Werckismic temperaments: Difference between revisions

Line 30:

: mapping generators: ~2, ~21/20, ~5

: ~~mapping generators~~: ~2, ~21/20, ~5

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0906{{c}}, ~21/20 = 83.0793{{c}}, ~5/4 = 386.1759{{c}}

: [[error map]]: {{val| +0.091 -0.250 +0.043 -0.914 +1.288 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.0654{{c}}, ~5/4 = 386.2710{{c}}

: error map: {{val| 0.000 -0.348 -0.043 -1.097 +1.084 }}

[[Minimax tuning]]:

* [[11-odd-limit]]

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 4/17 12/17 0 -6/17 6/17 }}, {{monzo| -27/17 4/17 0 -2/17 19/17 }}, {{monzo| 8/17 -10/17 0 5/17 12/17 }}, {{monzo| 0 0 0 0 1 }}]

: [[~~Eigenmonzo~~ basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7.11

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7.11

{{Optimal ET sequence|legend=1| 14c, 15, 29, 43, 57, 58, 72, 130, 202, 217, 289, 491 }}

[[Badness]]: 0.~~604 × 10-3~~

[[Badness]] (Sintel): 0.725

=== 13-limit ===

Line 49:

Line 54:

Mapping: {{mapping| 1 2 0 0 1 2 | 0 -6 0 7 2 -9 | 0 0 1 1 1 1 }}

: ~~mapping generators~~: ~2, ~21/20, ~5

Optimal tunings:

* WE: ~2 = 1200.0242{{c}}, ~21/20 = 83.0177{{c}}, ~5/4 = 386.6582{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.0144{{c}}, ~5/4 = 386.6802{{c}}

Minimax tuning:

Line 56:

Line 63:

: unchanged-interval (eigenmonzo) basis: 2.5.13/7

{{Optimal ET sequence|legend=1| 14cf, 15, 29, 43, 58, 72, 87, 130, 217, 289 }}

{{Optimal ET sequence|legend=0| 14cf, 15, 29, 43, 58, 72, 87, 130, 217, 289 }}

Badness: 0.~~540 × 10-3~~

Badness (Sintel): 0.505

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

Line 69:

Line 76:

: mapping generators: ~567/400, ~3, ~5

: ~~mapping generators~~: ~567/400, ~3, ~5

[[Optimal tuning]]s:

* [[WE]]: ~567/400 = 600.1005{{c}}, ~3/2 = 701.3045{c}}, ~5/4 = 386.3934{{c}}

: [[error map]]: {{val| +0.201 -0.449 +0.482 -0.353 }}

* [[CWE]]: ~567/400 = 600.0000{{c}}, ~3/2 = 701.2691{{c}}, ~5/4 = 386.5935{{c}}

: error map: {{val| 0.000 -0.686 +0.280 -0.715 }}

{{Optimal ET sequence|legend=1| 12, 26, 34, 46, 58, 72, 118, 130, 202, 320, 450, 522, 972bd, 1174bd }}

[[Badness]]: 0.~~501 × 10-3~~

[[Badness]] (Sintel): 2.21

=== 11-limit ===

Line 82:

Line 94:

: mapping generators: ~99/70, ~3, ~5

[[Optimal tuning]]s:

* [[WE]]: ~99/70 = 600.1002{{c}}, ~3/2 = 701.3812{c}}, ~5/4 = 386.3126{{c}}

: [[error map]]: {{val| +0.200 -0.373 +0.400 -0.823 +0.536 }}

* [[CWE]]: ~99/70 = 600.0000{{c}}, ~3/2 = 701.3457{{c}}, ~5/4 = 386.5123{{c}}

: error map: {{val| 0.000 -0.609 +0.199 -1.184 +0.144 }}

{{Optimal ET sequence|legend=1| 12, 26, 44ce, 46, 58, 72, 118, 130, 190, 248, 320 }}

[[Badness]]: 0.~~419 × 10-3~~

[[Badness]] (Sintel): 0.503

=== 13-limit ===

Line 94:

Line 113:

Mapping: {{mapping| 2 0 0 9 12 11 | 0 1 0 -4 -6 -7 | 0 0 1 2 3 4 }}

{{Optimal ET sequence|legend=1| 12f, 14cf, 26, 46, 58, 72, 130 }}

Optimal tunings:

* WE: ~55/39 = 600.0052{{c}}, ~3/2 = 701.4439{c}}, ~5/4 = 387.1861{{c}}

* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 701.4418{{c}}, ~5/4 = 387.1949{{c}}

