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

* [[WE]]: ~2 = 1200.2902{{c}}, ~75/64 = 271.6929{{c}}

: [[error map]]: {{val| +0.290 -0.104 -0.522 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~75/64 = 271.6394{{c}}

: error map: {{val| 0.000 -0.479 -1.232 }}

* [[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, 1586bccc }}

[[Badness]] (Sintel): 0.957

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

* 2401/2400, giving the {{nowrap| 31 & 243 }} temperament (quadrawell), 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 its 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 marvel comma or 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:

: [[error map]]: {{val| +0.019 -1.364 -0.795 +3.297 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/6 = 271.5097{{c}}

[[Minimax tuning]]:

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

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

* [[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

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

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

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

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

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

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

[[Badness]] (Sintel): 0.525

Subgroup: 2.3.5.7.11

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

 

* WE: ~2 = 1200.5989{{c}}, ~7/6 = 271.5616{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 271.4552{{c}}

 

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

 

Tuning ranges:

* 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 }}

 

* WE: ~2 = 1200.3621{{c}}, ~7/6 = 271.6283{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 271.5477{{c}}

 

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]

 

{{Optimal ET sequence|legend=0| 22, 31, 53, 84e }}

 

Badness (Sintel): 0.815

 

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 }}

 

Optimal tunings:  

* WE: ~2 = 1201.8031{{c}}, ~7/6 = 271.7083{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 271.3846{{c}}

 

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

 

Badness (Sintel): 0.954

 

==== Winston ====

Subgroup: 2.3.5.7.11.13

 

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

 

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

 

Optimal tunings:  

* WE: ~2 = 1200.2846{{c}}, ~7/6 = 271.1524{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 271.1032{{c}}

 

Tuning ranges:

* 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]

 

 

Badness (Sintel): 0.824

 

==== Doublethink ====

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-14-cot.

 

Subgroup: 2.3.5.7.11.13

 

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

 

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

 

* WE: ~2 = 1200.6876{{c}}, ~13/12 = 135.8006{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 135.7410{{c}}

 

* 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]

 

 

 

=== Newspeak ===

In newspeak, the simplicity of obtaining ~[[11/8]] by stacking the generator ~[[7/6]] twice (as in basic 11-limit orwell) is sacrificed to gain accuracy for larger equal temperaments (such as [[84edo]] and [[115edo]]), at the cost of much higher complexity: it is reached only after stacking the generator 33 times and octave-reducing. Newspeak intersects with undecimal orwell at [[31edo]].

 

 

Comma list: 225/224, 441/440, 1728/1715

 

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

 

* WE: ~2 = 1200.2072{{c}}, ~7/6 = 271.3353{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 271.2952{{c}}

 

Tuning ranges:

* 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 (Sintel): 1.04

 

Subgroup: 2.3.5.7.11

 

 

Mapping: {{mapping| 1 -7 6 -7 -18 | 0 14 -6 16 35 }}

: mapping generators: ~2, ~55/36

 

Optimal tunings:  

* WE: ~2 = 1200.0194{{c}}, ~55/36 = 735.7641{{c}}

* CWE: ~2 = 1200.000{{c}}, ~55/36 = 735.7527{{c}}

 

 

Badness (Sintel): 1.27

 

== Sabric ==

The sabric temperament tempers out the ragisma, [[4375/4374]], and may be described as the {{nowrap| 53 & 190 }} temperament. It was named by [[Xenllium]] in 2021 for its relation to the Sabra2 tuning (generator: 271.607278 cents).

 

 

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

 

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

 

* [[WE]]: ~2 = 1200.3056{{c}}, ~75/64 = 271.6760{{c}}

: [[error map]]: {{val| +0.306 -0.223 -0.425 +0.049 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~75/64 = 271.6110{{c}}

: error map: {{val| 0.000 -0.678 -1.147 -0.558 }}

 

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

 

[[Badness]] (Sintel): 2.24

 

== Triwell ==

Triwell tempers out the gamelisma, [[1029/1024]], and the triwellisma, [[235298/234375]]. It may be described as the {{nowrap| 31 & 159 }} temperament. It slices orwell's generator plus two octaves into three generators, and seven generators octave reduced make a ~8/7, which is the generator of [[slendric]]. Its ploidacot is 15-sheared-21-cot.

