Biyatismic clan: Difference between revisions

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 }}.

* ''[[Sonic]]'' (+55/54 or 100/99) → [[Porcupine rank three family #Sonic|Porcupine rank-3 family]]

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

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

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)]].  

{{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

[[Badness]]: 0.0297 × 10^-3

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.1530, ~11/10 = 157.0881

: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.9/7

[[Badness]]: 0.400 × 10^-3

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

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.8679, ~11/10 = 156.9582

{{Optimal ET sequence|legend=1| 15, 22, 31, 46, 53, 77, 99, 130e }}

Badness: 0.934 × 10^-3

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^-3

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

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 }}

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 }}

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

[[Badness]]: 0.699 × 10^-3

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^-3

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)³ with 8/7.

=== 7-limit (squalentine) ===

[[Subgroup]]: 2.3.5.7

: 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^-3

[[Projection pair]]s: 5 320000/64827 7 64000/9261 to 2.3.7/5

[[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

[[Badness]]: 0.583 × 10^-3

==== 13-limit ====

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

Badness: 1.456 × 10^-3

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]].

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 }}

Note that this extension requires the 29g val for 29edo, which has the sizes of 17/16 and 18/17 swapped.

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

* 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:

* Dirichlet: 0.931

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

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 }}

* 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:

* Dirichlet: 1.159

===== 23-limit =====

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 }}

* 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:

* Dirichlet: 1.084

==== Inanna ====

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^-3

==== Ishtar ====

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:Rank 3]]