~~/10~~ = 157.6099{{c}}

* [[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 }}

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[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.1693{{c}}, ~3/2 = 702.2521{{c}}, ~11~~/10~~ = 157.1102{{c}}

* [[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}}, ~11~~/10~~ = 157.1265{{c}}

* [[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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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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Optimal tunings:

* WE: ~2 = 1200.2411{{c}}, ~3/2 = 702.0090{{c}}, ~11~~/10~~ = 156.9897{{c}}

* 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}}, ~11~~/10~~ = 156.9568{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.8818{{c}}, ~12/11 = 156.9568{{c}}

~~~~

Minimax tuning:

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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 tunings:

* WE: ~2 = 1199.9251{{c}}, ~3/2 = 699.2984{{c}}, ~11~~/10~~ = 155.3569{{c}}

* 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}}, ~11~~/10~~ = 155.3484{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.2982{{c}}, ~12/11 = 155.3484{{c}}

~~~~

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* [[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 }}

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

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

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[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.4124{{c}}, ~3/2 = 701.1291{{c}}, ~11~~/10~~ = 155.8292{{c}}

* [[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}}, ~11~~/10~~ = 155.7399{{c}}

* [[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 }}

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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 tunings:

* WE: ~2 = 1198.9113{{c}}, ~3/2 = 700.9581{{c}}, ~11~~/10~~ = 154.7247{{c}}

* 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}}, ~11~~/10~~ = 154.9115{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.6409{{c}}, ~12/11 = 154.9115{{c}}

~~~~

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: 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:

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* [[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 }}

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[[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~~

~~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}}~~

<!-~~- * POTE:~~ ~~~2 = 1200.0000{{c}}, ~3~~/~~2 = 700.1158{{c}}~~, ~~~21/20 = 78~~.~~5211{{c}} -->~~

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

~~Badness~~ (~~Sintel~~): 1.36

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 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

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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 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 }}

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Badness (Sintel): 1.08

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

==== 17-limit ====

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

Subgroup: 2.3.5.7.11.13.17

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:

* 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 }}

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Badness (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:

* 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 }}

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Badness (Sintel): 1.16

===== 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:

* 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 }}

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Badness (Sintel): 1.08

==== Inanna ====

=== 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|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 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}}

~~~~

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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 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 }}

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== References ==

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

[[Category:Biyatismic clan| ]]

[[Category:Biyatismic| ]]

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

[[Category:Temperament clans]]

