Orwell extensions: Difference between revisions

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Undo revision 225002 by VectorGraphics (talk). Here only 22 and 31 are used with different warts for ease of comparison between these extensions. Plus 9 isn't a reasonable tuning for orwell.
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{{Breadcrumb|Orwell}}
[[Orwell]] has multiple competing [[extension]]s to the [[13-limit]]. This is evidenced by the fact that its [[support]]ing [[equal temperament]]s, [[22edo|22]] and [[31edo|31]], do less well in the 13-limit. The extensions are:  
[[Orwell]] has multiple competing [[extension]]s to the [[13-limit]]. This is evidenced by the fact that its [[support]]ing [[equal temperament]]s, [[22edo|22]] and [[31edo|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
* '''Tridecimal orwell''' ({{nowrap| 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
* '''Blair''' ({{nowrap| 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
* '''Winston''' ({{nowrap| 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 [[53edo|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.  
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 [[53edo|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 [[44edo|44]], 53, and [[62edo|62]].  
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 [[44edo|44]], 53, and [[62edo|62]].  
See [[Semicomma family #Orwell]], [[Semicomma family #Blair|#Blair]], and [[Semicomma family #Winston|#Winston]] for technical data.
== Interval chain ==
Odd harmonics 1–21 and their inverses are in '''bold'''.
{| class="wikitable center-1 right-2"
|-
! rowspan="3" | #
! rowspan="3" | Cents*
! colspan="4" | Approximate ratios
|-
! rowspan="2" | 11-limit
! colspan="3" | 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
|
|}
<nowiki>*</nowiki> in 11-limit CWE tuning


== Tuning spectra ==
== Tuning spectra ==
Line 15: Line 198:
|-
|-
! Edo<br>generators
! Edo<br>generators
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]
! Generator (¢)
! Generator (¢)
! Comments
! Comments
Line 30: Line 213:
|-
|-
|  
|  
| 14/11
| 11/7
| 269.585
| 269.585
|  
|  
|-
|-
|  
|  
| 12/11
| 11/6
| 270.127
| 270.127
|  
|  
Line 47: Line 230:
|  
|  
| 270.968
| 270.968
|  
| Lower bound of 9- to 15-odd-limit diamond monotone
|-
|-
|  
|  
Line 55: Line 238:
|-
|-
|  
|  
| 8/7
| 7/4
| 271.103
| 271.103
|  
|  
Line 72: Line 255:
| 1361367/1000000
| 1361367/1000000
| 271.326
| 271.326
| 7 limit least squares
| 7-odd-limit least squares
|-
|-
|  
|  
| 14/13
| 13/7
| 271.418
| 271.418
| 13 and 15 limit minimax
| 13- and 15-odd-limit minimax
|-
|-
| 19\84
| 19\84
|  
|  
| 271.429
| 271.429
|  
| 84e val
|-
|-
|  
|  
| {{vector|0 119 -46 20 -16}}
| {{monzo| 0 119 -46 20 -16 }}
| 271.445
| 271.445
| 11 limit least squares
| 11-odd-limit least squares
|-
|
| <math>x^{10} + 2x^3 = 8</math>
| 271.508
| equal beating
|-
|-
|  
|  
| 16/13
| 13/8
| 271.551
| 271.551
|  
|  
|-
|-
|  
|  
| {{vector|0 90 -41 14}}
| {{monzo| 0 90 -41 14 }}
| 271.561
| 271.561
| 9 limit least squares
| 9-odd-limit least squares
|-
|-
|  
|  
| 6/5
| 5/3
| 271.564
| 271.564
| 5 limit minimax
| 5-odd-limit minimax
|-
|-
|  
|  
| {{vector|0 -211 30 -47 -5 142}}
| {{monzo| 0 -211 30 -47 -5 142 }}
| 271.567
| 271.567
| 13 limit least squares
| 13-odd-limit least squares
|-
|-
|  
|  
| {{vector|0 -236 5 -51 -3 165}}
| {{monzo| 0 -236 5 -51 -3 165 }}
| 271.570
| 271.570
| 15 limit least squares
| 15-odd-limit least squares
|-
|-
|  
|  
| 1220703125/1033121304
| 1220703125/1033121304
| 271.590
| 271.590
| 5 limit least squares
| 5-odd-limit least squares
|-
|-
|  
|  
Line 135: Line 313:
|-
|-
|  
|  
| 18/13
| 13/9
| 271.618
| 271.618
|  
|  
|-
|-
|  
|  
| 10/9
| 9/5
| 271.623
| 271.623
| 9 limit minimax
| 9-odd-limit minimax
|-
|-
|  
|  
Line 152: Line 330:
|  
|  
| 271.698
| 271.698
|  
| Upper bound of 9- to 15-odd-limit diamond monotone
|-
|-
|  
|  
| 4/3
| 3/2
| 271.708
| 271.708
|  
|  
Line 165: Line 343:
|-
|-
|  
|  
| 16/15
| 15/8
| 272.067
| 272.067
|  
|  
Line 191: Line 369:


