Cataharry family: Difference between revisions

Line 6:

[[Subgroup]]: 2.3.5.7

[[Comma list]]: [[19683/19600]]

[[Comma list]]: 19683/19600

: mapping generators: ~2, ~140/81, ~5

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.1072{{c}}, ~140/81 = 950.6740{{c}}, ~5/4 = 386.3883{{c}}

: [[error map]]: {{val| +0.107 -0.607 +0.289 +0.423 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~140/81 = 950.6211{{c}}, ~5/4 = 386.4833{{c}}

: error map: {{val| 0.000 -0.713 +0.170 +0.280 }}

[[Minimax tuning]]:

Line 18:

Line 24:

{{Optimal ET sequence|legend=1| 19, 34d, 39d, 53, 72, 130, 183, 202, 385, 587 }}

[[Badness]] (~~Smith~~): 0.~~301 × 10-3~~

[[Badness]] (Sintel): 1.33

[[Projection pair]]s: <code>3 19600/6561 7 1033052339200000000/150094635296999121</code> to 2.5.81/7

Line 29:

Line 35:

: mapping generators: ~2, ~140/81, ~5

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0427{{c}}, ~140/81 = 950.6653{{c}}, ~5/4 = 386.3435{{c}}

: [[error map]]: {{val| +0.043 -0.624 +0.115 +0.647 +0.242 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~140/81 = 950.6425{{c}}, ~5/4 = 386.3889{{c}}

: error map: {{val| 0.000 -0.670 +0.075 +0.568 +0.139 }}

[[Minimax tuning]]:

* [[11-odd-limit]]

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 8/3 0 4/3 -2/3 -2/3 }}, {{monzo| 5 0 7/2 -2 -3/2 }}, {{monzo| 5 0 5/2 -1 -3/2 }}, {{monzo| 5 0 5/2 -2 -1/2 }}]

: [[~~Eigenmonzo~~ basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/5

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/5

{{Optimal ET sequence|legend=1| 19, 33cd, 39d, 53, 58, 72, 130, 183, 255, 385, 640 }}

[[Badness]] (~~Smith~~): 0.~~621 × 10-3~~

[[Badness]] (Sintel): 0.746

=== Madagascar ===

Line 51:

Line 63:

: 15/13 length = 0.3019, 26/25 length = 0.9477

: Angle (15/13, 26/25) = 81.6476

Optimal tunings:

* WE: ~2 = 1199.9446{{c}}, ~26/15 = 950.7332{{c}}, ~5/4 = 387.0575{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.7647{{c}}, ~5/4 = 387.0080{{c}}

Minimax tuning:

Line 59:

Line 75:

{{Optimal ET sequence|legend=0| 19, 33cdff, 39df, 53, 58, 72, 111, 130, 183, 313 }}

Badness (~~Smith~~): 0.~~560 × 10-3~~

Badness (Sintel): 0.524

Complexity spectrum: 15/13, 4/3, 13/10, 10/9, 6/5, 9/7, 18/13, 9/8, 5/4, 7/6, 13/12, 15/14, 16/15, 14/13, 8/7, 7/5, 16/13, 11/10, 15/11, 11/8, 12/11, 13/11, 11/9, 14/11

Projection pairs: 3 676/225 7 1357375919744/192216796875 11 16218292236328125/1491077447838784 13 8788/675 to 2.5.13/3

Projection pairs: <code>3 676/225 7 1357375919744/192216796875 11 16218292236328125/1491077447838784 13 8788/675</code> to 2.5.13/3

Scales: [[madagascar19]]

Line 73:

Line 89:

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.2224{{c}}, ~140/81 = 950.6271{{c}}, ~5/4 = 386.3424{{c}}

: [[error map]]: {{val| +0.222 -0.701 +0.474 -0.414 +0.559 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~140/81 = 950.5069{{c}}, ~5/4 = 386.5450{{c}}

: error map: {{val| 0.000 -0.941 +0.231 -0.809 +0.198 }}

{{Optimal ET sequence|legend=1| 19e, 33cdee, 39dee, 53e, 58, 72, 130, 202, 351, 423d, 481, 553 }}

