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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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[[Orwell|Orwell]] temperament has various extensions to the 13 limit.
{{Breadcrumb|Orwell}}


=Tuning Spectra=
[[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:
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.
* '''Tridecimal orwell''' ({{nowrap| 22 & 31 }}) – tempering out 99/98, 121/120, 176/175, and 275/273
* '''Blair''' ({{nowrap| 22 & 31f }}) – tempering out 65/64, 78/77, 91/90, and 99/98
* '''Winston''' ({{nowrap| 22f & 31 }}) – tempering out 66/65, 99/98, 105/104, and 121/120


==Tuning spectrum==
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.


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


Gencom mapping: [<1 0 3 1 3 8|, <0 7 -3 8 2 -19|]
See [[Semicomma family #Orwell]], [[Semicomma family #Blair|#Blair]], and [[Semicomma family #Winston|#Winston]] for technical data.


{| class="wikitable center-all"
== 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'''
|
|
|
|-
|-
! | ET<br>generator
| 1
! | [[eigenmonzo|eigenmonzo<br>(unchanged interval]])
| 271.46
! | subminor<br>third (¢)
| 7/6
! | comments
|  
|  
| 13/11, 15/13
|-
|-
| |
| 2
| | 7/6
| 542.91
| | 266.871
| '''11/8''', 15/11
| |
|  
| 18/13
| 35/26, 39/28
|-
|-
| |
| 3
| | 15/11
| 814.37
| | 268.475
| '''8/5'''
| |
|  
| 21/13, 52/33
| '''13/8'''
|-
|-
| |
| 4
| | 14/11
| 1085.82
| | 269.585
| '''15/8''', 28/15
| |
|  
| 13/7
| 24/13
|-
|-
| |
| 5
| | 12/11
| 157.28
| | 270.127
| 12/11, 11/10, 35/32
| |
|  
| 13/12
| 14/13
|-
|-
| |
| 6
| | 15/14
| 428.73
| | 270.139
| 14/11, 9/7, 32/25
| |
|  
|  
| 13/10, 33/26
|-
|-
| | 7\31
| 7
| |
| 700.19
| | 270.968
| '''3/2'''
| |
|  
| 52/35
|  
|-
|-
| |
| 8
| | 11/9
| 971.64
| | 271.049
| '''7/4'''
| |
|  
| 26/15
|  
|-
|-
| |
| 9
| | 8/7
| 43.10
| | 271.103
| 49/48, 36/35, 33/32
| |
| 40/39
| 27/26
| 26/25
|-
|-
| |
| 10
| | 7/5
| 314.55
| | 271.137
| 6/5
| |
|  
| 13/11
| 39/32
|-
|-
| |
| 11
| | 5/4
| 586.01
| | 271.229
| 7/5
| |
|  
| 39/28
| 18/13
|-
|-
| |
| 12
| | 1361367/1000000
| 857.46
| | 271.326
| 18/11
| | 7 limit least squares
| 64/39
| '''13/8'''
| 21/13
|-
|-
| |
| 13
| | 14/13
| 1128.92
| | 271.418
| 21/11, 27/14, 48/25
| | 13 and 15 limit minimax
| 25/13
|  
| 39/20
|-
|-
| | 19\84
| 14
| |
| 200.37
| | 271.429
| '''9/8''', 28/25
| |
|  
|  
|  
|-
