Orwell extensions: Difference between revisions

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Copypaste the interval table from orwell
→Tuning spectra: reduced ratios
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| 14/11
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| 12/11
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| 270.127
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| 14/13
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| 13- and 15-odd-limit minimax
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| 16/13
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| 5-odd-limit minimax
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| 271.618
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| 10/9
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| 271.623
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| 9-odd-limit minimax
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| 4/3
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| 16/13
| 13/8
| 270.044
| 270.044
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| 12/11
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| 270.127
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| 6/5
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| 271.564
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| 5-odd-limit minimax
| 5-odd-limit minimax
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| 10/9
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| 271.623
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| 9-odd-limit minimax
| 9-odd-limit minimax
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| 4/3
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| 16/15
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| 272.067
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| 18/13
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| 281.691
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| 14/13
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| 265.660
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| 18/13
| 13/9
| 269.398
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| 14/11
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| 269.585
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| 270.968
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| 8/7
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| 271.103
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| 7-odd-limit least squares
| 7-odd-limit least squares
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| 19\84
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| 19\84
| 271.429
| 271.429
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| 6/5
| 5/3
| 271.564
| 271.564
| 5-odd-limit minimax
| 5-odd-limit minimax
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| 10/9
| 9/5
| 271.623
| 271.623
| 9-odd-limit minimax
| 9-odd-limit minimax
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| 12\53
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| 12\53
| 271.698
| 271.698
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| 271.708
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Revision as of 07:42, 9 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.

Interval chain

Odd harmonics 1–21 and their inverses are in bold.

# Cents* Approximate ratios
11-limit 13-limit extensions
Tridecimal orwell Winston Blair
0 0.00 1/1
1 271.46 7/6 13/11, 15/13
2 542.91 11/8, 15/11 18/13 35/26, 39/28
3 814.37 8/5 21/13, 52/33 13/8
4 1085.82 15/8, 28/15 13/7 24/13
5 157.28 12/11, 11/10, 35/32 13/12 14/13
6 428.73 14/11, 9/7, 32/25 13/10, 33/26
7 700.19 3/2 52/35
8 971.64 7/4 26/15
9 43.10 49/48, 36/35, 33/32 40/39 27/26 26/25
10 314.55 6/5 13/11 39/32
11 586.01 7/5 39/28 18/13
12 857.46 18/11 64/39 13/8 21/13
13 1128.92 21/11, 27/14, 48/25 25/13 39/20
14 200.37 9/8, 28/25
15 471.83 21/16 13/10
16 743.28 49/32, 54/35 20/13
17 1014.74 9/5
18 86.19 21/20 26/25 27/26
19 357.65 27/22, 49/40 16/13 39/32
20 629.10 36/25 56/39
21 900.56 27/16, 42/25 22/13
22 1172.01 63/32 39/20

* in 11-limit CWE tuning

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
11/7 269.585
11/6 270.127
15/14 270.139
7\31 270.968
11/9 271.049
7/4 271.103
7/5 271.137
5/4 271.229
1361367/1000000 271.326 7-odd-limit least squares
13/7 271.418 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
13/8 271.551
[0 90 -41 14⟩ 271.561 9-odd-limit least squares
5/3 271.564 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
13/9 271.618
9/5 271.623 9-odd-limit minimax
15/13 271.641
12\53 271.698
3/2 271.708
13/11 271.942
15/8 272.067
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
13/7 267.925
15/11 268.475
13/11 268.921
15/13 269.032
11/7 269.585
13/8 270.044
11/6 270.127
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
7/4 271.103
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
5/3 271.564 5-odd-limit minimax
1220703125/1033121304 271.590 5-odd-limit least squares
9/5 271.623 9-odd-limit minimax
12\53 271.698
3/2 271.708
15/8 272.067
9/7 272.514
5\22 272.727
11/10 273.001
11/8 275.659
13/9 281.691

Blair

Edo
generators 		Eigenmonzo
(unchanged-interval) 		Generator (¢) Comments
15/13 247.741
13/12 265.357
13/7 265.660
7/6 266.871
15/11 268.475
13/9 269.398
11/7 269.585
11/6 270.127
15/14 270.139
7\31 270.968
11/9 271.049
7/4 271.103
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
5/3 271.564 5-odd-limit minimax
1220703125/1033121304 271.590 5-odd-limit least squares
9/5 271.623 9-odd-limit minimax
12\53 271.698
3/2 271.708
15/8 272.067
9/7 272.514
5\22 272.727
11/10 273.001
11/8 275.659
13/10 275.702
13/8 280.176
13/11 289.210
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