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 ==

'''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

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

[[Optimal tuning]]s:  

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

[[Tuning ranges]]:

* [[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

=== Overview to extensions ===

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).

== Orwell ==

{{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

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

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

: error map: {{val| 0.000 -1.387 -0.843 +3.252 }}

[[Minimax tuning]]:

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

: [[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)

* 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]

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

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

Optimal tunings:  

* 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 }}

: 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]

{{Optimal ET sequence|legend=0| 9, 22, 31, 53, 84e }}

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

Optimal tunings:  

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

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

* 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 }}

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

* 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 }}

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

Optimal tunings:  

* 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]

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

Badness (Sintel): 0.824

==== Doublethink ====

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

Optimal tunings:  

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

Tuning ranges:

* 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, 115ef }}

Badness (Sintel): 1.12

=== Newspeak ===

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

Optimal tunings:  

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

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

* 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 }}

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

 

 

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

 

 

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).

 

 

 

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

[[Optimal tuning]]s:  

* [[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 ==

[[Subgroup]]: 2.3.5.7

 

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

 

 

 

 

 

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

 

 

 

 

 

[[Subgroup]]: 2.3.5.7

 

 

 

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

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

 

 

 

Subgroup: 2.3.5.7.11

 

 

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

 

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

 

 

 

 

 

 

 

 

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

 

 

 

 

 

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

 

 

 

 

 

[[Subgroup]]: 2.3.5.7

 

 

 

: [[error map]]: {{val| +0.286 -0.151 -0.515 +0.084 }}

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

 

 

 

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

* 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 }}

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

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

Optimal tunings:  

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

{{Optimal ET sequence|legend=0| 22, …, 199d, 221e, 243e, 707bcdeee }}

Badness (Sintel): 2.60

[[Category:Temperament families]]

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