|-
| |
| 15
| | {{vector|0 119 -46 20 -16}}
| 471.83
| | 271.445
| '''21/16'''
| | 11 limit least squares
|  
| 13/10
|  
|-
|-
| |
| 16
| | <math>x^{10} + 2x^3 = 8</math>
| 743.28
| | 271.508
| 49/32, 54/35
| | equal beating
| 20/13
|  
|  
|-
|-
| |
| 17
| | 16/13
| 1014.74
| | 271.551
| 9/5
| |
|  
|  
|  
|-
|-
| |
| 18
| | {{vector|0 90 -41 14}}
| 86.19
| | 271.561
| 21/20
| | 9 limit least squares
|  
| 26/25
| 27/26
|-
|-
| |
| 19
| | 6/5
| 357.65
| | 271.564
| 27/22, 49/40
| | 5 limit minimax
| '''16/13'''
| 39/32
|  
|-
|-
| |
| 20
| | {{vector|0 -211 30 -47 -5 142}}
| 629.10
| | 271.567
| 36/25
| | 13 limit least squares
| 56/39
|  
|  
|-
|-
| |
| 21
| | {{vector|0 -236 5 -51 -3 165}}
| 900.56
| | 271.570
| 27/16, 42/25
| | 15 limit least squares
| 22/13
|  
|  
|-
|-
| |
| 22
| | 1220703125/1033121304
| 1172.01
| | 271.590
| 63/32
| | 5 limit least squares
|  
| 39/20
|  
|}
<nowiki>*</nowiki> 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 ===
{| class="wikitable center-all left-4"
|-
|-
| |
! Edo<br>generators
| | 13/12
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]
| | 271.593
! Generator (¢)
| |
! Comments
|-
|-
| |
|  
| | 13/10
| 7/6
| | 271.612
| 266.871
| |
|  
|-
|-
| |
|  
| | 18/13
| 15/11
| | 271.618
| 268.475
| |
|  
|-
|-
| |
|  
| | 10/9
| 11/7
| | 271.623
| 269.585
| | 9 limit minimax
|  
|-
|-
| |
|  
| | 15/13
| 11/6
| | 271.641
| 270.127
| |
|  
|-
|-
| | 12\53
|  
| |
| 15/14
| | 271.698
| 270.139
| |
|  
|-
|-
| |
| 7\31
| | 4/3
|  
| | 271.708
| 270.968
| |
| Lower bound of 9- to 15-odd-limit diamond monotone
|-
|-
| |
|  
| | 13/11
| 11/9
| | 271.942
| 271.049
| |
|  
|-
|-
| |
|  
| | 16/15
| 7/4
| | 272.067
| 271.103
| |
|  
|-
|-
| |
|  
| | 9/7
| 7/5
| | 272.514
| 271.137
| |
|  
|-
|-
| | 5\22
|  
| |
| 5/4
| | 272.727
| 271.229
| |
|  
|-
|-
| |
|  
| | 11/10
| 1361367/1000000
| | 273.001
| 271.326
| |
| 7-odd-limit least squares
|-
|-
| |
|  
| | 11/8
| 13/7
| | 275.659
| 271.418
| |
| 13- and 15-odd-limit minimax
|-
| 19\84
|
| 271.429
| 84e val
|-
|
| {{monzo| 0 119 -46 20 -16 }}
| 271.445
| 11-odd-limit least squares
|-
|
| 13/8
| 271.551
|
|-
|
| {{monzo| 0 90 -41 14 }}
| 271.561
| 9-odd-limit least squares
|-
|
| 5/3
| 271.564
| 5-odd-limit minimax
|-
|
| {{monzo| 0 -211 30 -47 -5 142 }}
| 271.567
| 13-odd-limit least squares
|-
|
| {{monzo| 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
|  
|}
|}


