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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: ~~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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== 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 = 700.8882, ~11/10 = 155.7756

{{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: {{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

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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]]

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-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 16: / 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 45: / 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 81: / 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 110: @@
 == 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 147: / Line 150: @@
 {{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 }}
@@ Line 157: / Line 162: @@
 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: {{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 }}
@@ Line 171: / 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 186: / 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 199: / 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 206: / 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 221: / 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 232: / 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]]