Semicomma family: Difference between revisions

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The 5-limit parent comma for the '''semicomma family''' is the [[semicomma]], 2109375/2097152 = {{monzo| -21 3 7 }}. This is the amount by which three pure 3/1 twelfths exceed seven pure 8/5 minor sixths.  
{{Technical data page}}
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 ==
'''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.
'''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
[[Subgroup]]: 2.3.5


[[Comma list]]: 2109375/2097152
[[Comma list]]: 2109375/2097152


[[Mapping]]: [{{val| 1 0 3}}, {{val| 0 7 -3}}]
{{Mapping|legend=1| 1 0 3 | 0 7 -3 }}


[[POTE generator]]: ~75/64 = 271.627
: mapping generators: ~2, ~75/64


[[Tuning ranges]]:  
[[Optimal tuning]]s:  
* valid range: [257.143, 276.923] (3\14 to 3\13)
* [[CTE]]: ~2 = 1200.000, ~75/64 = 271.670
* nice range: [271.229, 271.708]
: [[error map]]: {{val| 0.000 -0.264 -1.324 }}
* strict range: [271.229, 271.708]
* [[POTE]]: ~2 = 1200.000, ~75/64 = 271.627
: error map: {{val| 0.000 -0.564 -1.195 }}


{{Val list|legend=1| 22, 31, 53, 190, 243, 296, 645c }}
[[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)


[[Badness]]: 0.040807
{{Optimal ET sequence|legend=1| 22, 31, 53, 190, 243, 296, 645c }}


== Seven limit children ==
[[Badness]] (Smith): 0.040807
 
=== 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
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 31&159 temperament (triwell) with wedgie {{multival| 21 -9 -7 -63 -70 9 }}, or  
* 1029/1024, leading to the {{nowrap| 31 & 159 }} temperament (triwell), or
* 2401/2400, giving the 31&243 temperament (quadrawell) with wedgie {{multival| 28 -12 1 -84 -77 36 }}, or  
* 2401/2400, giving the {{nowrap| 31 & 243 }} temperament (quadrawell), or
* 4375/4374, giving the 53&243 temperament (sabric) with wedgie {{multival| 7 -3 61 -21 77 150 }}.
* 4375/4374, giving the {{nowrap| 53 & 243 }} temperament (sabric).


= Orwell =
== Orwell ==
{{main| 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's compatible with [[22edo|22]], [[31edo|31]], [[53edo|53]] and [[84edo|84]] equal, and may be described as the 22&31 temperament, or {{multival| 7 -3 8 -21 -7 27 }}. It's 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 limits, but it does use its second-best 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.
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.
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 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.
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
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val| 1 0 3 1 }}, {{val| 0 7 -3 8 }}]
{{Mapping|legend=1| 1 0 3 1 | 0 7 -3 8 }}
 
{{Multival|legend=1| 7 -3 8 -21 -7 27 }}


[[POTE generator]]: ~7/6 = 271.509
[[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 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:
* [[7-odd-limit]]
* [[7-odd-limit]]: ~7/6 = {{monzo| 2/11 0 -1/11 1/11 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 14/11 0 -7/11 7/11 }}, {{monzo| 27/11 0 3/11 -3/11 }}, {{monzo| 27/11 0 -8/11 8/11 }}]
: {{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]]s: 2, 7/5
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5
* 9-odd-limit
* [[9-odd-limit]]: ~7/6 = {{monzo| 3/17 2/17 -1/17 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 21/17 14/17 -7/17 0 }}, {{monzo| 42/17 -6/17 3/17 0 }}, {{monzo| 41/17 16/17 -8/17 0 }}]
: {{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]]s: 2, 10/9
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5


[[Tuning ranges]]:  
[[Tuning ranges]]:
* valid range: [266.667, 272.727] (2\9 to 5\22)
* 7-odd-limit [[diamond monotone]]: ~7/6 = [266.667, 272.727] (2\9 to 5\22)
* nice range: [266.871, 271.708]
* 9-odd-limit diamond monotone: ~7/6 = [270.968, 272.727] (7\31 to 5\22)
* strict range: [266.871, 271.708]
* 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''x<sup>3</sup> - 7''x'' - 48.
[[Algebraic generator]]: Sabra3, the real root of 12''x<sup>3</sup> - 7''x'' - 48.