=== Winston ===
=== Winston ===
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
|-
! Edo<br>generators
! Edo<br>generators
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]
! Generator (¢)
! Generator (¢)
! Comments
! Comments
Line 210: Line 387:
|-
|-
|  
|  
| 14/13
| 13/7
| 267.925
| 267.925
|  
|  
Line 230: Line 407:
|-
|-
|  
|  
| 14/11
| 11/7
| 269.585
| 269.585
|  
|  
|-
|-
|  
|  
| 16/13
| 13/8
| 270.044
| 270.044
|  
|  
|-
|-
|  
|  
| 12/11
| 11/6
| 270.127
| 270.127
|  
|  
Line 255: Line 432:
|-
|-
|  
|  
| {{vector|0 112 -67 20 -28 52}}
| {{monzo| 0 112 -67 20 -28 52 }}
| 270.860
| 270.860
| 15 limit least squares
| 15-odd-limit least squares
|-
|-
|  
|  
| {{vector|0 118 -61 16 -26 44}}
| {{monzo| 0 118 -61 16 -26 44 }}
| 270.933
| 270.933
| 13 limit least squares
| 13-odd-limit least squares
|-
|-
| 7\31
| 7\31
|  
|  
| 270.968
| 270.968
|  
| Lower bound of 9- to 15-odd-limit diamond monotone
|-
|-
|  
|  
| 11/9
| 11/9
| 271.049
| 271.049
| 13 and 15 limit minimax
| 13- and 15-odd-limit minimax
|-
|-
|  
|  
| 8/7
| 7/4
| 271.103
| 271.103
|  
|  
Line 292: Line 469:
| 1361367/1000000
| 1361367/1000000
| 271.326
| 271.326
| 7 limit least squares
| 7-odd-limit least squares
|-
|-
| 19\84
| 19\84
|  
|  
| 271.429
| 271.429
|  
| 84eff val
|-
|-
|  
|  
| {{vector|0 119 -46 20 -16}}
| {{monzo| 0 119 -46 20 -16 }}
| 271.445
| 271.445
| 11 limit least squares
| 11-odd-limit least squares
|-
|
| <math>x^{10} + 2x^3 = 8</math>
| 271.508
| equal beating
|-
|-
|  
|  
| {{vector|0 90 -41 14}}
| {{monzo| 0 90 -41 14 }}
| 271.561
| 271.561
| 9 limit least squares
| 9-odd-limit least squares
|-
|-
|  
|  
| 6/5
| 5/3
| 271.564
| 271.564
| 5 limit minimax
| 5-odd-limit minimax
|-
|-
|  
|  
| 1220703125/1033121304
| 1220703125/1033121304
| 271.590
| 271.590
| 5 limit least squares
| 5-odd-limit least squares
|-
|-
|  
|  
| 10/9
| 9/5
| 271.623
| 271.623
| 9 limit minimax
| 9-odd-limit minimax
|-
|-
| 12\53
| 12\53
|  
|  
| 271.698
| 271.698
|  
| 53f val
|-
|-
|  
|  
| 4/3
| 3/2
| 271.708
| 271.708
|  
|  
|-
|-
|  
|  
| 16/15
| 15/8
| 272.067
| 272.067
|  
|  
Line 352: Line 524:
|  
|  
| 272.727
| 272.727
|  
| 22f val, upper bound of 9- to 15-odd-limit diamond monotone
|-
|-
|  
|  
Line 365: Line 537:
|-
|-
|  
|  
| 18/13
| 13/9
| 281.691
| 281.691
|  
|  
Line 372: Line 544:
=== Blair ===
=== Blair ===