[[Badness]] (~~Smith~~): 0.~~988 × 10-3~~

[[Badness]] (Sintel): 1.19

=== 13-limit ===

Line 84:

Line 106:

Mapping: {{mapping| 1 0 0 -2 -7 -1 | 0 2 0 9 22 3 | 0 0 1 -1 -3 1 }}

Optimal tunings:

* WE: ~2 = 1200.1732{{c}}, ~26/15 = 950.6934{{c}}, ~5/4 = 387.0339{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.5975{{c}}, ~5/4 = 387.1756{{c}}

{{Optimal ET sequence|legend=0| 19e, 33cdeeff, 39deef, 53e, 58, 72, 130, 279, 409 }}

Badness (~~Smith~~): 0.~~786 × 10-3~~

Badness (Sintel): 0.735

== Hagrid ==

Line 96:

Line 122:

: mapping generators: ~99/70, ~140/81, ~5

[[Optimal tuning]]s:

* [[WE]]: ~99/70 = 600.0301{{c}}, ~140/81 = 950.5810{{c}}, ~5/4 = 386.1803{{c}}

: [[error map]]: {{val| +0.060 -0.793 -0.013 -0.019 +1.557 }}

* [[CWE]]: ~99/70 = 600.0000{{c}}, ~140/81 = 950.5516{{c}}, ~5/4 = 386.2351{{c}}

: error map: {{val| 0.000 -0.852 -0.079 -0.096 +1.440 }}

[[Badness]] (~~Smith~~): 1.~~116 × 10-3~~

[[Badness]] (Sintel): 1.34

=== 13-limit ===

Line 107:

Line 139:

Mapping: {{mapping| 2 0 0 -4 -1 -2 | 0 2 0 9 5 3 | 0 0 1 -1 0 1 }}

: ~~mapping generators~~: ~99/70, ~26/15, ~5

Optimal tunings:

* WE: ~55/39 = 599.9896{{c}}, ~26/15 = 950.6467{{c}}, ~5/4 = 387.0194{{c}}

* CWE: ~55/39 = 600.0000{{c}}, ~26/15 = 950.6571{{c}}, ~5/4 = 387.0028{{c}}

Badness (~~Smith~~): 0.~~876 × 10-3~~

Badness (Sintel): 0.819

== Draco ==

Line 119:

Line 154:

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.3870{{c}}, ~140/81 = 950.8204{{c}}, ~5/4 = 389.4825{{c}}

: [[error map]]: {{val| -0.613 -0.314 +1.943 +1.527 -2.000 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~140/81 = 951.2120{{c}}, ~5/4 = 389.3756{{c}}

: error map: {{val| 0.000 +0.469 +3.062 +2.706 -1.035 }}

{{Optimal ET sequence|legend=1| 19e, 24, 34d, 53, 58, 111, 280cd }}

[[Badness]] (~~Smith~~): 1.~~934 × 10-3~~

[[Badness]] (Sintel): 2.32

=== 13-limit ===

Line 130:

Line 171:

Mapping: {{mapping| 1 0 0 -2 -6 -1 | 0 2 0 9 9 3 | 0 0 1 -1 1 1 }}

Optimal tunings:

* WE: ~2 = 1199.3551{{c}}, ~26/15 = 950.7982{{c}}, ~5/4 = 389.3905{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.2095{{c}}, ~5/4 = 389.2432{{c}}

{{Optimal ET sequence|legend=0| 19e, 24, 34d, 53, 58, 111, 280cdf }}

Badness (~~Smith~~): 0.~~999 × 10-3~~

Badness (Sintel): 0.934

[[Category:Temperament families]]

[[Category:Cataharry family| ]]

[[Category:Rank 3]]