{{Optimal ET sequence|legend=0| 12f, 14cf, 26, 46, 58, 72, 130 }}

Badness: 0.~~636 × 10-3~~

Badness (Sintel): 0.595

== Trimyna ==

Line 106:

Line 129:

[[Comma list]]: 50421/50000

: mapping generators: ~2, ~10/7, ~5

: ~~mapping generators~~: ~2, ~7/5, ~5

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.1652{{c}}, ~10/7 = 620.4031{c}}, ~5/4 = 387.0990{{c}}

: [[error map]]: {{val| +0.165 -0.105 +1.116 -1.634 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 620.3427{{c}}, ~5/4 = 387.2591{{c}}

: error map: {{val| 0.000 -0.241 +0.945 -1.910 }}

{{Optimal ET sequence|legend=1| 27, 31, 58, 87, 118, 267d, 385d, 412d }} ~~<nowiki>~~*~~</nowiki>~~

{{Optimal ET sequence|legend=1| 27, 31, 58, 87, 118, 267d, 385d, 412d }} *

<nowiki>*</nowiki> [[optimal patent val]]: [[294edo|294]]

[[Badness]]: 0.~~870 × 10-3~~

[[Badness]] (Sintel): 3.84

[[Projection pair]]: 3 50000/16807 to 2.5.7

[[Projection pair]]: <code>3 50000/16807</code> to 2.5.7

=== 11-limit ===

Line 123:

Line 151:

[[Comma list]]: 441/440, 3388/3375

: mapping generators: ~2, ~10/7, ~5

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0909{{c}}, ~10/7 = 620.4267{c}}, ~5/4 = 387.4364{{c}}

: [[error map]]: {{val| +0.091 +0.088 +1.304 -1.544 -0.559 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 620.3901{{c}}, ~5/4 = 387.5065{{c}}

: error map: {{val| 0.000 -0.004 +1.193 -1.710 -0.691 }}

{{Optimal ET sequence|legend=1| 27e, 31, 58, 87, 118, 294, 412d }}

[[Badness]]: 0.~~512 × 10-3~~

[[Badness]] (Sintel): 0.711

==== 13-limit ====

Line 136:

Line 171:

Mapping: {{mapping| 1 4 0 0 5 -2 | 0 -5 0 1 -8 7 | 0 0 1 1 1 1 }}

{{Optimal ET sequence|legend=1| 27e, 29, 31, 56, 58, 87, 118, 145, 232 }}

Optimal tunings:

* WE: ~2 = 1199.6799{{c}}, ~10/7 = 620.4057{c}}, ~5/4 = 387.7593{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 620.5510{{c}}, ~5/4 = 387.5132{{c}}

{{Optimal ET sequence|legend=0| 27e, 29, 31, 56, 58, 87, 118, 145, 232 }}

Badness: 0.~~814 × 10-3~~

Badness (Sintel): 0.761

==== Trimy ====

Line 145:

Line 184:

Comma list: 441/440, 676/675, 847/845

Mapping: {{mapping| 1 4 0 ~~0 5 5~~ | 0 -10 0 2 -16 -15 | 0 0 1 1 1 1 }}

Mapping: {{mapping| 1 -6 0 2 -11 -10 | 0 10 0 -2 16 15 | 0 0 1 1 1 1 }}

: mapping generators: ~2, ~22/13, ~5

: ~~mapping generators~~: ~2, ~13/11, ~5

Optimal tunings:

* WE: ~2 = 1200.0924{{c}}, ~22/13 = 910.2591{c}}, ~5/4 = 387.4264{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~22/13 = 910.1939{{c}}, ~5/4 = 387.4946{{c}}

{{Optimal ET sequence|legend=1| 29, 54ceeff, 58, 87, 145, 149, 207, 236d, 294 }}

{{Optimal ET sequence|legend=0| 29, 54ceeff, 58, 87, 145, 149, 207, 236d, 294 }}

Badness: 1.~~478 × 10-3~~

Badness (Sintel): 1.38

[[Category:Temperament collections]]

