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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:  
Temperaments discussed elsewhere are:  
* ''[[Sonic]]'', {55/54, 100/99} → [[Porcupine rank three family #Sonic|Porcupine rank-3 family]]
* ''[[Sonic]]'' (+55/54 or 100/99) → [[Porcupine rank three family #Sonic|Porcupine rank-3 family]]
* ''[[Urania]]'', {81/80, 121/120} → [[Didymus rank three family #Urania|Didymus rank-3 family]]
* ''[[Urania]]'' (+81/80) → [[Didymus rank three family #Urania|Didymus rank-3 family]]
* ''[[Big Brother]]'', {99/98, 121/120} → [[Nuwell family #Big Brother|Nuwell family]]
* ''[[Big brother]]'' (+99/98) → [[Nuwell family #big Brother|Nuwell family]]
* ''[[Oxpecker]]'', {121/120, 126/125} → [[Starling family #Oxpecker|Starling family]]
* ''[[Bisector]]'' (+245/243) → [[Sensamagic family #Bisector|Sensamagic family]]
* [[Zeus]], {121/120, 176/175} → [[Porwell family #Zeus|Porwell family]]
* ''[[Artemis]]'', {121/120, 225/224} → [[Marvel family #Artemis|Marvel family]]
* ''[[Bisector]]'', {121/120, 245/243} → [[Sensamagic family #Bisector|Sensamagic family]]


Considered below are aphrodite, and the no-7 subgroup temperament, protomere, but we can begin by looking at the rank-4 temperament.  
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)]].  


== Biyatismic ==
== Protomere ==
Subgroup: 2.3.5.7.11
[[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 generators: ~2, ~3, ~11/10
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.4578, ~11/10 = 157.7466
 
{{Optimal ET sequence|legend=1| 7, 15, 22, 31, 46, 53, 137e, 183ee, 190ee }}
 
[[Badness]]: 0.0297 × 10<sup>-3</sup>
 
== Zeus ==
{{Main| Zeus }}
{{See also| Porwell family #Zeus }}


[[Comma list]]: [[121/120]]
[[Subgroup]]: 2.3.5.7.11


[[Mapping]]: [{{val| 1 0 1 0 2 }}, {{val| 0 1 1 0 1 }}, {{val| 0 0 -2 0 -1 }}, {{val| 0 0 0 1 0 }}]
[[Comma list]]: 121/120, 176/175


Mapping generators: ~2, ~3, ~11/10, ~7
{{Mapping|legend=1| 1 0 1 4 2 | 0 1 1 -1 1 | 0 0 -2 3 1 }}


[[POTE generator]]s: ~3/2 = 701.4578, ~11/10 = 157.7466, ~7/4 = 966.9589
[[Mapping to lattice]]: [{{val| 0 1 -1 2 0 }}, {{val| 0 1 1 -1 1 }}]


{{Val list|legend=1| 14c, 15, 22, 31, 46, 53, 60e, 68, 77, 91e, 99, 130e, 159ee, 190ee }}
Lattice basis:
: 11/10, 11/8
: Angle (11/10, 11/8) = 87.464 degrees


[[Badness]]: 0.0345 × 10<sup>-6</sup>
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.1530, ~11/10 = 157.0881


== Protomere ==
[[Minimax tuning]]:
Subgroup: 2.3.5.11
* [[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|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>
 
[[Projection pair]]s: 5 600/121 7 2662/375 11 120/11 to 2.3.11/5
 
Zeus11[22] [[hobbit]] [[transversal]]
: 33/32, 16/15, 11/10, 8/7, 64/55, 77/64, 5/4, 14/11, 4/3,
: 11/8, 45/32, 16/11, 3/2, 11/7, 8/5, 5/3, 55/32, 7/4,
: 11/6, 15/8, 64/33, 2
 
Zeus11[24] hobbit transversal
: 33/32, 16/15, 11/10, 9/8, 8/7, 77/64, 11/9, 5/4, 21/16, 4/3,
: 11/8, 45/32, 16/11, 3/2, 32/21, 8/5, 18/11, 5/3, 7/4, 16/9,
: 11/6, 15/8, 64/33, 2
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
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 to lattice: [{{val| 0 1 -1 2 0 -3 }}, {{val| 0 1 1 -1 1 -2 }}]
 
Lattice basis:
: 11/10 length = 0.7898, 11/8 length = 1.002
: Angle (11/10, 11/8) = 106.7439 degrees
 
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.8679, ~11/10 = 156.9582
 
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 }}]
: 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 }}]
: unchanged-interval (eigenmonzo) basis: 2.3.7/5
 