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


[[Badness]]: 0.020735
[[Badness]] (Smith): 0.020735


== 11-limit ==
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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


Mapping: [{{val| 1 0 3 1 3 }}, {{val| 0 7 -3 8 2 }}]
Mapping: {{mapping| 1 0 3 1 3 | 0 7 -3 8 2 }}


POTE generator: ~7/6 = 271.426
Optimal tunings:
* CTE: ~2 = 1200.000, ~7/6 = 271.560
* POTE: ~2 = 1200.000, ~7/6 = 271.426


Minimax tuning:  
Minimax tuning:
* 11-odd-limit
* 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 }}]
: [{{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 }}]
: Eigenmonzos: 2, 7/5
: Unchanged-interval (eigenmonzo) basis: 2.7/5


Tuning ranges:  
Tuning ranges:
* valid range: [270.968, 272.727] (7\31 to 5\22)
* 11-odd-limit diamond monotone: ~7/6 = [270.968, 272.727] (7\31 to 5\22)
* nice range: [266.871, 275.659]
* 11-odd-limit diamond tradeoff: ~7/6 = [266.871, 275.659]
* strict range: [270.968, 272.727]


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


Badness: 0.015231
Badness (Smith): 0.015231


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


Mapping: [{{val| 1 0 3 1 3 8 }}, {{val| 0 7 -3 8 2 -19 }}]
Mapping: {{mapping| 1 0 3 1 3 8 | 0 7 -3 8 2 -19 }}


POTE generator: ~7/6 = 271.546
Optimal tunings:
* CTE: ~2 = 1200.000, ~7/6 = 271.556
* POTE: ~2 = 1200.000, ~7/6 = 271.546


Tuning ranges:  
Tuning ranges:
* valid range: [270.968, 271.698] (7\31 to 12\53)
* 13- and 15-odd-limit diamond monotone: ~7/6 = [270.968, 271.698] (7\31 to 12\53)
* nice range: [266.871, 275.659]
* 13- and 15-odd-limit diamond tradeoff: ~7/6 = [266.871, 275.659]
* strict range: [270.968, 271.698]


Vals: {{Val list| 22, 31, 53, 84e, 137e }}
{{Optimal ET sequence|legend=0| 22, 31, 53, 84e }}


Badness: 0.019718
Badness (Smith): 0.019718


=== Blair ===
==== Blair ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


Mapping: [{{val| 1 0 3 1 3 3 }}, {{val| 0 7 -3 8 2 3 }}]
Mapping: {{mapping| 1 0 3 1 3 3 | 0 7 -3 8 2 3 }}


POTE generator: ~7/6 = 271.301
Optimal tunings:
* CTE: ~2 = 1200.000, ~7/6 = 271.747
* POTE: ~2 = 1200.000, ~7/6 = 271.301


Vals: {{Val list| 9, 22, 31f }}
{{Optimal ET sequence|legend=0| 9, 22, 31f }}


Badness: 0.023086
Badness (Smith): 0.023086


=== Winston ===
==== Winston ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


Mapping: [{{val| 1 0 3 1 3 1 }}, {{val| 0 7 -3 8 2 12 }}]
Mapping: {{mapping| 1 0 3 1 3 1 | 0 7 -3 8 2 12 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~7/6 = 271.163
* POTE: ~2 = 1200.000, ~7/6 = 271.088


POTE generator: ~7/6 = 271.088
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]


Tuning ranges:
{{Optimal ET sequence|legend=0| 9, 22f, 31 }}
* valid range: [270.968, 272.727] (7\31 to 5\22)
* nice range: [266.871, 281.691]
* strict range: [270.968, 272.727]


Vals: {{Val list| 22f, 31 }}
Badness (Smith): 0.019931


Badness: 0.019931
==== 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-tetradecacot.  


