Biyatismic clan: Difference between revisions

(3 intermediate revisions by the same user not shown)

Line 20:

* [[WE]]: ~2 = 1200.6628{{c}}, ~3/2 = 701.8452{{c}}, ~11/10 = 157.8337{{c}}

: [[error map]]: {{val| +0.663 +0.553 +1.190 -5.318 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9848{{c}}, ~11~~/10~~ = 157.6099{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9848{{c}}, ~12/11 = 157.6099{{c}}

: error map: {{val| 0.000 +0.030 +0.451 -6.943 }}

Line 44:

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.1693{{c}}, ~3/2 = 702.2521{{c}}, ~11~~/10~~ = 157.1102{{c}}

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

: [[error map]]: {{val| +0.169 +0.466 +2.057 +0.761 -5.668 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2478{{c}}, ~11~~/10~~ = 157.1265{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2478{{c}}, ~12/11 = 157.1265{{c}}

: error map: {{val| 0.000 +0.293 +1.681 +0.306 -6.197 }}

Line 84:

Optimal tunings:

* WE: ~2 = 1200.2411{{c}}, ~3/2 = 702.0090{{c}}, ~11~~/10~~ = 156.9897{{c}}

* 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}}, ~11~~/10~~ = 156.9568{{c}}

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

Minimax tuning:

Line 112:

Optimal tunings:

* WE: ~2 = 1199.9251{{c}}, ~3/2 = 699.2984{{c}}, ~11~~/10~~ = 155.3569{{c}}

* WE: ~2 = 1199.9251{{c}}, ~3/2 = 699.2984{{c}}, ~12/11 = 155.3569{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.2982{{c}}, ~11~~/10~~ = 155.3484{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.2982{{c}}, ~12/11 = 155.3484{{c}}

Line 176:

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.4124{{c}}, ~3/2 = 701.1291{{c}}, ~11~~/10~~ = 155.8292{{c}}

* [[WE]]: ~2 = 1200.4124{{c}}, ~3/2 = 701.1291{{c}}, ~12/11 = 155.8292{{c}}

: [[error map]]: {{val| +0.412 -0.414 +3.982 -1.435 -4.781 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.2359{{c}}, ~11~~/10~~ = 155.7399{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.2359{{c}}, ~12/11 = 155.7399{{c}}

: error map: {{val| 0.000 -0.719 +3.442 -2.029 -5.822 }}

Line 193:

Optimal tunings:

* WE: ~2 = 1198.9113{{c}}, ~3/2 = 700.9581{{c}}, ~11~~/10~~ = 154.7247{{c}}

* WE: ~2 = 1198.9113{{c}}, ~3/2 = 700.9581{{c}}, ~12/11 = 154.7247{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.6409{{c}}, ~11~~/10~~ = 154.9115{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.6409{{c}}, ~12/11 = 154.9115{{c}}

Line 291:

[[Badness]] (Sintel): 0.701

===~~= 13-limit =~~===

=== Eros ===

~~Subgroup: 2.~~3.~~5.7.11.13~~

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.

~~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 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~~ }}

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.

~~Badness (Sintel): 1~~.36

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.

~~==== 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

Line 323:

Line 312:

Badness (Sintel): 1.08

===== 17-limit =====

==== 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

Line 340:

Line 327:

Badness (Sintel): 0.931

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

==== 19-limit ====

Subgroup: 2.3.5.7.11.13.17.19

Line 355:

Line 342:

Badness (Sintel): 1.16

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

==== 23-limit ====

Subgroup: 2.3.5.7.11.13.17.19.23

Line 370:

Line 357:

Badness (Sintel): 1.08

==== Inanna ====

=== Astarte ===

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

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 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 ===

Subgroup: 2.3.5.7.11.13

Line 385:

Line 389:

