Orwell extensions: Difference between revisions

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Line 72: Line 72:
| 1361367/1000000
| 1361367/1000000
| 271.326
| 271.326
| 7 limit least squares
| 7-odd-limit least squares
|-
|-
|  
|  
| 14/13
| 14/13
| 271.418
| 271.418
| 13 and 15 limit minimax
| 13- and 15-odd-limit minimax
|-
|-
| 19\84
| 19\84
Line 85: Line 85:
|-
|-
|  
|  
| {{vector|0 119 -46 20 -16}}
| {{monzo| 0 119 -46 20 -16 }}
| 271.445
| 271.445
| 11 limit least squares
| 11-odd-limit least squares
|-
|-
|  
|  
| <math>x^{10} + 2x^3 = 8</math>
| ''f''<sup>10</sup> + 2''f''<sup>3</sup> - 8 = 0
| 271.508
| 271.508
| equal beating
| Equal beating tuning
|-
|-
|  
|  
Line 100: Line 100:
|-
|-
|  
|  
| {{vector|0 90 -41 14}}
| {{monzo| 0 90 -41 14 }}
| 271.561
| 271.561
| 9 limit least squares
| 9-odd-limit least squares
|-
|-
|  
|  
| 6/5
| 6/5
| 271.564
| 271.564
| 5 limit minimax
| 5-odd-limit minimax
|-
|-
|  
|  
| {{vector|0 -211 30 -47 -5 142}}
| {{monzo| 0 -211 30 -47 -5 142 }}
| 271.567
| 271.567
| 13 limit least squares
| 13-odd-limit least squares
|-
|-
|  
|  
| {{vector|0 -236 5 -51 -3 165}}
| {{monzo| 0 -236 5 -51 -3 165 }}
| 271.570
| 271.570
| 15 limit least squares
| 15-odd-limit least squares
|-
|-
|  
|  
| 1220703125/1033121304
| 1220703125/1033121304
| 271.590
| 271.590
| 5 limit least squares
| 5-odd-limit least squares
|-
|-
|  
|  
Line 142: Line 142:
| 10/9
| 10/9
| 271.623
| 271.623
| 9 limit minimax
| 9-odd-limit minimax
|-
|-
|  
|  
Line 255: Line 255:
|-
|-
|  
|  
| {{vector|0 112 -67 20 -28 52}}
| {{monzo| 0 112 -67 20 -28 52 }}
| 270.860
| 270.860
| 15 limit least squares
| 15-odd-limit least squares
|-
|-
|  
|  
| {{vector|0 118 -61 16 -26 44}}
| {{monzo| 0 118 -61 16 -26 44 }}
| 270.933
| 270.933
| 13 limit least squares
| 13-odd-limit least squares
|-
|-
| 7\31
| 7\31
Line 272: Line 272:
| 11/9
| 11/9
| 271.049
| 271.049
| 13 and 15 limit minimax
| 13- and 15-odd-limit minimax
|-
|-
|  
|  
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| 1361367/1000000
| 1361367/1000000
| 271.326
| 271.326
| 7 limit least squares
| 7-odd-limit least squares
|-
|-
| 19\84
| 19\84
Line 300: Line 300:
|-
|-
|  
|  
| {{vector|0 119 -46 20 -16}}
| {{monzo| 0 119 -46 20 -16 }}
| 271.445
| 271.445
| 11 limit least squares
| 11-odd-limit least squares
|-
|-
|  
|  
| <math>x^{10} + 2x^3 = 8</math>
| ''f''<sup>10</sup> + 2''f''<sup>3</sup> - 8 = 0
| 271.508
| 271.508
| equal beating
| Equal beating tuning
|-
|-
|  
|  
| {{vector|0 90 -41 14}}
| {{monzo| 0 90 -41 14 }}
| 271.561
| 271.561
| 9 limit least squares
| 9-odd-limit least squares
|-
|-
|  
|  
| 6/5
| 6/5
| 271.564
| 271.564
| 5 limit minimax
| 5-odd-limit minimax
|-
|-
|  
|  
| 1220703125/1033121304
| 1220703125/1033121304
| 271.590
| 271.590
| 5 limit least squares
| 5-odd-limit least squares
|-
|-
|  
|  
| 10/9
| 10/9
| 271.623
| 271.623
| 9 limit minimax
| 9-odd-limit minimax
|-
|-
| 12\53
| 12\53
Line 372: Line 372:
=== Blair ===
=== Blair ===