{{Optimal ET sequence|legend=1| 15, 22, 31, 46, 53, 77, 99, 130e }}
 
Badness: 0.934 × 10<sup>-3</sup>
 
Projection pairs: 5 600/121 7 2662/375 11 120/11 13 1280/99 to 2.3.11/5
 
Zeus13[22] hobbit transversal
: 260/243, 88/81, 11/10, 44/39, 162/143, 11/9, 16/13, 320/243, 4/3, 1040/729, 13/9, 729/520, 3/2, 99/65, 44/27, 18/11, 1280/729, 16/9, 11/6, 24/13, 243/130, 2
 
=== Tinia ===
Subgroup: 2.3.5.7.11.13
 
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 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 699.3420, ~11/10 = 155.3666
 
{{Optimal ET sequence|legend=1| 7, 9, 15, 22f, 24, 31 }}
 
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>


[[Comma list]]: 121/120
=== Woodpecker ===
Subgroup: 2.3.5.7.11.13


[[Sval]] [[mapping]]: [{{val| 1 0 1 2 }}, {{val| 0 1 1 1 }}, {{val| 0 0 -2 -1 }}]
Comma list: 66/65, 121/120, 126/125


Sval mapping generators: ~2, ~3, ~11/10
Mapping: {{mapping| 1 0 1 2 2 2 | 0 1 1 1 1 1 | 0 0 -2 -6 -1 1 }}


[[POTE generator]]s: ~3/2 = 701.4578, ~11/10 = 157.7466
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.5946, ~11/10 = 154.8652


{{Val list|legend=1| 7, 15, 22, 31, 46, 53, 137e, 183ee, 190ee }}
{{Optimal ET sequence|legend=1| 7d, 8d, 15, 23de, 24d, 31 }}


[[Badness]]: 0.0297 × 10<sup>-3</sup>
Badness: 1.093 × 10<sup>-3</sup>


== Aphrodite ==
== Aphrodite ==
Line 46: Line 174:


=== 7-limit (squalentine) ===
=== 7-limit (squalentine) ===
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val| 1 0 1 3 }}, {{val| 0 1 1 0 }}, {{val| 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


{{Val list|legend=1| 14c, 15, 29, 31, 46, 60, 77, 91, 122, 137d, 168d }}
[[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 }}


[[Badness]]: 0.943 × 10<sup>-3</sup>
[[Badness]]: 0.943 × 10<sup>-3</sup>
Line 61: Line 191:


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


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


[[Mapping]]: [{{val| 1 0 1 3 2 }}, {{val| 0 1 1 0 1 }}, {{val| 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


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


[[Badness]]: 0.583 × 10<sup>-3</sup>
[[Badness]]: 0.583 × 10<sup>-3</sup>
Line 78: Line 210:
Comma list: 121/120, 351/350, 441/440
Comma list: 121/120, 351/350, 441/440


Mapping: [{{val| 1 0 1 3 2 6 }}, {{val| 0 1 1 0 1 -1 }}, {{val| 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 }}


Vals: {{Val list| 14cf, 31, 45ef, 46, 77, 122ee, 137def, 168deef }}
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 }}


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


==== Eros ====
==== 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
Subgroup: 2.3.5.7.11.13


Comma list: 121/120, 196/195, 352/351
Comma list: 121/120, 196/195, 352/351


Mapping: [{{val| 1 0 1 3 2 7 }}, {{val| 0 1 1 0 1 -2 }}, {{val| 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


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


Badness: 1.150 × 10<sup>-3</sup>
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 ====
==== Inanna ====
Line 100: Line 291:
Comma list: 105/104, 121/120, 275/273
Comma list: 105/104, 121/120, 275/273


Mapping: [{{val| 1 0 1 3 2 1 }}, {{val| 0 1 1 0 1 2 }}, {{val| 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 }}


Vals: {{Val list| 14cf, 15, 29, 31, 45ef, 60e }}
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 }}


Badness: 1.077 × 10<sup>-3</sup>
Badness: 1.077 × 10<sup>-3</sup>
Line 111: Line 304:
Comma list: 91/90, 121/120, 441/440
Comma list: 91/90, 121/120, 441/440


Mapping: [{{val| 1 0 1 3 2 -1 }}, {{val| 0 1 1 0 1 3 }}, {{val| 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 }}


Vals: {{Val list| 14cf, 15, 17c, 29, 31f, 46, 106deff, 121def }}
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>
Badness: 1.151 × 10<sup>-3</sup>
== 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:Rank 3]]
[[Category:Rank 3]]
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