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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|53EDO]] or [[84edo|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
 
[[Optimal tuning]]s:
* [[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 }}


[[Tuning ranges]]:
[[Tuning ranges]]:
* 5-odd-limit [[diamond monotone]]: ~75/64 = [257.143, 276.923] (3\14 to 3\13)
* 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]
* 5-odd-limit [[diamond tradeoff]]: ~75/64 = [271.229, 271.708] (1/3-comma to 2/7-comma)
* 5-odd-limit diamond monotone and tradeoff: ~75/64 = [271.229, 271.708]


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


[[Badness]]: 0.040807
[[Badness]] (Smith): 0.040807


=== Seven limit children ===
=== 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|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|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]]: ~7/6 = {{monzo| 2/11 0 -1/11 1/11 }}
* [[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 (unchanged intervals): 2, 7/5
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5
* 9-odd-limit: ~7/6 = {{monzo| 3/17 2/17 -1/17 }}
* [[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 (unchanged intervals): 2, 10/9
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5


[[Tuning ranges]]:
[[Tuning ranges]]:
Line 59: Line 67:
* 7-odd-limit [[diamond tradeoff]]: ~7/6 = [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]
* 9-odd-limit diamond tradeoff: ~7/6 = [266.871, 272.514]
* 7-odd-limit diamond monotone and tradeoff: ~7/6 = [266.871, 271.708]
* 9-odd-limit diamond monotone and tradeoff: ~7/6 = [270.968, 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 ===
Line 73: Line 79:
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: ~7/6 = {{monzo| 2/11 0 -1/11 1/11 }}
* 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 (unchanged intervals): 2, 7/5
: Unchanged-interval (eigenmonzo) basis: 2.7/5


Tuning ranges:
Tuning ranges:
* 11-odd-limit diamond monotone: ~7/6 = [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)
* 11-odd-limit diamond tradeoff: ~7/6 = [266.871, 275.659]
* 11-odd-limit diamond tradeoff: ~7/6 = [266.871, 275.659]
* 11-odd-limit diamond monotone and tradeoff: ~7/6 = [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 ====
Line 96: Line 103:
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:
* 13- and 15-odd-limit diamond monotone: ~7/6 = [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)
* 13- and 15-odd-limit diamond tradeoff: ~7/6 = [266.871, 275.659]
* 13- and 15-odd-limit diamond tradeoff: ~7/6 = [266.871, 275.659]
* 13- and 15-odd-limit diamond monotone and tradeoff: ~7/6 = [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 ====
Line 114: Line 122:
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 ====
Line 127: Line 137:
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 }}


POTE generator: ~7/6 = 271.088
Optimal tunings:
* CTE: ~2 = 1200.000, ~7/6 = 271.163
* POTE: ~2 = 1200.000, ~7/6 = 271.088


Tuning ranges:
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 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]
* 13- and 15-odd-limit diamond tradeoff: ~7/6 = [266.871, 281.691]
* 13- and 15-odd-limit diamond monotone and tradeoff: ~7/6 = [270.968, 272.727]


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


Badness: 0.019931
Badness (Smith): 0.019931


==== Doublethink ====
==== 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
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:
* 13- and 15-odd-limit diamond monotone: ~13/12 = [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)
* 13- and 15-odd-limit diamond tradeoff: ~13/12 = [128.298, 138.573]
* 13- and 15-odd-limit diamond tradeoff: ~13/12 = [128.298, 138.573]
* 13- and 15-odd-limit diamond monotone and tradeoff: ~13/12 = [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 ===
Line 163: Line 177:
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:
* 11-odd-limit diamond monotone: ~7/6 = [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)
* 11-odd-limit diamond tradeoff: ~7/6 = [266.871, 272.514]
* 11-odd-limit diamond tradeoff: ~7/6 = [266.871, 272.514]
* 11-odd-limit diamond monotone and tradeoff: ~7/6 = [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


== Sabric ==
=== Borwell ===
The ''sabric'' temperament (53&amp;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: 2.3.5.7.11


Subgroup: 2.3.5.7
Comma list: 225/224, 243/242, 1728/1715


[[Comma list]]: 4375/4374, 2109375/2097152
Mapping: {{mapping| 1 7 0 9 17 | 0 -14 6 -16 -35 }}


[[Mapping]]: [{{val| 1 0 3 -11 }}, {{val| 0 7 -3 61 }}]
: mapping generators: ~2, ~72/55


{{Multival|legend=1| 7 -3 61 -21 77 150 }}
Optimal tunings:
* CTE: ~2 = 1200.000, ~55/36 = 735.754
* POTE: ~2 = 1200.000, ~55/36 = 735.752


[[POTE generator]]: ~75/64 = 271.607
{{Optimal ET sequence|legend=0| 31, 75e, 106, 137 }}


{{Val list|legend=1| 53, 137d, 190, 243 }}
Badness (Smith): 0.038377


[[Badness]]: 0.088355
== 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).