{| class="wikitable left-4"
{| class="wikitable center-all left-4"
|-
|-
! Edo<br>generators
! Edo<br>generators
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]
! Generator (¢)
! Generator (¢)
! Comments
! Comments
Line 390: Line 562:
|-
|-
|  
|  
| 14/13
| 13/7
| 265.660
| 265.660
|  
|  
Line 405: Line 577:
|-
|-
|  
|  
| 18/13
| 13/9
| 269.398
| 269.398
|  
|  
|-
|-
|  
|  
| 14/11
| 11/7
| 269.585
| 269.585
|  
|  
|-
|-
|  
|  
| 12/11
| 11/6
| 270.127
| 270.127
|  
|  
Line 424: Line 596:
|  
|  
|-
|-
| 7\31
|  
|  
| 7\31
| 270.968
| 270.968
|  
| 31f val
|-
|-
|  
|  
Line 435: Line 607:
|-
|-
|  
|  
| 8/7
| 7/4
| 271.103
| 271.103
|  
|  
Line 441: Line 613:
|  
|  
| 7/5
| 7/5
| 271.137 (7, 11, 13 and 15 limit minimax)
| 271.137
|
| 7-, 11-, 13- and 15-odd-limit minimax
|-
|-
|  
|  
Line 450: Line 622:
|-
|-
|  
|  
| |0 148 -49 29 -19 -11&gt;
| {{monzo| 0 148 -49 29 -19 -11 }}
| 271.231 (15 limit least squares)
| 271.231
|
| 15-odd-limit least squares
|-
|-
|  
|  
| |0 145 -52 25 -17 -10&gt;
| {{monzo| 0 145 -52 25 -17 -10 }}
| 271.261 (13 limit least squares)
| 271.261
|
| 13-odd-limit least squares
|-
|-
|  
|  
| 1361367/1000000
| 1361367/1000000
| 271.326 (7 limit least squares)
| 271.326
|
| 7-odd-limit least squares
|-
|-
| 19\84
|  
|  
| 19\84
| 271.429
| 271.429
|  
| 84efff val
|-
|-
|  
|  
| |0 119 -46 20 -16&gt;
| {{monzo| 0 119 -46 20 -16 }}
| 271.445 (11 limit least squares)
| 271.445
|
| 11-odd-limit least squares
|-
|-
|  
|  
| x^10 + 2x^3 = 8
| {{monzo| 0 90 -41 14 }}
| 271.508 (equal beating)
| 271.561
|  
| 9-odd-limit least squares
|-
|-
|  
|  
| |0 90 -41 14&gt;
| 5/3
| 271.561 (9 limit least squares)
| 271.564
|
| 5-odd-limit minimax
|-
|
| 6/5
| 271.564 (5 limit minimax)
|
|-
|-
|  
|  
| 1220703125/1033121304
| 1220703125/1033121304
| 271.590 (5 limit least squares)
| 271.590
|
| 5-odd-limit least squares
|-
|-
|  
|  
| 10/9
| 9/5
| 271.623 (9 limit minimax)
| 271.623
|
| 9-odd-limit minimax
|-
|-
| 12\53
|  
|  
| 12\53
| 271.698
| 271.698
|  
| 53ff val
|-
|-
|  
|  
| 4/3
| 3/2
| 271.708
| 271.708
|  
|  
|-
|-
|  
|  
| 16/15
| 15/8
| 272.067
| 272.067
|  
|  
Line 519: Line 686:
|  
|  
|-
|-
| 5\22
|  
|  
| 5\22
| 272.727
| 272.727
|  
|  
Line 540: Line 707:
|-
|-
|  
|  
| 16/13
| 13/8
| 280.176
| 280.176
|  
|  
Line 552: Line 719:
[[Category:Orwell]]
[[Category:Orwell]]
[[Category:Temperament extensions]]
[[Category:Temperament extensions]]
[[Category:Rank-2 temperaments]]

Latest revision as of 13:20, 2 March 2026

< 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
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