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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 ==
== Interval chain ==
Odd harmonics 1–21 and their inverses are in '''bold'''.
Odd harmonics 1–21 and their inverses are in '''bold'''.
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|-
|-
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|-
|-
! Edo<br>generators
! Edo<br>generators
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]
! Generator (¢)
! Generator (¢)
! Comments
! Comments
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|  
|  
| 270.968
| 270.968
|  
| Lower bound of 9- to 15-odd-limit diamond monotone
|-
|-
|  
|  
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| 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
|-
|-
|  
|  
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|  
|  
| 271.698
| 271.698
|  
| Upper bound of 9- to 15-odd-limit diamond monotone
|-
|-
|  
|  
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|-
|-
! Edo<br>generators
! Edo<br>generators
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]
! Generator (¢)
! Generator (¢)
! Comments
! Comments
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|  
|  
| 270.968
| 270.968
|  
| Lower bound of 9- to 15-odd-limit diamond monotone
|-
|-
|  
|  
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| 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
|-
|-
|  
|  
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|  
|  
| 272.727
| 272.727
| 22f val
| 22f val, upper bound of 9- to 15-odd-limit diamond monotone
|-
|-
|  
|  
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|-
|-
! Edo<br>generators
! Edo<br>generators
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]
! Generator (¢)
! Generator (¢)
! Comments
! Comments
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| 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
|-
|-
|  
|  
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[[Category:Orwell]]
[[Category:Orwell]]
[[Category:Temperament extensions]]
[[Category:Temperament extensions]]
[[Category:Rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/Orwell_extensions"