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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 75: Line 258:
|-
|-
|  
|  
| 14/13
| 13/7
| 271.418
| 271.418
| 13- and 15-odd-limit minimax
| 13- and 15-odd-limit minimax
Line 82: Line 265:
|  
|  
| 271.429
| 271.429
|  
| 84e val
|-
|-
|  
|  
Line 90: Line 273:
|-
|-
|  
|  
| ''f''<sup>10</sup> + 2''f''<sup>3</sup> - 8 = 0
| 13/8
| 271.508
| Equal beating tuning
|-
|
| 16/13
| 271.551
| 271.551
|  
|  
Line 105: Line 283:
|-
|-
|  
|  
| 6/5
| 5/3
| 271.564
| 271.564
| 5-odd-limit minimax
| 5-odd-limit minimax
Line 135: Line 313:
|-
|-
|  
|  
| 18/13
| 13/9
| 271.618
| 271.618
|  
|  
|-
|-
|  
|  
| 10/9
| 9/5
| 271.623
| 271.623
| 9-odd-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 267: Line 444:
|  
|  
| 270.968
| 270.968
|  
| Lower bound of 9- to 15-odd-limit diamond monotone
|-
|-
|  
|  
Line 275: Line 452:
|-
|-
|  
|  
| 8/7
| 7/4
| 271.103
| 271.103
|  
|  
Line 297: Line 474:
|  
|  
| 271.429
| 271.429
|  
| 84eff val
|-
|-
|  
|  
Line 303: Line 480:
| 271.445
| 271.445
| 11-odd-limit least squares
| 11-odd-limit least squares
|-
|
| ''f''<sup>10</sup> + 2''f''<sup>3</sup> - 8 = 0
| 271.508
| Equal beating tuning
|-
|-
|  
|  
Line 315: Line 487:
|-
|-
|  
|  
| 6/5
| 5/3
| 271.564
| 271.564
| 5-odd-limit minimax
| 5-odd-limit minimax
Line 325: Line 497:
|-
|-
|  
|  
| 10/9
| 9/5
| 271.623
| 271.623
| 9-odd-limit minimax
| 9-odd-limit minimax
Line 332: Line 504:
|  
|  
| 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 375: Line 547:
|-
|-
! 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 464: Line 636:
| 7-odd-limit least squares
| 7-odd-limit least squares
|-
|-
| 19\84
|  
|  
| 19\84
| 271.429
| 271.429
|  
| 84efff val
|-
|-
|  
|  
Line 473: Line 645:
| 271.445
| 271.445
| 11-odd-limit least squares
| 11-odd-limit least squares
|-
|
| ''f''<sup>10</sup> + 2''f''<sup>3</sup> - 8 = 0
| 271.508
| Equal beating tuning
|-
|-
|  
|  
Line 485: Line 652:
|-
|-
|  
|  
| 6/5
| 5/3
| 271.564
| 271.564
| 5-odd-limit minimax
| 5-odd-limit minimax
Line 495: Line 662:
|-
|-
|  
|  
| 10/9
| 9/5
| 271.623
| 271.623
| 9-odd-limit minimax
| 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]]
Retrieved from "https://en.xen.wiki/w/Orwell_extensions"