{| class="wikitable left-4"
{| class="wikitable center-all left-4"
|-
|-
! Edo<br>generators
! Edo<br>generators
Line 441: Line 441:
|  
|  
| 7/5
| 7/5
| 271.137 (7, 11, 13 and 15 limit minimax)
| 271.137
|
| 7-, 11-, 13- and 15-odd-limit minimax
|-
|-
|  
|  
Line 450: Line 450:
|-
|-
|  
|  
| |0 148 -49 29 -19 -11&gt;
| {{monzo| 0 148 -49 29 -19 -11 }}
| 271.231 (15 limit least squares)
| 271.231
|
| 15-odd-limit least squares
|-
|-
|  
|  
| |0 145 -52 25 -17 -10&gt;
| {{monzo| 0 145 -52 25 -17 -10 }}
| 271.261 (13 limit least squares)
| 271.261
|
| 13-odd-limit least squares
|-
|-
|  
|  
| 1361367/1000000
| 1361367/1000000
| 271.326 (7 limit least squares)
| 271.326
|
| 7-odd-limit least squares
|-
|-
|  
|  
Line 470: Line 470:
|-
|-
|  
|  
| |0 119 -46 20 -16&gt;
| {{monzo| 0 119 -46 20 -16 }}
| 271.445 (11 limit least squares)
| 271.445
|
| 11-odd-limit least squares
|-
|-
|  
|  
| x^10 + 2x^3 = 8
| ''f''<sup>10</sup> + 2''f''<sup>3</sup> - 8 = 0
| 271.508 (equal beating)
| 271.508
|  
| Equal beating tuning
|-
|-
|  
|  
| |0 90 -41 14&gt;
| {{monzo| 0 90 -41 14 }}
| 271.561 (9 limit least squares)
| 271.561
|
| 9-odd-limit least squares
|-
|-
|  
|  
| 6/5
| 6/5
| 271.564 (5 limit minimax)
| 271.564
|
| 5-odd-limit minimax
|-
|-
|  
|  
| 1220703125/1033121304
| 1220703125/1033121304
| 271.590 (5 limit least squares)
| 271.590
|
| 5-odd-limit least squares
|-
|-
|  
|  
| 10/9
| 10/9
| 271.623 (9 limit minimax)
| 271.623
|
| 9-odd-limit minimax
|-
|-
|  
|  

Revision as of 10:45, 3 November 2024

Orwell has multiple competing extensions to the 13-limit. This is evidenced by the fact that its supporting equal temperaments, 22 and 31, do less well in the 13-limit. The extensions are:

  • Orwell (22 & 31) – tempering out 99/98, 121/120, 176/175, and 275/273
  • Blair (22 & 31f) – tempering out 65/64, 78/77, 91/90, and 99/98
  • Winston (22f & 31) – tempering out 66/65, 99/98, 105/104, and 121/120

The most important of these is tridecimal orwell, which tempers out 352/351 and may also be characterized by tempering out 275/273 instead. It is supported by 53. However, it does come at the cost of a way increased complexity level. The other two extensions are of lower complexity, but in both cases the approximations are pretty poor. In winston, the ~13/8 is conflated with the ~18/11 and is generally tuned worse than in 31edo as a result of an improve ~18/11. In blair, the ~13/8 is conflated with the ~8/5 and is generally tuned worse than in 22edo as a result of an improved ~8/5.

Another possible path which relates a sense of compromise is to temper out 169/168, leading to doublethink. This has the effect of slicing the generator in two, and is supported by 44, 53, and 62.

Tuning spectra

These spectra suggest possible tuning choices. For 13-limit orwell, the 5-limit minimax tuning featuring pure 5/3 eigenmonzos seems like an excellent choice, as it is right in the middle of the least squares range and very close to 13-limit least squares. Pure 13's, using the 13/8 eigenmonzo, might also please some people. For blair, pure 5/4's using the 5/4 eigenmonzo tuning is very close to 15-odd-limit least squares and in general in the middle of the action. For winston, sticking with the 11/9 eigenmonzo minimax tuning seems reasonable.

Tridecimal orwell

Edo
generators 		Eigenmonzo
(unchanged-interval) 		Generator (¢) 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-odd-limit least squares
14/13 271.418 13- and 15-odd-limit minimax
19\84 271.429
[0 119 -46 20 -16⟩ 271.445 11-odd-limit least squares
f10 + 2f3 - 8 = 0 271.508 Equal beating tuning
16/13 271.551
[0 90 -41 14⟩ 271.561 9-odd-limit least squares
6/5 271.564 5-odd-limit minimax
[0 -211 30 -47 -5 142⟩ 271.567 13-odd-limit least squares
[0 -236 5 -51 -3 165⟩ 271.570 15-odd-limit least squares
1220703125/1033121304 271.590 5-odd-limit least squares
13/12 271.593
13/10 271.612
18/13 271.618
10/9 271.623 9-odd-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

Winston

Edo
generators 		Eigenmonzo
(unchanged-interval) 		Generator (¢) Comments
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-odd-limit least squares
[0 118 -61 16 -26 44⟩ 270.933 13-odd-limit least squares
7\31 270.968
11/9 271.049 13- and 15-odd-limit minimax
8/7 271.103
7/5 271.137
5/4 271.229
1361367/1000000 271.326 7-odd-limit least squares
19\84 271.429
[0 119 -46 20 -16⟩ 271.445 11-odd-limit least squares
f10 + 2f3 - 8 = 0 271.508 Equal beating tuning
[0 90 -41 14⟩ 271.561 9-odd-limit least squares
6/5 271.564 5-odd-limit minimax
1220703125/1033121304 271.590 5-odd-limit least squares
10/9 271.623 9-odd-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

Blair

Edo
generators 		Eigenmonzo
(unchanged-interval) 		Generator (¢) Comments
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-odd-limit minimax
5/4 271.229
[0 148 -49 29 -19 -11⟩ 271.231 15-odd-limit least squares
[0 145 -52 25 -17 -10⟩ 271.261 13-odd-limit least squares
1361367/1000000 271.326 7-odd-limit least squares
19\84 271.429
[0 119 -46 20 -16⟩ 271.445 11-odd-limit least squares
f10 + 2f3 - 8 = 0 271.508 Equal beating tuning
[0 90 -41 14⟩ 271.561 9-odd-limit least squares
6/5 271.564 5-odd-limit minimax
1220703125/1033121304 271.590 5-odd-limit least squares
10/9 271.623 9-odd-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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