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:
- 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 | 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 | ||
| 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 | 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 | |
| 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 | 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 |