==Spectrum of Winston Tunings by Eigenmonzos==
=== Winston ===
 
{| class="wikitable center-all left-4"
Gencom: [2 7/6; 66/65 99/98 105/104 121/120]
 
Gencom mapping: [&lt;1 0 3 1 3 1|, &lt;0 7 -3 8 2 12|]
 
{| class="wikitable"
|-
|-
! | Eigenmonzo
! Edo<br>generators
! | Subminor Third
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]
! Generator (¢)
! Comments
|-
|-
| | 7/6
|  
| | 266.871
| 7/6
| 266.871
|
|-
|-
| | 13/12
|  
| | 267.715
| 13/12
| 267.715
|
|-
|-
| | 14/13
|  
| | 267.925
| 13/7
| 267.925
|
|-
|-
| | 15/11
|  
| | 268.475
| 15/11
| 268.475
|
|-
|-
| | 13/11
|  
| | 268.921
| 13/11
| 268.921
|
|-
|-
| | 15/13
|  
| | 269.032
| 15/13
| 269.032
|
|-
|-
| | 14/11
|  
| | 269.585
| 11/7
| 269.585
|
|-
|-
| | 16/13
|  
| | 270.044
| 13/8
| 270.044
|
|-
|-
| | 12/11
|  
| | 270.127
| 11/6
| 270.127
|
|-
|-
| | 15/14
|  
| | 270.139
| 15/14
| 270.139
|
|-
|-
| | 13/10
|  
| | 270.281
| 13/10
| 270.281
|
|-
|-
| | |0 112 -67 20 -28 52&gt;
|  
| | 270.860 (15 limit least squares)
| {{monzo| 0 112 -67 20 -28 52 }}
| 270.860
| 15-odd-limit least squares
|-
|-
| | |0 118 -61 16 -26 44&gt;
|  
| | 270.933 (13 limit least squares)
| {{monzo| 0 118 -61 16 -26 44 }}
| 270.933
| 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
|  
| | 271.049 (13 and 15 limit minimax)
| 11/9
| 271.049
| 13- and 15-odd-limit minimax
|-
|-
| | 8/7
|  
| | 271.103
| 7/4
| 271.103
|
|-
|-
| | 7/5
|  
| | 271.137
| 7/5
| 271.137
|
|-
|-
| | 5/4
|  
| | 271.229
| 5/4
| 271.229
|
|-
|-
| | 1361367/1000000
|  
| | 271.326 (7 limit least squares)
| 1361367/1000000
| 271.326
| 7-odd-limit least squares
|-
|-
| | 19\84
| 19\84
| | 271.429
|  
| 271.429
| 84eff val
|-
|-
| | |0 119 -46 20 -16&gt;
|  
| | 271.445 (11 limit least squares)
| {{monzo| 0 119 -46 20 -16 }}
| 271.445
| 11-odd-limit least squares
|-
|-
| | x^10 + 2x^3 = 8
|  
| | 271.508 (equal beating)
| {{monzo| 0 90 -41 14 }}
| 271.561
| 9-odd-limit least squares
|-
|-
| | |0 90 -41 14&gt;
|  
| | 271.561 (9 limit least squares)
| 5/3
| 271.564
| 5-odd-limit minimax
|-
|-
| | 6/5
|  
| | 271.564 (5 limit minimax)
| 1220703125/1033121304
| 271.590
| 5-odd-limit least squares
|-
|-
| | 1220703125/1033121304
|  
| | 271.590 (5 limit least squares)
| 9/5
| 271.623
| 9-odd-limit minimax
|-
|-
| | 10/9
| 12\53
| | 271.623 (9 limit minimax)
|  
| 271.698
| 53f val
|-
|-
| | 12\53
|  
| | 271.698
| 3/2
| 271.708
|
|-
|-
| | 4/3
|  
| | 271.708
| 15/8
| 272.067
|  
|-
|-
| | 16/15
|  
| | 272.067
| 9/7
| 272.514
|
|-
|-
| | 9/7
| 5\22
| | 272.514
|  
| 272.727
| 22f val, upper bound of 9- to 15-odd-limit diamond monotone
|-
|-
| | 5\22
|  
| | 272.727
| 11/10
| 273.001
|  
|-
|-
| | 11/10
|  
| | 273.001
| 11/8
| 275.659
|  
|-
|-
| | 11/8
|  
| | 275.659
| 13/9
|-
| 281.691
| | 18/13
|
| | 281.691
|}
|}


==Spectrum of Blair Tunings by Eigenmonzos==
=== Blair ===


Gencom: [2 7/6; 65/64 78/77 91/90 99/98]
{| class="wikitable center-all left-4"
 
Gencom mapping: [&lt;1 0 3 1 3 3|, &lt;0 7 -3 8 2 3|]
 