@@ Line 30: / Line 30: @@
 {{Mapping|legend=1| 1 2 0 0 1 | 0 -6 0 7 2 | 0 0 1 1 1 }}
+: mapping generators: ~2, ~21/20, ~5
-: mapping generators: ~2, ~21/20, ~5
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0906{{c}}, ~21/20 = 83.0793{{c}}, ~5/4 = 386.1759{{c}}
+: [[error map]]: {{val| +0.091 -0.250 +0.043 -0.914 +1.288 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.0654{{c}}, ~5/4 = 386.2710{{c}}
+: error map: {{val| 0.000 -0.348 -0.043 -1.097 +1.084 }}
 [[Minimax tuning]]:
 * [[11-odd-limit]]
 : [{{monzo| 1 0 0 0 0 }}, {{monzo| 4/17 12/17 0 -6/17 6/17 }}, {{monzo| -27/17 4/17 0 -2/17 19/17 }}, {{monzo| 8/17 -10/17 0 5/17 12/17 }}, {{monzo| 0 0 0 0 1 }}]
-: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7.11
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7.11
 {{Optimal ET sequence|legend=1| 14c, 15, 29, 43, 57, 58, 72, 130, 202, 217, 289, 491 }}
-[[Badness]]: 0.604 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 0.725
 === 13-limit ===
@@ Line 49: / Line 54: @@
 Mapping: {{mapping| 1 2 0 0 1 2 | 0 -6 0 7 2 -9 | 0 0 1 1 1 1 }}
-: mapping generators: ~2, ~21/20, ~5
+Optimal tunings:
+* WE: ~2 = 1200.0242{{c}}, ~21/20 = 83.0177{{c}}, ~5/4 = 386.6582{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.0144{{c}}, ~5/4 = 386.6802{{c}}
 Minimax tuning:
@@ Line 56: / Line 63: @@
 : unchanged-interval (eigenmonzo) basis: 2.5.13/7
-{{Optimal ET sequence|legend=1| 14cf, 15, 29, 43, 58, 72, 87, 130, 217, 289 }}
+{{Optimal ET sequence|legend=0| 14cf, 15, 29, 43, 58, 72, 87, 130, 217, 289 }}
-Badness: 0.540 × 10<sup>-3</sup>
+Badness (Sintel): 0.505
 Complexity spectrum: 11/10, 15/13, 14/11, 4/3, 7/5, 5/4, 11/8, 18/13, 15/11, 13/12, 13/10, 6/5, 8/7, 16/15, 12/11, 13/11, 9/8, 16/13, 15/14, 10/9, 7/6, 11/9, 14/13, 9/7
@@ Line 69: / Line 76: @@
 {{Mapping|legend=1| 2 0 0 9 | 0 1 0 -4 | 0 0 1 2 }}
+: mapping generators: ~567/400, ~3, ~5
-: mapping generators: ~567/400, ~3, ~5
+[[Optimal tuning]]s:
+* [[WE]]: ~567/400 = 600.1005{{c}}, ~3/2 = 701.3045{c}}, ~5/4 = 386.3934{{c}}
+: [[error map]]: {{val| +0.201 -0.449 +0.482 -0.353 }}
+* [[CWE]]: ~567/400 = 600.0000{{c}}, ~3/2 = 701.2691{{c}}, ~5/4 = 386.5935{{c}}
+: error map: {{val| 0.000 -0.686 +0.280 -0.715 }}
 {{Optimal ET sequence|legend=1| 12, 26, 34, 46, 58, 72, 118, 130, 202, 320, 450, 522, 972bd, 1174bd }}
-[[Badness]]: 0.501 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 2.21
 === 11-limit ===
@@ Line 82: / Line 94: @@
 {{Mapping|legend=1| 2 0 0 9 12 | 0 1 0 -4 -6 | 0 0 1 2 3 }}
+: mapping generators: ~99/70, ~3, ~5
+[[Optimal tuning]]s:
+* [[WE]]: ~99/70 = 600.1002{{c}}, ~3/2 = 701.3812{c}}, ~5/4 = 386.3126{{c}}
+: [[error map]]: {{val| +0.200 -0.373 +0.400 -0.823 +0.536 }}
+* [[CWE]]: ~99/70 = 600.0000{{c}}, ~3/2 = 701.3457{{c}}, ~5/4 = 386.5123{{c}}
+: error map: {{val| 0.000 -0.609 +0.199 -1.184 +0.144 }}
 {{Optimal ET sequence|legend=1| 12, 26, 44ce, 46, 58, 72, 118, 130, 190, 248, 320 }}
-[[Badness]]: 0.419 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 0.503
 === 13-limit ===
@@ Line 94: / Line 113: @@
 Mapping: {{mapping| 2 0 0 9 12 11 | 0 1 0 -4 -6 -7 | 0 0 1 2 3 4 }}
-{{Optimal ET sequence|legend=1| 12f, 14cf, 26, 46, 58, 72, 130 }}
+Optimal tunings:
