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This is a collection of rank-3 temperaments tempering 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]]
* ''[[Aphrodite]]'', {121/120, 441/440} → [[Werckismic temperaments #Aphrodite|Werckismic temperaments]]


[[Category:Regular temperament theory]]
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)]].
[[Category:Temperament clan]]
 
[[Category:Biyatismic temperaments| ]] <!-- main article -->
== Protomere ==
[[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 }}
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 121/120, 176/175
 
{{Mapping|legend=1| 1 0 1 4 2 | 0 1 1 -1 1 | 0 0 -2 3 1 }}
 
[[Mapping to lattice]]: [{{val| 0 1 -1 2 0 }}, {{val| 0 1 1 -1 1 }}]
 
Lattice basis:
: 11/10, 11/8
: Angle (11/10, 11/8) = 87.464 degrees
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.1530, ~11/10 = 157.0881
 
[[Minimax tuning]]:
* [[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>
 
=== 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 ==
Aphrodite tempers out the squalentine comma, 64827/64000, in the 7-limit. Its generators can be taken to be 2, 3, and 21/20, and it equates (21/20)<sup>3</sup> with 8/7.
 
=== 7-limit (squalentine) ===
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 64827/64000
 
{{Mapping|legend=1| 1 0 1 3 | 0 1 1 0 | 0 0 -4 -3 }}
 
: 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 }}
 
[[Badness]]: 0.943 × 10<sup>-3</sup>
 
[[Projection pair]]s: 5 320000/64827 7 64000/9261 to 2.3.7/5
 
=== 11-limit ===
[[Subgroup]]: 2.3.5.7.11
 
[[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 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 }}
 
[[Badness]]: 0.583 × 10<sup>-3</sup>
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
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 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>
 
==== 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 }}
 
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 ====
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 }}
 
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>
 
==== 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 }}
 
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]]
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Biyatismic_clan"