Biyatismic clan: Difference between revisions

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The '''biyatismic clan''' of rank-3 temperaments 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 }}.

Temperaments discussed elsewhere are:

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* ''[[Urania]]'' (+81/80) → [[Didymus rank three family #Urania|Didymus rank-3 family]]

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

* ''[[Oxpecker]]'' (+126/125) → [[Starling family #Oxpecker|Starling family]]

* ''[[Artemis]]'' (+225/224) → [[Marvel family #Artemis|Marvel family]]

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

Considered below are zeus, 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, 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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: ~~sval mapping~~ generators: ~2, ~3, ~11/10

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

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.4578, ~11/10 = 157.7466

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* [[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 }}

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* 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 }}

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Badness: 0.808 × 10-3

== 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

[[Comma list]]: 121/120, 225/224

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 699.8719, ~11/10 = 158.3232

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

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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 }}

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 698.7090, ~11/10 = 158.7117

=== Diana ===

Subgroup: 2.3.5.7.11.13

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 }}

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.9789, ~11/10 = 159.0048

{{Optimal ET sequence|legend=1| 22, 29, 31, 53, 82e, 84e, 113e, 166ee }}

== Oxpecker ==

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 121/120, 126/125

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 700.8882, ~11/10 = 155.7756

{{Optimal ET sequence|legend=1| 7d, 8d, 15, 23de, 24d, 31, 46, 77 }}

[[Badness]]: 0.699 × 10-3

=== Woodpecker ===

Subgroup: 2.3.5.7.11.13

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 }}

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.5946, ~11/10 = 154.8652

Badness: 1.093 × 10-3

== Aphrodite ==

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[[Comma list]]: 64827/64000

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

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

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

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

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

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

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

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

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

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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 }}

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

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.1158, ~21/20 = 78.5211

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

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==== 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

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

Badness: 1.150 × 10-3

===== 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 }}

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

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

Badness:

* Smith: 0.979 × 10-3

* Dirichlet: 0.931

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

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: {{mapping| 1 0 1 3 2 7 6 9 | 0 1 1 0 1 -2 -1 -3 | 0 0 -4 -3 -2 -2 -5 0 }}

Optimal tunings:

* CTE: ~2 = 1\1, ~3/2 = 701.5642, ~22/21 = 78.2353

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

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

Badness:

* Smith: 1.13 × 10-3

* Dirichlet: 1.159

===== 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 }}

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

Badness:

* Smith: 0.939 × 10-3

* Dirichlet: 1.084

==== Inanna ====

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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: {{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

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

Badness: 1.151 × 10-3

== Notes ==

[[Category:Temperament clans]]

[[Category:Pages with mostly numerical content]]

[[Category:Biyatismic clan| ]]

[[Category:Biyatismic| ]]

[[Category:Rank 3]]