{| class="wikitable"
|-
! | Eigenmonzo
! | Subminor Third
|-
|-
| | 15/13
! Edo<br>generators
| | 247.741
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]
! Generator (¢)
! Comments
|-
|-
| | 13/12
|  
| | 265.357
| 15/13
| 247.741
|  
|-
|-
| | 14/13
|  
| | 265.660
| 13/12
| 265.357
|
|-
|-
| | 7/6
|  
| | 266.871
| 13/7
| 265.660
|  
|-
|-
| | 15/11
|  
| | 268.475
| 7/6
| 266.871
|  
|-
|-
| | 18/13
|  
| | 269.398
| 15/11
| 268.475
|  
|-
|-
| | 14/11
|  
| | 269.585
| 13/9
| 269.398
|
|-
|-
| | 12/11
|  
| | 270.127
| 11/7
| 269.585
|  
|-
|-
| | 15/14
|  
| | 270.139
| 11/6
| 270.127
|
|-
|-
| | 7\31
|  
| | 270.968
| 15/14
| 270.139
|
|-
|-
| | 11/9
| 7\31
| | 271.049
|  
| 270.968
| 31f val
|-
|-
| | 8/7
|  
| | 271.103
| 11/9
| 271.049
|
|-
|-
| | 7/5
|  
| | 271.137 (7, 11, 13 and 15 limit minimax)
| 7/4
| 271.103
|
|-
|-
| | 5/4
|  
| | 271.229
| 7/5
| 271.137
| 7-, 11-, 13- and 15-odd-limit minimax
|-
|-
| | |0 148 -49 29 -19 -11&gt;
|  
| | 271.231 (15 limit least squares)
| 5/4
| 271.229
|
|-
|-
| | |0 145 -52 25 -17 -10&gt;
|  
| | 271.261 (13 limit least squares)
| {{monzo| 0 148 -49 29 -19 -11 }}
| 271.231
| 15-odd-limit least squares
|-
|-
| | 1361367/1000000
|  
| | 271.326 (7 limit least squares)
| {{monzo| 0 145 -52 25 -17 -10 }}
| 271.261
| 13-odd-limit least squares
|-
|-
| | 19\84
|  
| | 271.429
| 1361367/1000000
| 271.326
| 7-odd-limit least squares
|-
|-
| | |0 119 -46 20 -16&gt;
| 19\84
| | 271.445 (11 limit least squares)
|  
| 271.429
| 84efff val
|-
|-
| | x^10 + 2x^3 = 8
|  
| | 271.508 (equal beating)
| {{monzo| 0 119 -46 20 -16 }}
| 271.445
| 11-odd-limit least squares
|-
|-
| | |0 90 -41 14&gt;
|  
| | 271.561 (9 limit least squares)
| {{monzo| 0 90 -41 14 }}
| 271.561
| 9-odd-limit least squares
|-
|-
| | 6/5
|  
| | 271.564 (5 limit minimax)
| 5/3
| 271.564
| 5-odd-limit minimax
|-
|-
| | 1220703125/1033121304
|  
| | 271.590 (5 limit least squares)
| 1220703125/1033121304
| 271.590
| 5-odd-limit least squares
|-
|-
| | 10/9
|  
| | 271.623 (9 limit minimax)
| 9/5
| 271.623
| 9-odd-limit minimax
|-
|-
| | 12\53
| 12\53
| | 271.698
|  
| 271.698
| 53ff val
|-
|-
| | 4/3
|  
| | 271.708
| 3/2
| 271.708
|
|-
|-
| | 16/15
|  
| | 272.067
| 15/8
| 272.067
|
|-
|-
| | 9/7
|  
| | 272.514
| 9/7
| 272.514
|
|-
|-
| | 5\22
| 5\22
| | 272.727
|  
| 272.727
|
|-
|-
| | 11/10
|  
| | 273.001
| 11/10
| 273.001
|
|-
|-
| | 11/8
|  
| | 275.659
| 11/8
| 275.659
|
|-
|-
| | 13/10
|  
| | 275.702
| 13/10
| 275.702
|
|-
|-
| | 16/13
|  
| | 280.176
| 13/8
| 280.176
|
|-
|-
| | 13/11
|  
| | 289.210
| 13/11
| 289.210
|
|}
|}
[[Category:blair]]
 
[[Category:eigenmonzo]]
[[Category:Orwell]]
[[Category:orwell]]
[[Category:Temperament extensions]]
[[Category:spectrum]]
[[Category:Rank-2 temperaments]]
[[Category:winston]]
Retrieved from "https://en.xen.wiki/w/Orwell_extensions"