@@ Line 6: / Line 6: @@
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: [[19683/19600]]
+[[Comma list]]: 19683/19600
 {{Mapping|legend=1| 1 0 0 -2 | 0 2 0 9 | 0 0 1 -1 }}
 : mapping generators: ~2, ~140/81, ~5
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.1072{{c}}, ~140/81 = 950.6740{{c}}, ~5/4 = 386.3883{{c}}
+: [[error map]]: {{val| +0.107 -0.607 +0.289 +0.423 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~140/81 = 950.6211{{c}}, ~5/4 = 386.4833{{c}}
+: error map: {{val| 0.000 -0.713 +0.170 +0.280 }}
 [[Minimax tuning]]:
@@ Line 18: / Line 24: @@
 {{Optimal ET sequence|legend=1| 19, 34d, 39d, 53, 72, 130, 183, 202, 385, 587 }}
-[[Badness]] (Smith): 0.301 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 1.33
 [[Projection pair]]s: <code>3 19600/6561 7 1033052339200000000/150094635296999121</code> to 2.5.81/7
@@ Line 29: / Line 35: @@
 {{Mapping|legend=1| 1 0 0 -2 6 | 0 2 0 9 -12 | 0 0 1 -1 3 }}
 : mapping generators: ~2, ~140/81, ~5
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0427{{c}}, ~140/81 = 950.6653{{c}}, ~5/4 = 386.3435{{c}}
+: [[error map]]: {{val| +0.043 -0.624 +0.115 +0.647 +0.242 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~140/81 = 950.6425{{c}}, ~5/4 = 386.3889{{c}}
+: error map: {{val| 0.000 -0.670 +0.075 +0.568 +0.139 }}
 [[Minimax tuning]]:
 * [[11-odd-limit]]
 : [{{monzo| 1 0 0 0 0 }}, {{monzo| 8/3 0 4/3 -2/3 -2/3 }}, {{monzo| 5 0 7/2 -2 -3/2 }}, {{monzo| 5 0 5/2 -1 -3/2 }}, {{monzo| 5 0 5/2 -2 -1/2 }}]
-: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/5
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/5
 {{Optimal ET sequence|legend=1| 19, 33cd, 39d, 53, 58, 72, 130, 183, 255, 385, 640 }}
-[[Badness]] (Smith): 0.621 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 0.746
 === Madagascar ===
@@ Line 51: / Line 63: @@
 : 15/13 length = 0.3019, 26/25 length = 0.9477
 : Angle (15/13, 26/25) = 81.6476
+Optimal tunings:
+* WE: ~2 = 1199.9446{{c}}, ~26/15 = 950.7332{{c}}, ~5/4 = 387.0575{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.7647{{c}}, ~5/4 = 387.0080{{c}}
 Minimax tuning:
@@ Line 59: / Line 75: @@
 {{Optimal ET sequence|legend=0| 19, 33cdff, 39df, 53, 58, 72, 111, 130, 183, 313 }}
-Badness (Smith): 0.560 × 10<sup>-3</sup>
+Badness (Sintel): 0.524
 Complexity spectrum: 15/13, 4/3, 13/10, 10/9, 6/5, 9/7, 18/13, 9/8, 5/4, 7/6, 13/12, 15/14, 16/15, 14/13, 8/7, 7/5, 16/13, 11/10, 15/11, 11/8, 12/11, 13/11, 11/9, 14/11
-Projection pairs: 3 676/225 7 1357375919744/192216796875 11 16218292236328125/1491077447838784 13 8788/675 to 2.5.13/3
+Projection pairs: <code>3 676/225 7 1357375919744/192216796875 11 16218292236328125/1491077447838784 13 8788/675</code> to 2.5.13/3
 Scales: [[madagascar19]]
@@ Line 73: / Line 89: @@
 {{Mapping|legend=1| 1 0 0 -2 -7 | 0 2 0 9 22 | 0 0 1 -1 -3 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.2224{{c}}, ~140/81 = 950.6271{{c}}, ~5/4 = 386.3424{{c}}