@@ Line 1: / Line 1: @@
-The '''biyatismic clan''' of rank-3 temperaments tempers out the [[biyatisma]], 121/120 = {{monzo| -3 -1 -1 0 2 }}.
+{{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 }}.
 Temperaments discussed elsewhere are:
@@ Line 5: / Line 6: @@
 * ''[[Urania]]'' (+81/80) → [[Didymus rank three family #Urania|Didymus rank-3 family]]
 * ''[[Big brother]]'' (+99/98) → [[Nuwell family #big Brother|Nuwell family]]
-* ''[[Oxpecker]]'' (+126/125) → [[Starling family #Oxpecker|Starling family]]
-* ''[[Artemis]]'' (+225/224) → [[Marvel family #Artemis|Marvel family]]
 * ''[[Bisector]]'' (+245/243) → [[Sensamagic family #Bisector|Sensamagic family]]
-Considered below are zeus, 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, aphrodite, and the no-7 subgroup temperament, protomere. For the rank-4 biyatismic temperament, see [[Rank-4 temperament #Biyatismic (121/120)]].
 == Protomere ==
@@ Line 18: / Line 17: @@
 {{Mapping|legend=2| 1 0 1 2 | 0 1 1 1 | 0 0 -2 -1 }}
-: sval mapping generators: ~2, ~3, ~11/10
+: Mapping generators: ~2, ~3, ~11/10
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.4578, ~11/10 = 157.7466
@@ Line 47: / Line 46: @@
 * [[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 }}
@@ Line 83: / Line 82: @@
 * 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 }}
@@ Line 109: / Line 108: @@
 Badness: 0.808 × 10<sup>-3</sup>
+== 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
+[[Comma list]]: 121/120, 225/224
+{{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 ET sequence|legend=1| 9, 15d, 16d, 20, 22, 31, 53, 82e, 84e, 113e, 144ee }}
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+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 }}
+Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 698.7090, ~11/10 = 158.7117
+{{Optimal ET sequence|legend=1| 9, 20, 22f, 29, 31 }}
+=== Diana ===
+Subgroup: 2.3.5.7.11.13
+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 }}
+Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.9789, ~11/10 = 159.0048
+{{Optimal ET sequence|legend=1| 22, 29, 31, 53, 82e, 84e, 113e, 166ee }}
+== Oxpecker ==
+[[Subgroup]]: 2.3.5.7.11
+[[Comma list]]: 121/120, 126/125
+{{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 ET sequence|legend=1| 7d, 8d, 15, 23de, 24d, 31, 46, 77 }}
+[[Badness]]: 0.699 × 10<sup>-3</sup>
+=== Woodpecker ===
+Subgroup: 2.3.5.7.11.13
+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 }}
+Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.5946, ~11/10 = 154.8652
+{{Optimal ET sequence|legend=1| 7d, 8d, 15, 23de, 24d, 31 }}
+Badness: 1.093 × 10<sup>-3</sup>
 == Aphrodite ==
@@ Line 118: / Line 178: @@
 [[Comma list]]: 64827/64000
-{{Mapping|legend=1| 1 0 1 3 | 0 1 1 0 | 0 0 4 3 }}
+{{Mapping|legend=1| 1 0 1 3 | 0 1 1 0 | 0 0 -4 -3 }}
+: Mapping generators: ~2, ~3, ~21/20
-: mapping generators: ~2, ~3, ~21/20
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 700.2144, ~21/20 = 78.5694
 {{Optimal ET sequence|legend=1| 14c, 15, 29, 31, 46, 60, 77, 91, 122, 137d, 168d }}
@@ Line 133: / Line 195: @@
 [[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|legend=1| 1 0 1 3 2 | 0 1 1 0 1 | 0 0 -4 -3 -2 }}
+: Mapping generators: ~2, ~3, ~22/21
-: mapping generators: ~2, ~3, ~22/21
+Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.3200, ~21/20 = 78.6421
 {{Optimal ET sequence|legend=1| 14c, 15, 29, 31, 46, 60e, 77, 91e, 137de, 168dee }}
@@ Line 146: / Line 210: @@
 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 }}
+Mapping: {{mapping| 1 0 1 3 2 6 | 0 1 1 0 1 -1 | 0 0 -4 -3 -2 -11 }}
+Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.1158, ~21/20 = 78.5211
 {{Optimal ET sequence|legend=1| 14cf, 31, 45ef, 46, 77, 122ee, 137def, 168deef }}
@@ Line 153: / Line 219: @@
 ==== 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
 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: {{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 ET sequence|legend=1| 17c, 29, 31, 46, 60e, 77, 106de, 183dee }}
 Badness: 1.150 × 10<sup>-3</sup>
+===== 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 }}
+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
+Optimal ET sequence: {{Optimal ET sequence| 17cg, 29g, 31, 46, 60e, 77, 106de }}
+Badness:
+* Smith: 0.979 × 10<sup>-3</sup>
+* Dirichlet: 0.931
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list: 121/120, 154/153, 196/195, 286/285, 352/351
+Mapping: {{mapping| 1 0 1 3 2 7 6 9 | 0 1 1 0 1 -2 -1 -3 | 0 0 -4 -3 -2 -2 -5 0 }}
+Optimal tunings:
+* CTE: ~2 = 1\1, ~3/2 = 701.5642, ~22/21 = 78.2353
+* CWE: ~2 = 1\1, ~3/2 = 701.6963, ~22/21 = 78.6479
+Optimal ET sequence: {{Optimal ET sequence| 17cg, 29g, 31, 46, 60e, 75dfgh, 77, 106de }}
+Badness:
+* Smith: 1.13 × 10<sup>-3</sup>
+* Dirichlet: 1.159
+===== 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 }}
+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 ET sequence: {{Optimal ET sequence| 17cg, 29g, 31, 46, 60e, 75dfgh, 77, 106de }}
+Badness:
+* Smith: 0.939 × 10<sup>-3</sup>
+* Dirichlet: 1.084
 ==== Inanna ====
@@ Line 168: / Line 291: @@
 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: {{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 ET sequence|legend=1| 14cf, 15, 29, 31, 45ef, 60e }}
@@ Line 179: / Line 304: @@
 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: {{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 ET sequence|legend=1| 14cf, 15, 17c, 29, 31f, 46, 106deff, 121def }}
 Badness: 1.151 × 10<sup>-3</sup>
+== Notes ==
 [[Category:Temperament clans]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Biyatismic clan| ]] <!-- main article -->
 [[Category:Biyatismic| ]] <!-- key article -->
 [[Category:Rank 3]]