=== Borwell ===
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11


Comma list: 225/224, 243/242, 1728/1715
[[Comma list]]: 4375/4374, 2109375/2097152


Mapping: [{{val| 1 7 0 9 17 }}, {{val| 0 -14 6 -16 -35 }}]
{{Mapping|legend=1| 1 0 3 -11 | 0 7 -3 61 }}


POTE generator: ~55/36 = 735.752
[[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 }}


Vals: {{Val list| 31, 106, 137, 442bd }}
{{Optimal ET sequence|legend=1| 53, 137d, 190, 243, 1511bccd }}


Badness: 0.038377
[[Badness]] (Smith): 0.088355


== Triwell ==
== Triwell ==
The triwell temperament (31&amp;159) slices orwell major sixth ~128/75 into three generators, nine of which give the fifth harmonic.
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
[[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 ===
Line 228: Line 253:
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) 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 ===
Line 258: Line 289:
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 }}


POTE generator: ~8/7 = 232.083
Optimal tunings:
* CTE: ~2 = 1200.000, ~8/7 = 232.065
* POTE: ~2 = 1200.000, ~8/7 = 232.083


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


Badness: 0.036493
Badness (Smith): 0.036493


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


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val| 1 14 -3 6 }}, {{val| 0 -35 15 -9 }}]
{{Mapping|legend=1| 1 14 -3 6 | 0 -35 15 -9 }}


{{Multival|legend=1| 35 -15 9 -105 -84 63 }}
: mapping generators: ~2, ~2625/2048


[[POTE generator]]: ~2625/2048 = 425.673
[[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 }}


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


[[Badness]]: 0.143488
[[Badness]] (Smith): 0.143488


=== 11-limit ===
=== 11-limit ===
Line 288: Line 325:
Comma list: 540/539, 1375/1372, 2100875/2097152
Comma list: 540/539, 1375/1372, 2100875/2097152


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


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


Vals: {{Val list| 31, 172e, 203e, 234, 265, 296, 919bc, 1215bcc, 1511bcc }}
{{Optimal ET sequence|legend=0| 31, 234, 265, 296, 919bc }}


Badness: 0.052774
Badness (Smith): 0.052774


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


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val| 1 0 3 0 }}, {{val| 0 35 -15 62 }}]
{{Mapping|legend=1| 1 0 3 0 | 0 35 -15 62 }}


{{Multival|legend=1| 35 -15 62 -105 0 186 }}
: mapping generators: ~2, ~405/392


[[POTE generator]]: ~405/392 = 54.324
[[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 }}


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


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


=== 11-limit ===
=== 11-limit ===
Line 318: Line 361:
Comma list: 540/539, 4375/4356, 2109375/2097152
Comma list: 540/539, 4375/4356, 2109375/2097152


Mapping: [{{val| 1 0 3 0 5 }}, {{val| 0 35 -15 62 -34 }}]
Mapping: {{mapping| 1 0 3 0 5 | 0 35 -15 62 -34 }}


POTE generator: ~33/32 = 54.334
Optimal tunings:
* CTE: ~2 = 1200.000, ~33/32 = 54.338
* POTE: ~2 = 1200.000, ~33/32 = 54.334


Vals: {{Val list| 22, 221, 243, 265, 773ce, 1038ccee, 1303ccee }}
{{Optimal ET sequence|legend=0| 22, 221, 243, 265 }}


Badness: 0.097202
Badness (Smith): 0.097202


=== Quinbetter ===
=== Quinbetter ===
Line 331: Line 376:
Comma list: 385/384, 24057/24010, 43923/43750
Comma list: 385/384, 24057/24010, 43923/43750


Mapping: [{{val| 1 0 3 0 4 }}, {{val| 0 35 -15 62 -12 }}]
Mapping: {{mapping| 1 0 3 0 4 | 0 35 -15 62 -12 }}


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


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


Badness: 0.078657
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]]
Retrieved from "https://en.xen.wiki/w/Semicomma_family"