Orwell extensions: Difference between revisions

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→Spectrum of Orwell Tunings by Eigenmonzos: improve and standardize tuning spectra tables
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These spectra suggest possible tuning choices. For 13-limit orwell, the 5-limit minimax tuning featuring pure 6/5 eigenmonzos seems like an excellent choice, as it's right in the middle of the least squares range and very close to 13-limit least squares. Pure 13s, using the 16/13 eigenmonzo, might also please some people. For blair, pure 5/4s using the 5/4 eigenmonzo tuning is very close to 15-limit least squares and in general in the middle of the action. For winston, sticking with the 11/9 eigenmonzo minimax tuning seems reasonable.
These spectra suggest possible tuning choices. For 13-limit orwell, the 5-limit minimax tuning featuring pure 6/5 eigenmonzos seems like an excellent choice, as it's right in the middle of the least squares range and very close to 13-limit least squares. Pure 13s, using the 16/13 eigenmonzo, might also please some people. For blair, pure 5/4s using the 5/4 eigenmonzo tuning is very close to 15-limit least squares and in general in the middle of the action. For winston, sticking with the 11/9 eigenmonzo minimax tuning seems reasonable.


==Spectrum of Orwell Tunings by Eigenmonzos==
==Tuning spectrum==


Gencom: [2 7/6; 99/98 121/120 176/175 275/273]
Gencom: [2 7/6; 99/98 121/120 176/175 275/273]
Line 10: Line 10:
Gencom map: [<1 0 3 1 3 8|, <0 7 -3 8 2 -19|]
Gencom map: [<1 0 3 1 3 8|, <0 7 -3 8 2 -19|]


{| class="wikitable"
{| class="wikitable center-all"
|-
|-
! | Eigenmonzo
! | ET<br>generator
! | Subminor Third
! | [[eigenmonzo|eigenmonzo<br>(unchanged interval]])
! | subminor<br>third (¢)
! | comments
|-
|-
| |
| | 7/6
| | 7/6
| | 266.871
| | 266.871
| |
|-
|-
| |
| | 15/11
| | 15/11
| | 268.475
| | 268.475
| |
|-
|-
| |
| | 14/11
| | 14/11
| | 269.585
| | 269.585
| |
|-
|-
| |
| | 12/11
| | 12/11
| | 270.127
| | 270.127
| |
|-
|-
| |
| | 15/14
| | 15/14
| | 270.139
| | 270.139
| |
|-
|-
| | 7\31
| | 7\31
| |
| | 270.968
| | 270.968
| |
|-
|-
| |
| | 11/9
| | 11/9
| | 271.049
| | 271.049
| |
|-
|-
| |
| | 8/7
| | 8/7
| | 271.103
| | 271.103
| |
|-
|-
| |
| | 7/5
| | 7/5
| | 271.137
| | 271.137
| |
|-
|-
| |
| | 5/4
| | 5/4
| | 271.229
| | 271.229
| |
|-
|-
| |
| | 1361367/1000000
| | 1361367/1000000
| | 271.326 (7 limit least squares)
| | 271.326
| | 7 limit least squares
|-
|-
| |
| | 14/13
| | 14/13
| | 271.418 (13 and 15 limit minimax)
| | 271.418
| | 13 and 15 limit minimax
|-
|-
| | 19\84
| | 19\84
| |
| | 271.429
| | 271.429
| |
|-
|-
| | |0 119 -46 20 -16&gt;
| |
| | 271.445 (11 limit least squares)
| | {{vector|0 119 -46 20 -16}}
| | 271.445
| | 11 limit least squares
|-
|-
| | x^10 + 2x^3 = 8
| |
| | 271.508 (equal beating)
| | <math>x^{10} + 2x^3 = 8</math>
| | 271.508
| | equal beating
|-
|-
| |
| | 16/13
| | 16/13
| | 271.551
| | 271.551
| |
|-
|-
| | |0 90 -41 14&gt;
| |
| | 271.561 (9 limit least squares)
| | {{vector|0 90 -41 14}}
| | 271.561
| | 9 limit least squares
|-
|-
| |
| | 6/5
| | 6/5
| | 271.564 (5 limit minimax)
| | 271.564
| | 5 limit minimax
|-
|-
| | |0 -211 30 -47 -5 142&gt;
| |
| | 271.567 (13 limit least squares)
| | {{vector|0 -211 30 -47 -5 142}}
| | 271.567
| | 13 limit least squares
|-
|-
| | |0 -236 5 -51 -3 165&gt;
| |
| | 271.570 (15 limit least squares)
| | {{vector|0 -236 5 -51 -3 165}}
| | 271.570
| | 15 limit least squares
|-
|-
| |
| | 1220703125/1033121304
| | 1220703125/1033121304
| | 271.590 (5 limit least squares)
| | 271.590
| | 5 limit least squares
|-
|-
| |
| | 13/12
| | 13/12
| | 271.593
| | 271.593
| |
|-
|-
| |
| | 13/10
| | 13/10
| | 271.612
| | 271.612
| |
|-
|-
| |
| | 18/13
| | 18/13
| | 271.618
| | 271.618
| |
|-
|-
| |
| | 10/9
| | 10/9
| | 271.623 (9 limit minimax)
| | 271.623
| | 9 limit minimax
|-
|-
| |
| | 15/13
| | 15/13
| | 271.641
| | 271.641
| |
|-
|-
| | 12\53
| | 12\53
| |
| | 271.698
| | 271.698
| |
|-
|-
| |
| | 4/3
| | 4/3
| | 271.708
| | 271.708
| |
|-
|-
| |
| | 13/11
| | 13/11
| | 271.942
| | 271.942
| |
|-
|-
| |
| | 16/15
| | 16/15
| | 272.067
| | 272.067
| |
|-
|-
| |
| | 9/7
| | 9/7
| | 272.514
| | 272.514
| |
|-
|-
| | 5\22
| | 5\22
| |
| | 272.727
| | 272.727
| |
|-
|-
| |
| | 11/10
| | 11/10
| | 273.001
| | 273.001
| |
|-
|-
| |
| | 11/8
| | 11/8
| | 275.659
| | 275.659
| |
|}
|}


Revision as of 18:26, 4 October 2021

Orwell temperament has various extensions to the 13 limit.

