Semicomma family: Difference between revisions

The [[5-limit]] parent [[comma]] for the '''semicomma family''' of [[regular temperament|temperaments]] is the [[semicomma]] ({{monzo|legend=1| -21 3 7 }}, [[ratio]]: 2109375/2097152). This is the amount by which three pure 3/1 twelfths exceed seven pure 8/5 minor sixths.

'''Orson''', first discovered by [[Erv Wilson]]{{citation needed}}, is the [[5-limit]] temperament [[tempering out]] the semicomma. It has a [[generator]] of [[~]][[75/64]], seven of which give the [[3/1|perfect twelfth]]; its [[ploidacot]] is alpha-heptacot. The generator 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

[[Comma list]]: 2109375/2097152

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

: mapping generators: ~2, ~75/64

* [[CTE]]: ~2 = 1200.000, ~75/64 = 271.670

: [[error map]]: {{val| 0.000 -0.264 -1.324 }}

* [[POTE]]: ~2 = 1200.000, ~75/64 = 271.627

: error map: {{val| 0.000 -0.564 -1.195 }}

* 5-odd-limit [[diamond monotone]]: ~75/64 = [257.143, 276.923] (3\14 to 3\13)

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

{{Optimal ET sequence|legend=1| 22, 31, 53, 190, 243, 296, 645c }}

[[Badness]] (Smith): 0.040807

The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 7-limit family member we are looking at. Adding 65625/65536 (or 225/224) leads to orwell, but we could also add

* 1029/1024, leading to the {{nowrap| 31 & 159 }} temperament (triwell), or

* 4375/4374, giving the {{nowrap| 53 & 243 }} temperament (sabric).

{{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 {{nowrap| 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

[[Comma list]]: 225/224, 1728/1715

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

[[Optimal tuning]]s:  

* [[CTE]]: ~2 = 1200.000, ~7/6 = 271.513

: [[error map]]: {{val| 0.000 -1.364 -0.853 +3.278 }}

* [[POTE]]: ~2 = 1200.000, ~7/6 = 271.509

: error map: {{val| 0.000 -1.394 -0.840 +3.243 }}

* [[7-odd-limit]]: ~7/6 = {{monzo| 2/11 0 -1/11 1/11 }}

* [[9-odd-limit]]: ~7/6 = {{monzo| 3/17 2/17 -1/17 }}

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

[[Tuning ranges]]:

* 7-odd-limit [[diamond monotone]]: ~7/6 = [266.667, 272.727] (2\9 to 5\22)

* 7-odd-limit [[diamond tradeoff]]: ~7/6 = [266.871, 271.708]

[[Algebraic generator]]: Sabra3, the real root of 12''x³ - 7''x'' - 48.

{{Optimal ET sequence|legend=1| 9, 22, 31, 53, 84, 137, 221d, 358d }}

 

[[Badness]] (Smith): 0.020735

 

=== 11-limit ===

Subgroup: 2.3.5.7.11

 

Comma list: 99/98, 121/120, 176/175

 

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

 

* CTE: ~2 = 1200.000, ~7/6 = 271.560

* POTE: ~2 = 1200.000, ~7/6 = 271.426

 

Minimax tuning:

* 11-odd-limit: ~7/6 = {{monzo| 2/11 0 -1/11 1/11 }}

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 14/11 0 -7/11 7/11 0 }}, {{monzo| 27/11 0 3/11 -3/11 0 }}, {{monzo| 27/11 0 -8/11 8/11 0 }}, {{monzo| 37/11 0 -2/11 2/11 0 }}]

: Unchanged-interval (eigenmonzo) basis: 2.7/5

 

* 11-odd-limit diamond monotone: ~7/6 = [270.968, 272.727] (7\31 to 5\22)

* 11-odd-limit diamond tradeoff: ~7/6 = [266.871, 275.659]

 

 

 

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

 

Comma list: 99/98, 121/120, 176/175, 275/273

 

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

 

* CTE: ~2 = 1200.000, ~7/6 = 271.556

* POTE: ~2 = 1200.000, ~7/6 = 271.546

 

Tuning ranges:

* 13- and 15-odd-limit diamond monotone: ~7/6 = [270.968, 271.698] (7\31 to 12\53)

* 13- and 15-odd-limit diamond tradeoff: ~7/6 = [266.871, 275.659]

 

 

 

==== Blair ====

Subgroup: 2.3.5.7.11.13

 

Comma list: 65/64, 78/77, 91/90, 99/98

 

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

 

* CTE: ~2 = 1200.000, ~7/6 = 271.747

* POTE: ~2 = 1200.000, ~7/6 = 271.301

 

{{Optimal ET sequence|legend=0| 9, 22, 31f }}

 

Badness (Smith): 0.023086

 

==== Winston ====

 

Comma list: 66/65, 99/98, 105/104, 121/120

 

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

 

* CTE: ~2 = 1200.000, ~7/6 = 271.163

* POTE: ~2 = 1200.000, ~7/6 = 271.088

 

* 13- and 15-odd-limit diamond monotone: ~7/6 = [270.968, 272.727] (7\31 to 5\22)

* 13- and 15-odd-limit diamond tradeoff: ~7/6 = [266.871, 281.691]

 

{{Optimal ET sequence|legend=0| 9, 22f, 31 }}

 

Badness (Smith): 0.019931

 

Doublethink is a weak extension of orwell to the 13-limit. It splits the generator of ~7/6 into two [[13/12]]~[[14/13]]'s by tempering out their difference, [[169/168]]. Its ploidacot is alpha-tetradecacot.  