=== Doublethink ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


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


POTE generator: ~13/12 = 135.723
Optimal tunings:
* CTE: ~2 = 1200.000, ~13/12 = 135.811
* POTE: ~2 = 1200.000, ~13/12 = 135.723


Tuning ranges:  
Tuning ranges:
* valid range: [135.484, 136.364] (7\62 to 5\44)
* 13- and 15-odd-limit diamond monotone: ~13/12 = [135.484, 136.364] (7\62 to 5\44)
* nice range: [128.298, 138.573]
* 13- and 15-odd-limit diamond tradeoff: ~13/12 = [128.298, 138.573]
* strict range: [135.484, 136.364]


Vals: {{Val list| 9, 35bd, 44, 53, 62, 115ef, 168eef }}
{{Optimal ET sequence|legend=0| 9, 35bd, 44, 53, 62, 115ef }}


Badness: 0.027120
Badness (Smith): 0.027120


== Newspeak ==
=== Newspeak ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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


Mapping: [{{val| 1 0 3 1 -4 }}, {{val| 0 7 -3 8 33 }}]
Mapping: {{mapping| 1 0 3 1 -4 | 0 7 -3 8 33 }}


POTE generator: ~7/6 = 271.288
Optimal tunings:
* CTE: ~2 = 1200.000, ~7/6 = 271.316
* POTE: ~2 = 1200.000, ~7/6 = 271.288


Tuning ranges:  
Tuning ranges:
* valid range: [270.968, 271.698] (7\31 to 12\53)
* 11-odd-limit diamond monotone: ~7/6 = [270.968, 271.698] (7\31 to 12\53)
* nice range: [266.871, 272.514]
* 11-odd-limit diamond tradeoff: ~7/6 = [266.871, 272.514]
* strict range: [270.968, 271.698]


Vals: {{Val list| 31, 84, 115, 376b, 491bd, 606bde }}
{{Optimal ET sequence|legend=0| 22e, 31, 84, 115 }}


Badness: 0.031438
Badness (Smith): 0.031438


== Borwell ==
=== Borwell ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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


Mapping: [{{val| 1 7 0 9 17 }}, {{val| 0 -14 6 -16 -35 }}]
Mapping: {{mapping| 1 7 0 9 17 | 0 -14 6 -16 -35 }}
 
: mapping generators: ~2, ~72/55
 
Optimal tunings:
* 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
 
== Sabric ==
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 }}


POTE generator: ~55/36 = 735.752
{{Optimal ET sequence|legend=1| 53, 137d, 190, 243, 1511bccd }}


Vals: {{Val list| 31, 106, 137, 442bd }}
[[Badness]] (Smith): 0.088355


Badness: 0.038377
== Triwell ==
The triwell temperament ({{nowrap| 31 & 159 }}) slices orwell major sixth ~128/75 into three generators, nine of which give the 5th harmonic.


= Triwell =
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7


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


[[Mapping]]: [{{val| 1 7 0 1 }}, {{val| 0 -21 9 7 }}]
{{Mapping|legend=1| 1 7 0 1 | 0 -21 9 7 }}


{{Multival|legend=1| 21 -9 -7 -63 -70 9 }}
: mapping generators: ~2, ~448/375


[[POTE generator]]: ~448/375 = 309.472
[[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
: error map: {{val| 0.000 -0.872 -1.063 -2.520 }}


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


[[Badness]]: 0.080575
[[Badness]] (Smith): 0.080575


== 11-limit ==
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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


Mapping: [{{val| 1 7 0 1 13 }}, {{val| 0 -21 9 7 -37 }}]
Mapping: {{mapping| 1 7 0 1 13 | 0 -21 9 7 -37 }}


POTE generator: ~448/375 = 309.471
Optimal tunings:
* CTE: ~2 = 1200.000, ~448/375 = 309.444
* POTE: ~2 = 1200.000, ~448/375 = 309.471


Vals: {{Val list| 31, 97, 128, 159, 190 }}
{{Optimal ET sequence|legend=0| 31, 97, 128, 159, 190 }}


Badness: 0.029807
Badness (Smith): 0.029807


= Quadrawell =
== Quadrawell ==
The ''quadrawell'' temperament (31&amp;212, named by [[User:Xenllium|Xenllium]]) has an [[8/7]] generator of about 232 cents, twelve of which give the fifth harmonic.
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
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val| 1 7 0 3 }}, {{val| 0 -28 12 -1 }}]
{{Mapping|legend=1| 1 7 0 3 | 0 -28 12 -1 }}


