Orwell/Extensions
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:
- Tridecimal 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. Supported by 53, it has the highest accuracy in its approximation of 13/8, but also the highest complexity. The other two extensions have lower complexity, but also lower accuracy. In winston, ~13/8 is conflated with ~18/11 and is generally tuned worse than in 31edo as a result of an improved ~18/11. In blair, ~13/8 is conflated with ~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.
See Semicomma family #Orwell, #Blair, and #Winston for technical data.
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 |
Unchanged interval (eigenmonzo) |
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 | Lower bound of 9- to 15-odd-limit diamond monotone | |
| 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 | 84e val | |
| [0 119 -46 20 -16⟩ | 271.445 | 11-odd-limit least squares | |
| 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 | Upper bound of 9- to 15-odd-limit diamond monotone | |
| 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 |
Unchanged interval (eigenmonzo) |
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 | Lower bound of 9- to 15-odd-limit diamond monotone | |
| 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 | 84eff val | |
| [0 119 -46 20 -16⟩ | 271.445 | 11-odd-limit least squares | |
| [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 | 53f val | |
| 3/2 | 271.708 | ||
| 15/8 | 272.067 | ||
| 9/7 | 272.514 | ||
| 5\22 | 272.727 | 22f val, upper bound of 9- to 15-odd-limit diamond monotone | |
| 11/10 | 273.001 | ||
| 11/8 | 275.659 | ||
| 13/9 | 281.691 |
Blair
| Edo generators |
Unchanged interval (eigenmonzo) |
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 | 31f val | |
| 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 | 84efff val | |
| [0 119 -46 20 -16⟩ | 271.445 | 11-odd-limit least squares | |
| [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 | 53ff val | |
| 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 |