 

[[Subgroup]]: 2.3.5.7

 

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

 

{{Mapping|legend=1| 1 -14 9 8 | 0 21 -9 -7 }}

: mapping generators: ~2, ~375/224

 

* [[WE]]: ~2 = 1200.4763{{c}}, ~375/224 = 890.8812{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~375/224 = 890.5312{{c}}

: error map: {{val| 0.000 -0.799 -1.095 -2.545 }}

 

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

 

 

 

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

 

Mapping: {{mapping| 1 -14 9 8 -24 | 0 21 -9 -7 37 }}

 

* WE: ~2 = 1200.4804{{c}}, ~375/224 = 890.8854{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~375/224 = 890.5344{{c}}

 

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

 

 

== Quadrawell ==

Quadrawell tempers out [[2401/2400]] and may be described as the {{nowrap| 31 & 212 }} temperament. It has a [[7/4]] generator of about 968 cents, four of which minus three octaves give the original generator of orwell. It can also be viewed as [[2.5.7|2.5.7-subgroup]] [[mothra]] with a different mapping of prime [[3/1|3]]. Its ploidacot is 22-sheared-28-cot.

 

 

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

 

{{Mapping|legend=1| 1 -21 12 2 | 0 28 -12 1 }}

: mapping generators: ~2, ~7/4

 

* [[WE]]: ~2 = 1200.3006{{c}}, ~7/4 = 968.1489{{c}}

: [[error map]]: {{val| +0.301 -0.098 -0.493 -0.076 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 967.9090{{c}}

: error map: {{val| 0.000 -0.503 -1.222 -0.917 }}

 

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

 

[[Badness]] (Sintel): 1.92

 

=== 11-limit ===

Subgroup: 2.3.5.7.11

 

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

 

Mapping: {{mapping| 1 -21 12 2 -28 | 0 28 -12 1 39 }}

 

* WE: ~2 = 1200.3622{{c}}, ~7/4 = 968.2089{{c}}

 

{{Optimal ET sequence|legend=0| 31, 119, 150, 181, 212, 455ee, 667cdee }}

 

Badness (Sintel): 1.21

 

== Rainwell ==

The rainwell temperament tempers out the mirkwai comma, [[16875/16807]], and the rainy comma, [[2100875/2097152]]. It may be described as the {{nowrap| 31 & 265 }} temperament. Its ploidacot is 22-sheared-35-cot.

 

 

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

 

{{Mapping|legend=1| 1 -21 12 -3 | 0 35 -15 9 }}

 

: mapping generators: ~2, ~2625/2048

 

* [[WE]]: ~2 = 1200.2032{{c}}, ~2401/1536 = 774.4577{{c}}

: [[error map]]: {{val| +0.203 -0.204 -0.740 +0.683 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~2401/1536 = 774.3282{{c}}

: error map: {{val| 0.000 -0.469 -1.236 +0.128 }}

 

{{Optimal ET sequence|legend=1| 31, 172, 203, 234, 265, 296 }}

 

 

=== 11-limit ===

Subgroup: 2.3.5.7.11

 

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

 

Mapping: {{mapping| 1 -21 12 -3 -43 | 0 35 -15 9 72 }}

 

* WE: ~2 = 1200.1915{{c}}, ~2205/1408 = 774.4451{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~2205/1408 = 774.3233{{c}}

 

{{Optimal ET sequence|legend=0| 31, 234, 265, 296, 919bc }}

 

 

== Quinwell ==

The quinwell temperament tempers out the wizma, [[420175/419904]], and may be described as the {{nowrap| 22 & 243 }} temperament. It slices orwell's generator into five quartertones. Its ploidacot is alpha-35-cot.

 

[[Subgroup]]: 2.3.5.7

 

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

 

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

: mapping generators: ~2, ~405/392

 

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1200.2860{{c}}, ~405/392 = 54.3373{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~405/392 = 54.3273{{c}}

: error map: {{val| 0.000 -0.501 -1.223 -0.536 }}

 

{{Optimal ET sequence|legend=1| 22, …, 199d, 221, 243, 751c, 994cd, 1237bccd, 1480bccd }}

 

[[Badness]] (Sintel): 4.27

 

=== 11-limit ===

Subgroup: 2.3.5.7.11

 

Comma list: 540/539, 4375/4356, 2109375/2097152

 

 

* WE: ~2 = 1200.0642{{c}}, ~33/32 = 54.3395{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 54.3369{{c}}

 

{{Optimal ET sequence|legend=0| 22, 221, 243, 265 }}

 

Badness (Sintel): 3.21

 

Subgroup: 2.3.5.7.11

 

Comma list: 385/384, 24057/24010, 43923/43750

 

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

 

Optimal tunings:  

* WE: ~2 = 1200.0642{{c}}, ~405/392 = 54.3373{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~405/392 = 54.3192{{c}}

 

 

Badness (Sintel): 2.60

 

[[Category:Temperament families]]

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

[[Category:Rank 2]]

[[Category:Orson]]

[[Category:Orwell]]