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

@@ Line 7: / Line 7: @@
 * ''[[Bisector]]'' (+245/243) → [[Sensamagic family #Bisector|Sensamagic family]]
-Considered below are zeus, artemis, oxpecker, big brother, 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
@@ Line 20: / Line 24: @@
 * [[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}}, ~11/10 = 157.6099{{c}}
+* [[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 }}
-<!-- * [[POTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4578{{c}}, ~11/10 = 157.7466{{c}} -->
 {{Optimal ET sequence|legend=1| 7, 15, 22, 31, 46, 53, 137e, 183ee, 190ee }}
@@ Line 45: / Line 48: @@
 [[Optimal tuning]]s:
-* [[WE]]: ~2 = 1200.1693{{c}}, ~3/2 = 702.2521{{c}}, ~11/10 = 157.1102{{c}}
+* [[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}}, ~11/10 = 157.1265{{c}}
+* [[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 }}
-<!-- * [[POTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.1530{{c}}, ~11/10 = 157.0881{{c}} -->
 [[Minimax tuning]]:
@@ Line 77: / 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 86: / Line 88: @@
 Optimal tunings:
-* WE: ~2 = 1200.2411{{c}}, ~3/2 = 702.0090{{c}}, ~11/10 = 156.9897{{c}}
+* 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}}, ~11/10 = 156.9568{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.8818{{c}}, ~12/11 = 156.9568{{c}}
-<!-- * POTE: ~2 = 1200.0000{{c}}, ~3/2 = 701.8679{{c}}, ~11/10 = 156.9582{{c}} -->
 Minimax tuning:
@@ Line 112: / 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 tunings:
-* WE: ~2 = 1199.9251{{c}}, ~3/2 = 699.2984{{c}}, ~11/10 = 155.3569{{c}}
+* 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}}, ~11/10 = 155.3484{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.2982{{c}}, ~12/11 = 155.3484{{c}}
-<!-- * POTE: ~2 = 1200.0000{{c}}, ~3/2 = 699.3420{{c}}, ~11/10 = 155.3666{{c}} -->
 {{Optimal ET sequence|legend=0| 7, 9, 15, 22f, 24, 31 }}
@@ Line 137: / Line 137: @@
 * [[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 }}
-<!-- * [[POTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 699.8719{{c}}, ~11/10 = 158.3232{{c}} -->
 {{Optimal ET sequence|legend=1| 9, 15d, 16d, 20, 22, 31, 53, 60e, 84e, 91e, 113e, 144ee }}
@@ Line 148: / 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}}
-<!-- * 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 }}
@@ Line 164: / 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}}
-<!-- * 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 }}
@@ Line 183: / Line 180: @@
 [[Optimal tuning]]s:
-* [[WE]]: ~2 = 1200.4124{{c}}, ~3/2 = 701.1291{{c}}, ~11/10 = 155.8292{{c}}
+* [[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}}, ~11/10 = 155.7399{{c}}
+* [[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 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 700.8882{{c}}, ~11/10 = 155.7756{{c}} -->
 {{Optimal ET sequence|legend=1| 7d, 8d, 15, 23de, 24d, 31, 46, 77 }}
@@ Line 198: / 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 tunings:
-* WE: ~2 = 1198.9113{{c}}, ~3/2 = 700.9581{{c}}, ~11/10 = 154.7247{{c}}
+* 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}}, ~11/10 = 154.9115{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.6409{{c}}, ~12/11 = 154.9115{{c}}
-<!-- * POTE: ~2 = 1200.0000{{c}}, ~3/2 = 701.5946{{c}}, ~11/10 = 154.8652{{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|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 236: / 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:
@@ Line 261: / Line 290: @@
 * [[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 }}
-<!-- * POTE: ~2 = 1200.0000{{c}}, ~3/2 = 700.3200{{c}}, ~22/21 = 78.6421{{c}} -->
 {{Optimal ET sequence|legend=1| 14c, 15, 29, 31, 46, 60e, 77, 91e, 137de, 168dee }}
@@ Line 267: / Line 295: @@
 [[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
-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}}
-<!-- * POTE: ~2 = 1200.0000{{c}}, ~3/2 = 700.1158{{c}}, ~21/20 = 78.5211{{c}} -->
-{{Optimal ET sequence|legend=0| 14cf, 29ff, 31, 45ef, 46, 77, 122ee, 137def, 168deef }}
-Badness (Sintel): 1.36
+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 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 290: / 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 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}}
-<!-- * POTE: ~2 = 1200.0000{{c}}, ~3/2 = 701.5014{{c}}, ~21/20 = 78.6143{{c}} -->
 {{Optimal ET sequence|legend=0| 17c, 29, 31, 46, 60e, 77, 106de, 183dee }}
@@ Line 301: / Line 316: @@
 Badness (Sintel): 1.08
-===== 17-limit =====
+==== 17-limit ====
-Note that this extension requires the 29g val for 29edo, which has the sizes of 17/16 and 18/17 swapped.
 Subgroup: 2.3.5.7.11.13.17
 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:
 * 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}}
-<!-- * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9299{{c}}, ~22/21 = 78.2539{{c}} -->
 {{Optimal ET sequence|legend=0| 17cg, 29g, 31, 46, 60e, 77, 106de }}
@@ Line 319: / Line 331: @@
 Badness (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:
 * 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}}
-<!-- * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 701.5642{{c}}, ~22/21 = 78.2353{{c}} -->
 {{Optimal ET sequence|legend=0| 17cg, 29g, 31, 46, 60e, 75dfgh, 77, 106de }}
@@ Line 335: / Line 346: @@
 Badness (Sintel): 1.16
-===== 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:
 * 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}}
-<!-- * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7115{{c}}, ~23/22 = 78.2054{{c}} -->
 {{Optimal ET sequence|legend=0| 17cg, 29g, 31, 46, 60e, 75dfgh, 77, 106de }}
@@ Line 351: / Line 361: @@
 Badness (Sintel): 1.08
-==== Inanna ====
+=== 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|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 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}}
-<!-- * POTE: ~2 = 1200.0000{{c}}, ~3/2 = 698.7754{{c}}, ~21/20 = 79.6096{{c}} -->
 {{Optimal ET sequence|legend=0| 14cf, 15, 29, 31, 45ef, 60e }}
@@ Line 367: / Line 393: @@
 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 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}}
-<!-- * POTE: ~2 = 1200.0000{{c}}, ~3/2 = 703.3952{{c}}, ~21/20 = 78.9578{{c}} -->
 {{Optimal ET sequence|legend=0| 14cf, 15, 17c, 29, 31f, 46, 106deff, 121def }}
@@ Line 385: / Line 410: @@
 == 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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