+* WE: ~55/39 = 600.0052{{c}}, ~3/2 = 701.4439{c}}, ~5/4 = 387.1861{{c}}
+* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 701.4418{{c}}, ~5/4 = 387.1949{{c}}
+{{Optimal ET sequence|legend=0| 12f, 14cf, 26, 46, 58, 72, 130 }}
-Badness: 0.636 × 10<sup>-3</sup>
+Badness (Sintel): 0.595
 == Trimyna ==
@@ Line 106: / Line 129: @@
 [[Comma list]]: 50421/50000
-{{Mapping|legend=1| 1 4 0 0 | 0 -5 0 1 | 0 0 1 1 }}
+{{Mapping|legend=1| 1 -1 0 1 | 0 5 0 -1 | 0 0 1 1 }}
+: mapping generators: ~2, ~10/7, ~5
-: mapping generators: ~2, ~7/5, ~5
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.1652{{c}}, ~10/7 = 620.4031{c}}, ~5/4 = 387.0990{{c}}
+: [[error map]]: {{val| +0.165 -0.105 +1.116 -1.634 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 620.3427{{c}}, ~5/4 = 387.2591{{c}}
+: error map: {{val| 0.000 -0.241 +0.945 -1.910 }}
-{{Optimal ET sequence|legend=1| 27, 31, 58, 87, 118, 267d, 385d, 412d }} <nowiki>*</nowiki>
+{{Optimal ET sequence|legend=1| 27, 31, 58, 87, 118, 267d, 385d, 412d }} *
 <nowiki>*</nowiki> [[optimal patent val]]: [[294edo|294]]
-[[Badness]]: 0.870 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 3.84
-[[Projection pair]]: 3 50000/16807 to 2.5.7
+[[Projection pair]]: <code>3 50000/16807</code> to 2.5.7
 === 11-limit ===
@@ Line 123: / Line 151: @@
 [[Comma list]]: 441/440, 3388/3375
-{{Mapping|legend=1| 1 4 0 0 5 | 0 -5 0 1 -8 | 0 0 1 1 1 }}
+{{Mapping|legend=1| 1 -1 0 1 -3 | 0 5 0 -1 8 | 0 0 1 1 1 }}
+: mapping generators: ~2, ~10/7, ~5
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0909{{c}}, ~10/7 = 620.4267{c}}, ~5/4 = 387.4364{{c}}
+: [[error map]]: {{val| +0.091 +0.088 +1.304 -1.544 -0.559 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 620.3901{{c}}, ~5/4 = 387.5065{{c}}
+: error map: {{val| 0.000 -0.004 +1.193 -1.710 -0.691 }}
 {{Optimal ET sequence|legend=1| 27e, 31, 58, 87, 118, 294, 412d }}
-[[Badness]]: 0.512 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 0.711
 ==== 13-limit ====
@@ Line 136: / Line 171: @@
 Mapping: {{mapping| 1 4 0 0 5 -2 | 0 -5 0 1 -8 7 | 0 0 1 1 1 1 }}
-{{Optimal ET sequence|legend=1| 27e, 29, 31, 56, 58, 87, 118, 145, 232 }}
+Optimal tunings:
+* WE: ~2 = 1199.6799{{c}}, ~10/7 = 620.4057{c}}, ~5/4 = 387.7593{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 620.5510{{c}}, ~5/4 = 387.5132{{c}}
+{{Optimal ET sequence|legend=0| 27e, 29, 31, 56, 58, 87, 118, 145, 232 }}
-Badness: 0.814 × 10<sup>-3</sup>
+Badness (Sintel): 0.761
 ==== Trimy ====
@@ Line 145: / Line 184: @@
 Comma list: 441/440, 676/675, 847/845
-Mapping: {{mapping| 1 4 0 0 5 5 | 0 -10 0 2 -16 -15 | 0 0 1 1 1 1 }}
+Mapping: {{mapping| 1 -6 0 2 -11 -10 | 0 10 0 -2 16 15 | 0 0 1 1 1 1 }}
+: mapping generators: ~2, ~22/13, ~5
-: mapping generators: ~2, ~13/11, ~5
+Optimal tunings:
+* WE: ~2 = 1200.0924{{c}}, ~22/13 = 910.2591{c}}, ~5/4 = 387.4264{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~22/13 = 910.1939{{c}}, ~5/4 = 387.4946{{c}}
-{{Optimal ET sequence|legend=1| 29, 54ceeff, 58, 87, 145, 149, 207, 236d, 294 }}
+{{Optimal ET sequence|legend=0| 29, 54ceeff, 58, 87, 145, 149, 207, 236d, 294 }}
-Badness: 1.478 × 10<sup>-3</sup>
+Badness (Sintel): 1.38
 [[Category:Temperament collections]]