{{Multival|legend=1| 28 -12 1 -84 -77 36 }}
: mapping generators: ~2, ~8/7


[[POTE generator]]: ~8/7 = 232.094
[[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 }}


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


[[Badness]]: 0.075754
[[Badness]] (Smith): 0.075754


== 11-limit ==
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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


Map: [{{val| 1 7 0 3 11 }}, {{val| 0 -28 12 -1 -39}}]
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
 
{{Optimal ET sequence|legend=0| 31, 119, 150, 181, 212, 455ee, 667cdee }}
 
Badness (Smith): 0.036493
 
== 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 }}
 
{{Optimal ET sequence|legend=1| 31, 172, 203, 234, 265, 296 }}
 
[[Badness]] (Smith): 0.143488
 
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~8/7 = 232.083
Comma list: 540/539, 1375/1372, 2100875/2097152


Vals: {{Val list| 31, 119, 150, 181, 212, 455ee, 667cdee }}
Mapping: {{mapping| 1 14 -3 6 29 | 0 -35 15 -9 -72 }}


Badness: 0.036493
Optimal tunings:  
* CTE: ~2 = 1200.000, ~2625/2048 = 425.671
* POTE: ~2 = 1200.000, ~2625/2048 = 425.679


= Sabric =
{{Optimal ET sequence|legend=0| 31, 234, 265, 296, 919bc }}
The ''sabric'' temperament (53&amp;190, named by [[User:Xenllium]]) 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
Badness (Smith): 0.052774


[[Comma list]]: 4375/4374, 2109375/2097152
== Quinwell ==
The quinwell temperament ({{nowrap| 22 & 243 }}) slices orwell minor third into five generators and tempers out the wizma, 420175/419904.


[[Mapping]]: [{{val| 1 0 3 -11 }}, {{val| 0 7 -3 61 }}]
[[Subgroup]]: 2.3.5.7


{{Multival|legend=1| 7 -3 61 -21 77 150 }}
[[Comma list]]: 420175/419904, 2109375/2097152


[[POTE generator]]: ~75/64 = 271.607
{{Mapping|legend=1| 1 0 3 0 | 0 35 -15 62 }}


{{Val list|legend=1| 53, 137d, 190, 243 }}
: mapping generators: ~2, ~405/392


[[Badness]]: 0.088355
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~405/392 = 54.335
: [[error map]]: {{val| 0.000 -0.233 -1.338 -0.061 }}
* [[POTE]]: ~2 = 1200.000, ~405/392 = 54.324
: error map: {{val| 0.000 -0.604 -1.178 -0.718 }}


= Rainwell =
{{Optimal ET sequence|legend=1| 22, …, 199d, 221, 243, 751c, 994cd, 1237bccd, 1480bccd }}
The ''rainwell'' temperament (31&amp;265, named by [[User:Xenllium]]) tempers out the mirkwai comma, 16875/16807 and the [[rainy comma]], 2100875/2097152.


Subgroup: 2.3.5.7
[[Badness]] (Smith): 0.168897


[[Comma list]]: 16875/16807, 2100875/2097152
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Mapping]]: [{{val| 1 14 -3 6 }}, {{val| 0 -35 15 -9 }}]
Comma list: 540/539, 4375/4356, 2109375/2097152


{{Multival|legend=1| 35 -15 9 -105 -84 63 }}
Mapping: {{mapping| 1 0 3 0 5 | 0 35 -15 62 -34 }}


[[POTE generator]]: ~2625/2048 = 425.673
Optimal tunings:
* CTE: ~2 = 1200.000, ~33/32 = 54.338
* POTE: ~2 = 1200.000, ~33/32 = 54.334


{{Val list|legend=1| 31, 172, 203, 234, 265, 296 }}
{{Optimal ET sequence|legend=0| 22, 221, 243, 265 }}


Badness: 0.143488
Badness (Smith): 0.097202


== 11-limit ==
=== Quinbetter ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 540/539, 1375/1372, 2100875/2097152
Comma list: 385/384, 24057/24010, 43923/43750


