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{{Technical data page}}
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]] [[Temperament|temperaments]] [[Tempering out|tempers out]] the [[biyatisma]], 121/120 = {{monzo| -3 -1 -1 0 2 }}.


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* [[11-odd-limit]]
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 11/9 10/9 -1/3 -2/9 0 }}, {{monzo| 22/9 2/9 1/3 -4/9 0 }}, {{monzo| 22/9 2/9 -2/3 5/9 0 }}, {{monzo| 10/3 2/3 0 -1/3 0 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 11/9 10/9 -1/3 -2/9 0 }}, {{monzo| 22/9 2/9 1/3 -4/9 0 }}, {{monzo| 22/9 2/9 -2/3 5/9 0 }}, {{monzo| 10/3 2/3 0 -1/3 0 }}]
: [[Eigenmonzo basis|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 }}
{{Optimal ET sequence|legend=1| 15, 22, 31, 46, 53, 68, 77, 99, 130e }}
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* 13-odd-limit  
* 13-odd-limit  
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 11/9 10/9 -1/3 -2/9 0 0 }}, {{monzo| 22/9 2/9 1/3 -4/9 0 0 }}, {{monzo| 22/9 2/9 -2/3 5/9 0 0 }}, {{monzo| 10/3 2/3 0 -1/3 0 0 }}, {{monzo| 14/3 -8/3 1 1/3 0 0 }}]
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 11/9 10/9 -1/3 -2/9 0 0 }}, {{monzo| 22/9 2/9 1/3 -4/9 0 0 }}, {{monzo| 22/9 2/9 -2/3 5/9 0 0 }}, {{monzo| 10/3 2/3 0 -1/3 0 0 }}, {{monzo| 14/3 -8/3 1 1/3 0 0 }}]
: eigenmonzo (unchanged-interval) basis: 2.9/5.9/7
: unchanged-interval (eigenmonzo) basis: 2.9/5.9/7
* 15-odd-limit
* 15-odd-limit
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 0 1 0 0 0 0 }}, {{monzo| 11/5 1/5 2/5 -2/5 0 0 }}, {{monzo| 11/5 1/5 -3/5 3/5 0 0 }}, {{monzo| 13/5 3/5 1/5 -1/5 0 0 }}, {{monzo| 38/5 -12/5 1/5 -1/5 0 0 }}]
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 0 1 0 0 0 0 }}, {{monzo| 11/5 1/5 2/5 -2/5 0 0 }}, {{monzo| 11/5 1/5 -3/5 3/5 0 0 }}, {{monzo| 13/5 3/5 1/5 -1/5 0 0 }}, {{monzo| 38/5 -12/5 1/5 -1/5 0 0 }}]
: 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=1| 15, 22, 31, 46, 53, 77, 99, 130e }}
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== Artemis ==
== 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
[[Subgroup]]: 2.3.5.7.11


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Badness: 1.456 × 10<sup>-3</sup>
Badness: 1.456 × 10<sup>-3</sup>


==== (13-limit) [[#Eros|Eros]] ====
==== Eros ====
This is the [[13-limit]] restriction of [[#Eros|23-limit Eros]], documented for readability in a separate section of this page.
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
Subgroup: 2.3.5.7.11.13
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Badness: 1.150 × 10<sup>-3</sup>
Badness: 1.150 × 10<sup>-3</sup>


==== Inanna ====
===== 17-limit =====
Subgroup: 2.3.5.7.11.13
Note that this extension requires the 29g val for 29edo, which has the sizes of 17/16 and 18/17 swapped.


Comma list: 105/104, 121/120, 275/273
Subgroup: 2.3.5.7.11.13.17


Mapping: {{mapping| 1 0 1 3 2 1 | 0 1 1 0 1 2 | 0 0 -4 -3 -2 -7 }}
Comma list: 121/120, 154/153, 196/195, 352/351


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 698.7754, ~21/20 = 79.6096
Mapping: {{mapping| 1 0 1 3 2 7 6 | 0 1 1 0 1 -2 -1 | 0 0 -4 -3 -2 -2 -5 }}


{{Optimal ET sequence|legend=1| 14cf, 15, 29, 31, 45ef, 60e }}
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 701.9299, ~22/21 = 78.2539
* CWE: ~2 = 1\1, ~3/2 = 701.7925, ~22/21 = 78.6203


Badness: 1.077 × 10<sup>-3</sup>
Optimal ET sequence: {{Optimal ET sequence| 17cg, 29g, 31, 46, 60e, 77, 106de }}


==== Ishtar ====
Badness:
Subgroup: 2.3.5.7.11.13
* Smith: 0.979 × 10<sup>-3</sup>
* Dirichlet: 0.931


Comma list: 91/90, 121/120, 441/440
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Mapping: {{mapping| 1 0 1 3 2 -1 | 0 1 1 0 1 3 | 0 0 -4 -3 -2 -1 }}
Comma list: 121/120, 154/153, 196/195, 286/285, 352/351


