Orwell extensions

Orwell temperament has various extensions to the 13 limit.

Tuning Spectra

These spectra suggest possible tuning choices. For 13-limit orwell, the 5-limit minimax tuning featuring pure 6/5 eigenmonzos seems like an excellent choice, as it's right in the middle of the least squares range and very close to 13-limit least squares. Pure 13s, using the 16/13 eigenmonzo, might also please some people. For blair, pure 5/4s using the 5/4 eigenmonzo tuning is very close to 15-limit least squares and in general in the middle of the action. For winston, sticking with the 11/9 eigenmonzo minimax tuning seems reasonable.

Tuning spectrum

Gencom: [2 7/6; 99/98 121/120 176/175 275/273]

Gencom mapping: [<1 0 3 1 3 8|, <0 7 -3 8 2 -19|]

ET generator	eigenmonzo (unchanged interval)	subminor third (¢)	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 limit least squares
	14/13	271.418	13 and 15 limit minimax
19\84		271.429
	[0 119 -46 20 -16⟩	271.445	11 limit least squares
	[math]\displaystyle{ x^{10} + 2x^3 = 8 }[/math]	271.508	equal beating
	16/13	271.551
	[0 90 -41 14⟩	271.561	9 limit least squares
	6/5	271.564	5 limit minimax
	[0 -211 30 -47 -5 142⟩	271.567	13 limit least squares
	[0 -236 5 -51 -3 165⟩	271.570	15 limit least squares
	1220703125/1033121304	271.590	5 limit least squares
	13/12	271.593
	13/10	271.612
	18/13	271.618
	10/9	271.623	9 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

Tuning Spectrum for Winston

Gencom: [2 7/6; 66/65 99/98 105/104 121/120]

Gencom mapping: [⟨1 0 3 1 3 1], ⟨0 7 -3 8 2 12]]

ET generator	eigenmonzo (unchanged interval)	subminor third (¢)	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 limit least squares
	[0 118 -61 16 -26 44⟩	270.933	13 limit least squares
7\31		270.968
	11/9	271.049	13 and 15 limit minimax
	8/7	271.103
	7/5	271.137
	5/4	271.229
	1361367/1000000	271.326	7 limit least squares
19\84		271.429
	[0 119 -46 20 -16⟩	271.445	11 limit least squares
	[math]\displaystyle{ x^{10} + 2x^3 = 8 }[/math]	271.508	equal beating
	[0 90 -41 14⟩	271.561	9 limit least squares
	6/5	271.564	5 limit minimax
	1220703125/1033121304	271.590	5 limit least squares
	10/9	271.623	9 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

Spectrum of Blair Tunings by Eigenmonzos

Gencom: [2 7/6; 65/64 78/77 91/90 99/98]

Gencom mapping: [<1 0 3 1 3 3|, <0 7 -3 8 2 3|]

Eigenmonzo	Subminor Third
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 limit minimax)
5/4	271.229
|0 148 -49 29 -19 -11>	271.231 (15 limit least squares)
|0 145 -52 25 -17 -10>	271.261 (13 limit least squares)
1361367/1000000	271.326 (7 limit least squares)
19\84	271.429
|0 119 -46 20 -16>	271.445 (11 limit least squares)
x^10 + 2x^3 = 8	271.508 (equal beating)
|0 90 -41 14>	271.561 (9 limit least squares)
6/5	271.564 (5 limit minimax)
1220703125/1033121304	271.590 (5 limit least squares)
10/9	271.623 (9 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