Tuning Spectra

These spectra suggest possible tuning choices. For 13-limit orwell, the 5-limit minimax tuning featuring pure 6/5 eigenmonzos seems like an excellent choice, as it's right in the middle of the least squares range and very close to 13-limit least squares. Pure 13s, using the 16/13 eigenmonzo, might also please some people. For blair, pure 5/4s using the 5/4 eigenmonzo tuning is very close to 15-limit least squares and in general in the middle of the action. For winston, sticking with the 11/9 eigenmonzo minimax tuning seems reasonable.

Tuning spectrum

Gencom: [2 7/6; 99/98 121/120 176/175 275/273]

Gencom map: [<1 0 3 1 3 8|, <0 7 -3 8 2 -19|]

ET
generator 		eigenmonzo
(unchanged interval) 		subminor
third (¢) 		comments
7/6 266.871
15/11 268.475
14/11 269.585
12/11 270.127
15/14 270.139
7\31 270.968
11/9 271.049
8/7 271.103
7/5 271.137
5/4 271.229
1361367/1000000 271.326 7 limit least squares
14/13 271.418 13 and 15 limit minimax
19\84 271.429
[0 119 -46 20 -16⟩ 271.445 11 limit least squares
[math]\displaystyle{ x^{10} + 2x^3 = 8 }[/math] 271.508 equal beating
16/13 271.551
[0 90 -41 14⟩ 271.561 9 limit least squares
6/5 271.564 5 limit minimax
[0 -211 30 -47 -5 142⟩ 271.567 13 limit least squares
[0 -236 5 -51 -3 165⟩ 271.570 15 limit least squares
1220703125/1033121304 271.590 5 limit least squares
13/12 271.593
13/10 271.612
18/13 271.618
10/9 271.623 9 limit minimax
15/13 271.641
12\53 271.698
4/3 271.708
13/11 271.942
16/15 272.067
9/7 272.514
5\22 272.727
11/10 273.001
11/8 275.659

Spectrum of Winston Tunings by Eigenmonzos

Gencom: [2 7/6; 66/65 99/98 105/104 121/120]

Gencom map: [<1 0 3 1 3 1|, <0 7 -3 8 2 12|]

Eigenmonzo Subminor Third
7/6 266.871
13/12 267.715
14/13 267.925
15/11 268.475
13/11 268.921
15/13 269.032
14/11 269.585
16/13 270.044
12/11 270.127
15/14 270.139
13/10 270.281
|0 112 -67 20 -28 52> 270.860 (15 limit least squares)
|0 118 -61 16 -26 44> 270.933 (13 limit least squares)
7\31 270.968
11/9 271.049 (13 and 15 limit minimax)
8/7 271.103
7/5 271.137
5/4 271.229
1361367/1000000 271.326 (7 limit least squares)
19\84 271.429
|0 119 -46 20 -16> 271.445 (11 limit least squares)
x^10 + 2x^3 = 8 271.508 (equal beating)
|0 90 -41 14> 271.561 (9 limit least squares)
6/5 271.564 (5 limit minimax)
1220703125/1033121304 271.590 (5 limit least squares)
10/9 271.623 (9 limit minimax)
12\53 271.698
4/3 271.708
16/15 272.067
9/7 272.514
5\22 272.727
11/10 273.001
11/8 275.659
18/13 281.691

Spectrum of Blair Tunings by Eigenmonzos

Gencom: [2 7/6; 65/64 78/77 91/90 99/98]

Gencom map: [<1 0 3 1 3 3|, <0 7 -3 8 2 3|]

Eigenmonzo Subminor Third
15/13 247.741
13/12 265.357
14/13 265.660
7/6 266.871
15/11 268.475
18/13 269.398
14/11 269.585
12/11 270.127
15/14 270.139
7\31 270.968
11/9 271.049
8/7 271.103
7/5 271.137 (7, 11, 13 and 15 limit minimax)
5/4 271.229
|0 148 -49 29 -19 -11> 271.231 (15 limit least squares)
|0 145 -52 25 -17 -10> 271.261 (13 limit least squares)
1361367/1000000 271.326 (7 limit least squares)
19\84 271.429
|0 119 -46 20 -16> 271.445 (11 limit least squares)
x^10 + 2x^3 = 8 271.508 (equal beating)
|0 90 -41 14> 271.561 (9 limit least squares)
6/5 271.564 (5 limit minimax)
1220703125/1033121304 271.590 (5 limit least squares)
10/9 271.623 (9 limit minimax)
12\53 271.698
4/3 271.708
16/15 272.067
9/7 272.514
5\22 272.727
11/10 273.001
11/8 275.659
13/10 275.702
16/13 280.176
13/11 289.210
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