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.3952, ~21/20 = 78.9578
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 }}


{{Optimal ET sequence|legend=1| 14cf, 15, 17c, 29, 31f, 46, 106deff, 121def }}
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 701.5642, ~22/21 = 78.2353
* CWE: ~2 = 1\1, ~3/2 = 701.6963, ~22/21 = 78.6479


Badness: 1.151 × 10<sup>-3</sup>
Optimal ET sequence: {{Optimal ET sequence| 17cg, 29g, 31, 46, 60e, 75dfgh, 77, 106de }}


== Eros ==
Badness:
This temperament is an extension of [[#Aphrodite]], specifically [[#(13-limit) Eros|(13-limit) Eros]], into the 23-limit, in which it fairs impressively as a rank 3 temperament.
* Smith: 1.13 × 10<sup>-3</sup>
* Dirichlet: 1.159


Due to having natural extensions to such a high prime limit and due to its unusual efficiency, it is documented here in its own section to keep other extensions of Aphrodite more readable.
===== 23-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23


=== 13-limit ===
Comma list: 121/120, 154/153, 161/160, 196/195, 286/285, 352/351
Subgroup: [[13-limit]]


Comma list: 121/120, 441/440, 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: {{mapping| 1 0 1 3 2 7 | 0 1 1 0 1 -2 | 0 0 -4 -3 -2 -2 }}
Optimal tunings:  
* CTE: ~2 = 1\1, ~3 = 1901.7115, ~23/22 = 78.2054
* CWE: ~2 = 1\1, ~3 = 1901.8010, ~23/22 = 78.7188


Optimal tuning ([[CTE]]): 2, ~3 = 1901.722, ~22/21 = 78.110
Optimal ET sequence: {{Optimal ET sequence| 17cg, 29g, 31, 46, 60e, 75dfgh, 77, 106de }}


[[Patent val]] EDO tunings: [[29edo|29]], [[31edo|31]], [[46edo|46]], [[77edo|77]], [[92edo|92]]
Badness:  
* Smith: 0.939 × 10<sup>-3</sup>
* Dirichlet: 1.084


Dirichlet badness: 1.075
==== Inanna ====
Subgroup: 2.3.5.7.11.13


=== 17-limit ===
Comma list: 105/104, 121/120, 275/273
Note that there is one more patent val EDO tuning in the 17-limit than in the 19- and 23-limit corresponding to using 46 EDO with a better mapping for 19. 29 disappears as it requires the 29g val.


Subgroup: [[17-limit]]
Mapping: {{mapping| 1 0 1 3 2 1 | 0 1 1 0 1 2 | 0 0 -4 -3 -2 -7 }}


Comma list: 121/120, 441/440, 352/351, 154/153
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 698.7754, ~21/20 = 79.6096


Mapping: {{mapping| 1 0 1 3 2 7 6 | 0 1 1 0 1 -2 -1 | 0 0 -4 -3 -2 -2 -5 }}
{{Optimal ET sequence|legend=1| 14cf, 15, 29, 31, 45ef, 60e }}


Optimal tuning ([[CTE]]): 2, ~3 = 1901.930, ~22/21 = 78.254
Badness: 1.077 × 10<sup>-3</sup>


[[Patent val]] EDO tunings: [[31edo|31]], [[46edo|46]], [[77edo|77]], [[92edo|92]]
==== Ishtar ====
Subgroup: 2.3.5.7.11.13


Dirichlet badness: 0.931
Comma list: 91/90, 121/120, 441/440


=== 19-limit ===
Mapping: {{mapping| 1 0 1 3 2 -1 | 0 1 1 0 1 3 | 0 0 -4 -3 -2 -1 }}
Subgroup: [[19-limit]]


Comma list: 441/440, 1617/1600, 196/195, 154/153, 286/285
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.3952, ~21/20 = 78.9578


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


Optimal tuning ([[CTE]]): 2, ~3/2 = 1901.564, ~22/21 = 78.235
Badness: 1.151 × 10<sup>-3</sup>
 
[[Patent val]] EDO tunings: [[31edo|31]], [[46edo|46]], [[77edo|77]]
 
Dirichlet badness: 1.159
 
=== 23-limit ===
Subgroup: [[23-limit]]
 
Comma list: 441/440, 1617/1600, 196/195, 154/153, 286/285, 161/160
 
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 }}
 
Optimal tuning ([[CTE]]): 2, ~3 = 1901.711, ~23/22 = 78.205
 
[[Patent val]] EDO tunings: [[31edo|31]], [[46edo|46]], [[77edo|77]]


Dirichlet badness: 1.084
== Notes ==


[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Pages with mostly numerical content]]
[[Category:Biyatismic clan| ]] <!-- main article -->
[[Category:Biyatismic clan| ]] <!-- main article -->
[[Category:Biyatismic| ]] <!-- key article -->
[[Category:Biyatismic| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
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