Badness (Sintel): 1.01

==== Ishtar ====

=== Ishtar ===

Subgroup: 2.3.5.7.11.13

@@ Line 20: / Line 20: @@
 * [[WE]]: ~2 = 1200.6628{{c}}, ~3/2 = 701.8452{{c}}, ~11/10 = 157.8337{{c}}
 : [[error map]]: {{val| +0.663 +0.553 +1.190 -5.318 }}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9848{{c}}, ~11/10 = 157.6099{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9848{{c}}, ~12/11 = 157.6099{{c}}
 : error map: {{val| 0.000 +0.030 +0.451 -6.943 }}
@@ Line 44: / Line 44: @@
 [[Optimal tuning]]s:
-* [[WE]]: ~2 = 1200.1693{{c}}, ~3/2 = 702.2521{{c}}, ~11/10 = 157.1102{{c}}
+* [[WE]]: ~2 = 1200.1693{{c}}, ~3/2 = 702.2521{{c}}, ~12/11 = 157.1102{{c}}
 : [[error map]]: {{val| +0.169 +0.466 +2.057 +0.761 -5.668 }}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2478{{c}}, ~11/10 = 157.1265{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2478{{c}}, ~12/11 = 157.1265{{c}}
 : error map: {{val| 0.000 +0.293 +1.681 +0.306 -6.197 }}
@@ Line 84: / Line 84: @@
 Optimal tunings:
-* WE: ~2 = 1200.2411{{c}}, ~3/2 = 702.0090{{c}}, ~11/10 = 156.9897{{c}}
+* 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}}, ~11/10 = 156.9568{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.8818{{c}}, ~12/11 = 156.9568{{c}}
 Minimax tuning:
@@ Line 112: / Line 112: @@
 Optimal tunings:
-* WE: ~2 = 1199.9251{{c}}, ~3/2 = 699.2984{{c}}, ~11/10 = 155.3569{{c}}
+* WE: ~2 = 1199.9251{{c}}, ~3/2 = 699.2984{{c}}, ~12/11 = 155.3569{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.2982{{c}}, ~11/10 = 155.3484{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.2982{{c}}, ~12/11 = 155.3484{{c}}
 {{Optimal ET sequence|legend=0| 7, 9, 15, 22f, 24, 31 }}
@@ Line 176: / Line 176: @@
 [[Optimal tuning]]s:
-* [[WE]]: ~2 = 1200.4124{{c}}, ~3/2 = 701.1291{{c}}, ~11/10 = 155.8292{{c}}
+* [[WE]]: ~2 = 1200.4124{{c}}, ~3/2 = 701.1291{{c}}, ~12/11 = 155.8292{{c}}
 : [[error map]]: {{val| +0.412 -0.414 +3.982 -1.435 -4.781 }}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.2359{{c}}, ~11/10 = 155.7399{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.2359{{c}}, ~12/11 = 155.7399{{c}}
 : error map: {{val| 0.000 -0.719 +3.442 -2.029 -5.822 }}
@@ Line 193: / Line 193: @@
 Optimal tunings:
-* WE: ~2 = 1198.9113{{c}}, ~3/2 = 700.9581{{c}}, ~11/10 = 154.7247{{c}}
+* WE: ~2 = 1198.9113{{c}}, ~3/2 = 700.9581{{c}}, ~12/11 = 154.7247{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.6409{{c}}, ~11/10 = 154.9115{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.6409{{c}}, ~12/11 = 154.9115{{c}}
 {{Optimal ET sequence|legend=0| 7d, 8d, 15, 23de, 24d, 31 }}
@@ Line 291: / Line 291: @@
 [[Badness]] (Sintel): 0.701
-==== 13-limit ====
+=== Eros ===
-Subgroup: 2.3.5.7.11.13
+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.
-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 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 }}
+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.
-Badness (Sintel): 1.36
+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.
-==== 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
@@ Line 323: / Line 312: @@
 Badness (Sintel): 1.08
-===== 17-limit =====
+==== 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
@@ Line 340: / Line 327: @@
 Badness (Sintel): 0.931
-===== 19-limit =====
+==== 19-limit ====
 Subgroup: 2.3.5.7.11.13.17.19
@@ Line 355: / Line 342: @@
 Badness (Sintel): 1.16
-===== 23-limit =====
+==== 23-limit ====
 Subgroup: 2.3.5.7.11.13.17.19.23
@@ Line 370: / Line 357: @@
 Badness (Sintel): 1.08
-==== Inanna ====
+=== Astarte ===
+This extension is catalogued as tridecimal aphrodite in [[Graham Breed]]'s temperament finder.
+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 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 ===
 Subgroup: 2.3.5.7.11.13
@@ Line 385: / Line 389: @@
 Badness (Sintel): 1.01
-==== Ishtar ====
+=== Ishtar ===
 Subgroup: 2.3.5.7.11.13