 

 

Comma list: 99/98, 121/120, 169/168, 176/175

 

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

 

* CTE: ~2 = 1200.000, ~13/12 = 135.811

* POTE: ~2 = 1200.000, ~13/12 = 135.723

 

* 13- and 15-odd-limit diamond monotone: ~13/12 = [135.484, 136.364] (7\62 to 5\44)

* 13- and 15-odd-limit diamond tradeoff: ~13/12 = [128.298, 138.573]

 

{{Optimal ET sequence|legend=0| 9, 35bd, 44, 53, 62, 115ef }}

 

 

Subgroup: 2.3.5.7.11

 

 

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

 

Optimal tunings:  

* CTE: ~2 = 1200.000, ~7/6 = 271.316

* POTE: ~2 = 1200.000, ~7/6 = 271.288

 

* 11-odd-limit diamond monotone: ~7/6 = [270.968, 271.698] (7\31 to 12\53)

* 11-odd-limit diamond tradeoff: ~7/6 = [266.871, 272.514]

 

{{Optimal ET sequence|legend=0| 22e, 31, 84, 115 }}

 

Badness (Smith): 0.031438

 

Subgroup: 2.3.5.7.11

 

Comma list: 225/224, 243/242, 1728/1715

 

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

 

: mapping generators: ~2, ~72/55

 

* CTE: ~2 = 1200.000, ~55/36 = 735.754

* POTE: ~2 = 1200.000, ~55/36 = 735.752

 

{{Optimal ET sequence|legend=0| 31, 75e, 106, 137 }}

 

Badness (Smith): 0.038377

 

The sabric temperament ({{nowrap| 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

 

[[Comma list]]: 4375/4374, 2109375/2097152

 

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

 

[[Optimal tuning]]s:

* [[CTE]]: ~2 = 1200.000, ~75/64 = 271.622

: [[error map]]: {{val| 0.000 -0.599 -1.180 +0.131 }}

* [[POTE]]: ~2 = 1200.000, ~75/64 = 271.607

: error map: {{val| 0.000 -0.707 -1.134 -0.808 }}

 

{{Optimal ET sequence|legend=1| 53, 137d, 190, 243, 1511bccd }}

 

 

The triwell temperament ({{nowrap| 31 & 159 }}) slices orwell major sixth ~128/75 into three generators, nine of which give the 5th harmonic.

 

[[Subgroup]]: 2.3.5.7

 

[[Comma list]]: 1029/1024, 235298/234375

 

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

 

: mapping generators: ~2, ~448/375

 

[[Optimal tuning]]s:

* [[CTE]]: ~2 = 1200.000, ~448/375 = 309.456

: [[error map]]: {{val| 0.000 -0.522 -1.213 -2.637 }}

* [[POTE]]: ~2 = 1200.000, ~448/375 = 309.472

 

{{Optimal ET sequence|legend=1| 31, 97, 128, 159, 190 }}

 

 

=== 11-limit ===

 

Comma list: 385/384, 441/440, 456533/455625

 

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

 

Optimal tunings:  

* CTE: ~2 = 1200.000, ~448/375 = 309.444

* POTE: ~2 = 1200.000, ~448/375 = 309.471

 

{{Optimal ET sequence|legend=0| 31, 97, 128, 159, 190 }}

 

 

== Quadrawell ==

The ''quadrawell'' temperament ({{nowrap| 31 & 212 }}) has an [[8/7]] generator of about 232 cents, twelve of which give the 5th harmonic.

 

[[Subgroup]]: 2.3.5.7

 

[[Comma list]]: 2401/2400, 2109375/2097152

 

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

 

: mapping generators: ~2, ~8/7

 

[[Optimal tuning]]s:  

* [[CTE]]: ~2 = 1200.000, ~8/7 = 232.082

: [[error map]]: {{val| 0.000 -0.255 -1.328 -0.908 }}

* [[POTE]]: ~2 = 1200.000, ~8/7 = 232.094

: error map: {{val| 0.000 -0.574 -1.191 -0.919 }}

 

{{Optimal ET sequence|legend=1| 31, 119, 150, 181, 212, 243, 698cd, 941cd }}

 

 

=== 11-limit ===

Subgroup: 2.3.5.7.11

 

Comma list: 385/384, 1375/1372, 14641/14580

 

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

 

Optimal tunings:  

* CTE: ~2 = 1200.000, ~8/7 = 232.065

* POTE: ~2 = 1200.000, ~8/7 = 232.083

 

 

 

== Rainwell ==

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

 

[[Subgroup]]: 2.3.5.7

 

[[Comma list]]: 16875/16807, 2100875/2097152

 

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

 

: mapping generators: ~2, ~2625/2048

 

[[Optimal tuning]]s:  

* [[CTE]]: ~2 = 1200.000, ~2625/2048 = 425.666

: [[error map]]: {{val| 0.000 -0.278 -1.318 0.177 }}

* [[POTE]]: ~2 = 1200.000, ~2625/2048 = 425.673

: error map: {{val| 0.000 -0.526 -1.212 0.113 }}

 

 

 

=== 11-limit ===

Subgroup: 2.3.5.7.11

 

Comma list: 540/539, 1375/1372, 2100875/2097152

 

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

 

* CTE: ~2 = 1200.000, ~2625/2048 = 425.671

* POTE: ~2 = 1200.000, ~2625/2048 = 425.679

 

 

Badness (Smith): 0.052774

 

== Quinwell ==

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

 

 

[[Comma list]]: 420175/419904, 2109375/2097152

 

 

: mapping generators: ~2, ~405/392

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[[Category:Semicomma family| ]] <!-- main article -->