Semicomma family: Difference between revisions

'''Orson''', first discovered by [[Erv Wilson]], is the [[5-limit]] temperament tempering out the semicomma. It has a [[generator]] of [[75/64]], which is sharper than [[7/6]] by [[225/224]] when untempered, and less sharp than that in any good orson tempering, for example [[53edo]] or [[84edo]]. These give tunings to the generator which are sharp of 7/6 by less than five [[cent]]s, making it hard to treat orson as anything other than an (at least) 7-limit system, leading to orwell.

[[Subgroup]]: 2.3.5

{{Mapping|legend=1| 1 0 3 | 0 7 -3 }}

 

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/64 = 271.627

* 5-odd-limit [[diamond tradeoff]]: ~75/64 = [271.229, 271.708] (1/3-comma to 2/7-comma)

=== Overview to extensions ===

{{Main| Orwell }}

So called because 19\84 (as a fraction of the octave) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps. It is compatible with [[22edo|22]], [[31edo|31]], [[53edo|53]] and [[84edo|84]] equal, and may be described as the 22 & 31 temperament. It is a good system in the [[7-limit]] and naturally extends into the [[11-limit]]. [[84edo]], with the 19\84 generator, provides a good tuning for the 5-, 7- and 11-limit, but it does use its second-closest approximation to 11. However, the 19\84 generator is remarkably close to the 11-limit [[POTE tuning]], as the generator is only 0.0024 cents sharper, and it is a good approximation to the 7-limit POTE generator also; hence 84 may be considered the most recommendable tuning in the 7-limit. [[53edo]] might be preferred in the 5-limit because of its nearly pure fifth and in the 11-limit because of it slightly better 11, though most of its 11-limit harmony is actually worse. Aside from the semicomma and 65625/65536, 7-limit orwell tempers out [[2430/2401]], the nuwell comma, [[1728/1715]], the orwellisma, [[225/224]], the septimal kleisma, and [[6144/6125]], the porwell comma.

The 11-limit version of orwell tempers out [[99/98]], which means that two of its sharpened 7/6 generators give a flattened 11/8, as well as 121/120, 176/175, 385/384 and 540/539. Despite lowered tuning accuracy, orwell comes into its own in the 11-limit, giving acceptable accuracy and relatively low complexity. Tempering out the orwellisma, 1728/1715, means that orwell interprets three stacked 7/6 generators as an 8/5, and the tempered 1-7/6-11/8-8/5 chord is natural to orwell.

Orwell has [[mos scale]]s of size 9, 13, 22 and 31. The 9-note mos is small enough to be retained in the mind as a genuine scale, is pleasing melodically, and has [[Retuning 12edo to Orwell9|considerable harmonic resources]] despite its absence of 5-limit triads. The 13-note mos has those, and of course the 22- and 31-note mos are very well supplied with everything.

[[Subgroup]]: 2.3.5.7

{{Mapping|legend=1| 1 0 3 1 | 0 7 -3 8 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/6 = 271.509

: {{monzo list| 1 0 0 0 | 14/11 0 -7/11 7/11 | 27/11 0 3/11 -3/11 | 27/11 0 -8/11 8/11 }}

: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/5

: {{monzo list| 1 0 0 0 | 21/17 14/17 -7/17 0 | 42/17 -6/17 3/17 0 | 41/17 16/17 -8/17 0 }}

: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/5

Mapping: {{mapping| 1 0 3 1 3 | 0 7 -3 8 2 }}

Optimal tuning (POTE): ~2 = 1\1, ~7/6 = 271.426

: Eigenmonzo (unchanged-interval) basis: 2.7/5

Mapping: {{mapping| 1 0 3 1 3 8 | 0 7 -3 8 2 -19 }}

Optimal tuning (POTE): ~2 = 1\1, ~7/6 = 271.546

Mapping: {{mapping| 1 0 3 1 3 3 | 0 7 -3 8 2 3 }}

Optimal tuning (POTE): ~2 = 1\1, ~7/6 = 271.301

Mapping: {{mapping| 1 0 3 1 3 1 | 0 7 -3 8 2 12 }}

Optimal tuning (POTE): ~2 = 1\1, ~7/6 = 271.088

Mapping: {{mapping| 1 0 3 1 3 2 | 0 14 -6 16 4 15 }}

Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 135.723

Mapping: {{mapping| 1 0 3 1 -4 | 0 7 -3 8 33 }}

Optimal tuning (POTE): ~2 = 1\1, ~7/6 = 271.288

Mapping: {{mapping| 1 7 0 9 17 | 0 -14 6 -16 -35 }}

Optimal tuning (POTE): ~2 = 1\1, ~55/36 = 735.752

The sabric temperament (53 & 190) tempers out the [[4375/4374|ragisma (4375/4374)]]. It is so named because it is closely related to the ''Sabra2 tuning'' (generator: 271.607278 cents).

[[Subgroup]]: 2.3.5.7

{{Mapping|legend=1| 1 0 3 -11 | 0 7 -3 61 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/64 = 271.607

The triwell temperament (31 & 159) slices orwell major sixth ~128/75 into three generators, nine of which give the fifth harmonic.

[[Subgroup]]: 2.3.5.7

{{Mapping|legend=1| 1 7 0 1 | 0 -21 9 7 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~448/375 = 309.472

Mapping: {{mapping| 1 7 0 1 13 | 0 -21 9 7 -37 }}

Optimal tuning (POTE): ~2 = 1\1, ~448/375 = 309.471

[[Subgroup]]: 2.3.5.7

{{Mapping|legend=1| 1 7 0 3 | 0 -28 12 -1 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 232.094

Mapping: {{mapping| 1 7 0 3 11 | 0 -28 12 -1 -39 }}

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 232.083

The ''rainwell'' temperament (31 & 265) tempers out the mirkwai comma, 16875/16807 and the [[rainy comma]], 2100875/2097152.

[[Subgroup]]: 2.3.5.7

{{Mapping|legend=1| 1 14 -3 6 | 0 -35 15 -9 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2625/2048 = 425.673

Mapping: {{mapping| 1 14 -3 6 29 | 0 -35 15 -9 -72 }}

Optimal tuning (POTE): ~2 = 1\1, ~2625/2048 = 425.679

The quinwell temperament (22 & 243) slices orwell minor third into five generators and tempers out the wizma, 420175/419904.

[[Subgroup]]: 2.3.5.7

{{Mapping|legend=1| 1 0 3 0 | 0 35 -15 62 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~405/392 = 54.324

Mapping: {{mapping| 1 0 3 0 5 | 0 35 -15 62 -34 }}

Optimal tuning (POTE): ~2 = 1\1, ~33/32 = 54.334

Mapping: {{mapping| 1 0 3 0 4 | 0 35 -15 62 -12 }}

Optimal tuning (POTE): ~2 = 1\1, ~405/392 = 54.316