Mapping: [{{val| 1 14 -3 6 29 }}, {{val| 0 -35 15 -9 -72 }}]
Mapping: {{mapping| 1 0 3 0 4 | 0 35 -15 62 -12 }}


POTE generator: ~2625/2048 = 425.679
Optimal tunings:
* CTE: ~2 = 1200.000, ~405/392 = 54.332
* POTE: ~2 = 1200.000, ~405/392 = 54.316


Vals: {{Val list| 31, 172e, 203e, 234, 265, 296, 919bc, 1215bcc, 1511bcc }}
{{Optimal ET sequence|legend=0| 22, …, 199d, 221e, 243e, 707bcdeee }}


Badness: 0.052774
Badness (Smith): 0.078657


[[Category:Regular temperament theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Pages with mostly numerical content]]
[[Category:Semicomma family| ]] <!-- main article -->
[[Category:Semicomma family| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Orson]]
[[Category:Orson]]
[[Category:Orwell]]
[[Category:Orwell]]

Latest revision as of 00:25, 24 June 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The 5-limit parent comma for the semicomma family of temperaments is the semicomma (monzo: [-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 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 cents, 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: [⟨1 0 3], ⟨0 7 -3]]

mapping generators: ~2, ~75/64

Optimal tunings:

  • CTE: ~2 = 1200.000, ~75/64 = 271.670
error map: ⟨0.000 -0.264 -1.324]
  • POTE: ~2 = 1200.000, ~75/64 = 271.627
error map: ⟨0.000 -0.564 -1.195]

Tuning ranges:

Optimal ET sequence: 22, 31, 53, 190, 243, 296, 645c

Badness (Smith): 0.040807

Overview to extensions

The second comma of the 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 31 & 159 temperament (triwell), or
  • 2401/2400, giving the 31 & 243 temperament (quadrawell), or
  • 4375/4374, giving the 53 & 243 temperament (sabric).

Orwell

Main article: 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 22, 31, 53 and 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 scales 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 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: [⟨1 0 3 1], ⟨0 7 -3 8]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/6 = 271.513
error map: ⟨0.000 -1.364 -0.853 +3.278]
  • POTE: ~2 = 1200.000, ~7/6 = 271.509
error map: ⟨0.000 -1.394 -0.840 +3.243]

Minimax tuning:

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

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

Optimal ET sequence: 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: [⟨1 0 3 1 3], ⟨0 7 -3 8 2]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/6 = 271.560
  • POTE: ~2 = 1200.000, ~7/6 = 271.426

Minimax tuning:

  • 11-odd-limit: ~7/6 = [2/11 0 -1/11 1/11⟩
[[1 0 0 0 0⟩, [14/11 0 -7/11 7/11 0⟩, [27/11 0 3/11 -3/11 0⟩, [27/11 0 -8/11 8/11 0⟩, [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]

Optimal ET sequence: 9, 22, 31, 53, 84e

Badness (Smith): 0.015231

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 3 1 3 8], ⟨0 7 -3 8 2 -19]]

Optimal tunings:

  • 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]

Optimal ET sequence: 22, 31, 53, 84e

Badness (Smith): 0.019718

Blair

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 3 1 3 3], ⟨0 7 -3 8 2 3]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/6 = 271.747
  • POTE: ~2 = 1200.000, ~7/6 = 271.301

Optimal ET sequence: 9, 22, 31f

Badness (Smith): 0.023086

Winston

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 3 1 3 1], ⟨0 7 -3 8 2 12]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/6 = 271.163
  • POTE: ~2 = 1200.000, ~7/6 = 271.088

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: 9, 22f, 31

Badness (Smith): 0.019931

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-tetradecacot.