+: [[error map]]: {{val| +0.222 -0.701 +0.474 -0.414 +0.559 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~140/81 = 950.5069{{c}}, ~5/4 = 386.5450{{c}}
+: error map: {{val| 0.000 -0.941 +0.231 -0.809 +0.198 }}
 {{Optimal ET sequence|legend=1| 19e, 33cdee, 39dee, 53e, 58, 72, 130, 202, 351, 423d, 481, 553 }}
-[[Badness]] (Smith): 0.988 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 1.19
 === 13-limit ===
@@ Line 84: / Line 106: @@
 Mapping: {{mapping| 1 0 0 -2 -7 -1 | 0 2 0 9 22 3 | 0 0 1 -1 -3 1 }}
+Optimal tunings:
+* WE: ~2 = 1200.1732{{c}}, ~26/15 = 950.6934{{c}}, ~5/4 = 387.0339{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.5975{{c}}, ~5/4 = 387.1756{{c}}
 {{Optimal ET sequence|legend=0| 19e, 33cdeeff, 39deef, 53e, 58, 72, 130, 279, 409 }}
-Badness (Smith): 0.786 × 10<sup>-3</sup>
+Badness (Sintel): 0.735
 == Hagrid ==
@@ Line 96: / Line 122: @@
 {{Mapping|legend=1| 2 0 0 -4 -1 | 0 2 0 9 5 | 0 0 1 -1 0 }}
 : mapping generators: ~99/70, ~140/81, ~5
+[[Optimal tuning]]s:
+* [[WE]]: ~99/70 = 600.0301{{c}}, ~140/81 = 950.5810{{c}}, ~5/4 = 386.1803{{c}}
+: [[error map]]: {{val| +0.060 -0.793 -0.013 -0.019 +1.557 }}
+* [[CWE]]: ~99/70 = 600.0000{{c}}, ~140/81 = 950.5516{{c}}, ~5/4 = 386.2351{{c}}
+: error map: {{val| 0.000 -0.852 -0.079 -0.096 +1.440 }}
 {{Optimal ET sequence|legend=1| 24, 34d, 48, 58, 72, 130, 202 }}
-[[Badness]] (Smith): 1.116 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 1.34
 === 13-limit ===
@@ Line 107: / Line 139: @@
 Mapping: {{mapping| 2 0 0 -4 -1 -2 | 0 2 0 9 5 3 | 0 0 1 -1 0 1 }}
-: mapping generators: ~99/70, ~26/15, ~5
+Optimal tunings:
+* WE: ~55/39 = 599.9896{{c}}, ~26/15 = 950.6467{{c}}, ~5/4 = 387.0194{{c}}
+* CWE: ~55/39 = 600.0000{{c}}, ~26/15 = 950.6571{{c}}, ~5/4 = 387.0028{{c}}
 {{Optimal ET sequence|legend=0| 24, 34d, 58, 72, 130 }}
-Badness (Smith): 0.876 × 10<sup>-3</sup>
+Badness (Sintel): 0.819
 == Draco ==
@@ Line 119: / Line 154: @@
 {{Mapping|legend=1| 1 0 0 -2 -6 | 0 2 0 9 9 | 0 0 1 -1 1 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.3870{{c}}, ~140/81 = 950.8204{{c}}, ~5/4 = 389.4825{{c}}
+: [[error map]]: {{val| -0.613 -0.314 +1.943 +1.527 -2.000 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~140/81 = 951.2120{{c}}, ~5/4 = 389.3756{{c}}
+: error map: {{val| 0.000 +0.469 +3.062 +2.706 -1.035 }}
 {{Optimal ET sequence|legend=1| 19e, 24, 34d, 53, 58, 111, 280cd }}
-[[Badness]] (Smith): 1.934 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 2.32
 === 13-limit ===
@@ Line 130: / Line 171: @@
 Mapping: {{mapping| 1 0 0 -2 -6 -1 | 0 2 0 9 9 3 | 0 0 1 -1 1 1 }}
+Optimal tunings:
+* WE: ~2 = 1199.3551{{c}}, ~26/15 = 950.7982{{c}}, ~5/4 = 389.3905{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.2095{{c}}, ~5/4 = 389.2432{{c}}
 {{Optimal ET sequence|legend=0| 19e, 24, 34d, 53, 58, 111, 280cdf }}
-Badness (Smith): 0.999 × 10<sup>-3</sup>
+Badness (Sintel): 0.934
 [[Category:Temperament families]]
 [[Category:Cataharry family| ]] <!-- main article -->
 [[Category:Rank 3]]