Orwell extensions

< Orwell

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