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 3 1 3 2], ⟨0 14 -6 16 4 15]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~13/12 = 135.811
  • POTE: ~2 = 1200.000, ~13/12 = 135.723

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: 9, 35bd, 44, 53, 62, 115ef

Badness (Smith): 0.027120

Newspeak

Subgroup: 2.3.5.7.11

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

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

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: 22e, 31, 84, 115

Badness (Smith): 0.031438

Borwell

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 7 0 9 17], ⟨0 -14 6 -16 -35]]

mapping generators: ~2, ~72/55

Optimal tunings:

  • CTE: ~2 = 1200.000, ~55/36 = 735.754
  • POTE: ~2 = 1200.000, ~55/36 = 735.752

Optimal ET sequence: 31, 75e, 106, 137

Badness (Smith): 0.038377

Sabric

The sabric temperament (53 & 190) tempers out the 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: [⟨1 0 3 -11], ⟨0 7 -3 61]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~75/64 = 271.622
error map: ⟨0.000 -0.599 -1.180 +0.131]
  • POTE: ~2 = 1200.000, ~75/64 = 271.607
error map: ⟨0.000 -0.707 -1.134 -0.808]

Optimal ET sequence: 53, 137d, 190, 243, 1511bccd

Badness (Smith): 0.088355

Triwell

The triwell temperament (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: [⟨1 7 0 1], ⟨0 -21 9 7]]

mapping generators: ~2, ~448/375

Optimal tunings:

  • CTE: ~2 = 1200.000, ~448/375 = 309.456
error map: ⟨0.000 -0.522 -1.213 -2.637]
  • POTE: ~2 = 1200.000, ~448/375 = 309.472
error map: ⟨0.000 -0.872 -1.063 -2.520]

Optimal ET sequence: 31, 97, 128, 159, 190

Badness (Smith): 0.080575

11-limit

Subgroup: 2.3.5.7.11

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

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: 31, 97, 128, 159, 190

Badness (Smith): 0.029807

Quadrawell

The quadrawell temperament (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: [⟨1 7 0 3], ⟨0 -28 12 -1]]

mapping generators: ~2, ~8/7

Optimal tunings:

  • CTE: ~2 = 1200.000, ~8/7 = 232.082
error map: ⟨0.000 -0.255 -1.328 -0.908]
  • POTE: ~2 = 1200.000, ~8/7 = 232.094
error map: ⟨0.000 -0.574 -1.191 -0.919]

Optimal ET sequence: 31, 119, 150, 181, 212, 243, 698cd, 941cd

Badness (Smith): 0.075754

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal ET sequence: 31, 119, 150, 181, 212, 455ee, 667cdee

Badness (Smith): 0.036493

Rainwell

The rainwell temperament (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: [⟨1 14 -3 6], ⟨0 -35 15 -9]]

mapping generators: ~2, ~2625/2048

Optimal tunings:

  • CTE: ~2 = 1200.000, ~2625/2048 = 425.666
error map: ⟨0.000 -0.278 -1.318 0.177]
  • POTE: ~2 = 1200.000, ~2625/2048 = 425.673
error map: ⟨0.000 -0.526 -1.212 0.113]

Optimal ET sequence: 31, 172, 203, 234, 265, 296

Badness (Smith): 0.143488

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 14 -3 6 29], ⟨0 -35 15 -9 -72]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~2625/2048 = 425.671
  • POTE: ~2 = 1200.000, ~2625/2048 = 425.679

Optimal ET sequence: 31, 234, 265, 296, 919bc

Badness (Smith): 0.052774

Quinwell

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

Subgroup: 2.3.5.7

Comma list: 420175/419904, 2109375/2097152

Mapping: [⟨1 0 3 0], ⟨0 35 -15 62]]

mapping generators: ~2, ~405/392

Optimal tunings:

  • CTE: ~2 = 1200.000, ~405/392 = 54.335
error map: ⟨0.000 -0.233 -1.338 -0.061]
  • POTE: ~2 = 1200.000, ~405/392 = 54.324
error map: ⟨0.000 -0.604 -1.178 -0.718]

Optimal ET sequence: 22, …, 199d, 221, 243, 751c, 994cd, 1237bccd, 1480bccd

Badness (Smith): 0.168897

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 3 0 5], ⟨0 35 -15 62 -34]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~33/32 = 54.338
  • POTE: ~2 = 1200.000, ~33/32 = 54.334

Optimal ET sequence: 22, 221, 243, 265

Badness (Smith): 0.097202

Quinbetter

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 3 0 4], ⟨0 35 -15 62 -12]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~405/392 = 54.332
  • POTE: ~2 = 1200.000, ~405/392 = 54.316

Optimal ET sequence: 22, …, 199d, 221e, 243e, 707bcdeee

Badness (Smith): 0.078657

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