Orwell extensions: Difference between revisions

{{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:

* '''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

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]].  

See [[Semicomma family #Orwell]], [[Semicomma family #Blair|#Blair]], and [[Semicomma family #Winston|#Winston]] for technical data.

Odd harmonics 1–21 and their inverses are in '''bold'''.

{| class="wikitable center-1 right-2"

! rowspan="3" | #

! rowspan="2" | 11-limit

! colspan="3" | 13-limit extensions

== 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.

{| class="wikitable center-all left-4"

! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]

| 7/6

| 266.871

| 13/12

| 13/7

| 15/11

| 268.475

| 13/11

| 15/13

| 11/7

| 269.585

| 13/8

| 11/6

| 270.127

| 15/14

| 13/10

| 270.860

| 7\31

| 270.968

| 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

| 19\84

| 271.429

| {{monzo| 0 119 -46 20 -16 }}

| 271.445

| 11-odd-limit least squares

| {{monzo| 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

| 3/2

| 271.708

| 15/8

| 272.067

| 9/7

| 272.514

| 5\22

| 272.727

| 11/10

| 273.001

| 11/8

| 275.659

| 13/9

=== Blair ===

[[Category:Orwell]]

[[Category:Temperament extensions]]

[[Category:Rank-2 temperaments]]