Biyatismic clan: Difference between revisions

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

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 701.4578{{c}}, ~11/10 = 157.7466{{c}}

[[Badness]] (Smith): 0.0297 × 10^-3

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 702.1530{{c}}, ~11/10 = 157.0881{{c}}

[[Badness]] (Smith): 0.400 × 10^-3

Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 701.8679{{c}}, ~11/10 = 156.9582{{c}}

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

Badness (Smith): 0.934 × 10^-3

Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 699.3420{{c}}, ~11/10 = 155.3666{{c}}

Badness (Smith): 0.808 × 10^-3

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 699.8719{{c}}, ~11/10 = 158.3232{{c}}

Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 698.7090{{c}}, ~11/10 = 158.7117{{c}}

{{Optimal ET sequence|legend=0| 9, 20, 22f, 29, 31 }}

Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 700.9789{{c}}, ~11/10 = 159.0048{{c}}

{{Optimal ET sequence|legend=0| 22, 29, 31, 53, 82e, 84e, 113e, 166ee }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 700.8882{{c}}, ~11/10 = 155.7756{{c}}

[[Badness]] (Smith): 0.699 × 10^-3

Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 701.5946{{c}}, ~11/10 = 154.8652{{c}}

Badness (Smith): 1.093 × 10^-3

{{Mapping|legend=1| 1 0 3 1 3 | 0 1 3 2 2 | 0 0 -4 -1 -2 }}

{{Optimal ET sequence|legend=1| 8d, 9, 22, 31, 53 }}

[[Projection pair]]s: 5 2401/486 11 98/9 to 2.3.7

{{Optimal ET sequence|legend=0| 8d, 9, 22f, 31 }}

Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 700.3200{{c}}, ~21/20 = 78.6421{{c}}

[[Badness]] (Smith): 0.583 × 10^-3

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

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

 

Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 700.1158{{c}}, ~21/20 = 78.5211{{c}}

 

 

 

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]].

Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 701.5014{{c}}, ~21/20 = 78.6143{{c}}

Badness (Smith): 1.150 × 10^-3

 

* CTE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9299{{c}}, ~22/21 = 78.2539{{c}}

Badness:

* Sintel: 0.931

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

* CTE: ~2 = 1200.0000{{c}}, ~3/2 = 701.5642{{c}}, ~22/21 = 78.2353{{c}}

Badness:

* Sintel: 1.159

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

* CTE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7115{{c}}, ~23/22 = 78.2054{{c}}

Badness:  

* Sintel: 1.084

==== Inanna ====

Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 698.7754{{c}}, ~21/20 = 79.6096{{c}}

Badness (Sintel): 1.077 × 10^-3

==== Ishtar ====

Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 703.3952{{c}}, ~21/20 = 78.9578{{c}}

Badness (Sintel): 1.151 × 10^-3

== Notes ==