Biyatismic clan: Difference between revisions

The '''biyatismic clan''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the biyatisma, [[121/120]].

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

* ''[[Urania]]'' (+81/80) → [[Rastmic rank-3 clan #Urania|Rastmic rank-3 clan]]

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

Considered below are zeus, artemis, oxpecker, kahoupokane, big brother, aphrodite, and the no-7 subgroup temperament, protomere. For the rank-4 biyatismic temperament, see [[Rank-4 temperament #Biyatismic (121/120)]].  

== Protomere ==

[[Subgroup]]: 2.3.5.11

 

 

 

 

{{Optimal ET sequence|legend=1| 7, 15, 22, 31, 46, 53, 137e, 183ee, 190ee }}

 

[[Badness]] (Sintel): 0.245

 

 

 

 

 

 

 

* [[WE]]: ~2 = 1200.1693{{c}}, ~3/2 = 702.2521{{c}}, ~12/11 = 157.1102{{c}}

 

 

 

 

 

 

 

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

* WE: ~2 = 1200.2411{{c}}, ~3/2 = 702.0090{{c}}, ~12/11 = 156.9897{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.8818{{c}}, ~12/11 = 156.9568{{c}}

: [{{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 }}]

: [{{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=0| 15f, 22, 31, 46, 53, 77, 99, 130e }}

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

== Kahoupokane ==

 

[[Subgroup]]: 2.3.5.7.11

 

 

 

* [[WE]]: ~2 = 1200.1911{{c}}, ~3/2 = 703.1412{{c}}, ~11/10 = 158.1068{{c}}

 

 

 

 

 

 

 

 

 

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 99/98, 121/120

: mapping generators: ~2, ~3, ~11/7

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1200.6559{{c}}, ~3/2 = 700.2627{{c}}, ~11/7 = 771.8821{{c}}

: error map: {{val| 0.000 -1.553 -0.039 +3.245 -7.980 }}

{{Optimal ET sequence|legend=1| 8d, 9, 14c, 17c, 22, 31, 53, 84e }}

[[Badness]] (Sintel): 0.609

[[Projection pair]]s: <code>5 2401/486, 11 98/9</code> to 2.3.7

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 99/98, 121/120

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

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.7360{{c}}, ~11/7 = 773.0154{{c}}

{{Optimal ET sequence|legend=0| 8d, 9, 14c, 17c, 22f, 31, 79cf }}

Badness (Sintel): 0.889

: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Squalentine]].''

[[Subgroup]]: 2.3.5.7.11

 

* [[WE]]: ~2 = 1201.0691{{c}}, ~3/2 = 700.9439{{c}}, ~22/21 = 78.7122{{c}}

: [[error map]]: {{val| +1.069 +0.058 +1.920 -1.755 -4.591 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.8990{{c}}, ~22/21 = 78.4412{{c}}

: error map: {{val| 0.000 -1.056 +0.820 -4.150 -7.301 }}

 

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

 

[[Badness]] (Sintel): 0.701

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 has the pleasing property that all the harmonics are on the negative side of the last generator. Specifically, -3 to 2 fifths and -5 to 0 [[~]][[23/22]]'s will provide odd harmonics 1–23 up to octave equivalence; you can think of this as a 6×6 grid, which is a recommendable place to start looking at its structure.

Tempering out the less accurate comma 121/120 can be seen as an implication of tempering out [[441/440]] ({{S|21}}), [[484/483]] ({{S|22}}), and [[529/528]] ({{S|23}}). Therefore, characteristic of any good tuning is the prime [[11/1|11]] being the flattest prime, with other primes having strictly less than 5{{cent}} of error.

This temperament was discovered by [[Scott Dakota]]. Note that the 17-limit extension requires the 29g val for 29edo, which has the sizes of 17/16 and 18/17 swapped.

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

 

 

 

 

 

 

 

 

 

 

 

 

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

 

 

 

 

 

 

 

 

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.8010{{c}}, ~23/22 = 78.7188{{c}}

{{Optimal ET sequence|legend=0| 17cg, 29g, 31, 46, 60e, 75dfgh, 77, 106de }}

Badness (Sintel): 1.08

This extension is catalogued as tridecimal aphrodite in [[Graham Breed]]'s temperament finder.  

Comma list: 121/120, 351/350, 441/440

 

Optimal tunings:  

* WE: ~2 = 1201.0656{{c}}, ~3/2 = 700.7374{{c}}, ~22/21 = 78.5908{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.7005{{c}}, ~22/21 = 78.3253{{c}}

{{Optimal ET sequence|legend=0| 14cf, 29ff, 31, 45ef, 46, 77, 122ee, 137def, 168deef }}

Badness (Sintel): 1.36

=== Inanna ===

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

Optimal tunings:  

* WE: ~2 = 1201.7881{{c}}, ~3/2 = 699.8166{{c}}, ~22/21 = 79.7282{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.5926{{c}}, ~22/21 = 79.3822{{c}}

 

=== Ishtar ===

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

 

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.7640{{c}}, ~22/21 = 78.8025{{c}}

 

{{Optimal ET sequence|legend=0| 14cf, 15, 17c, 29, 31f, 46, 106deff, 121def }}

Badness (Sintel): 1.08