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Semicomma family: Difference between revisions

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= Orson =
= Orson =
'''Orson''', the [[5-limit]] temperament tempering out the semicomma, 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''', the [[5-limit]] temperament tempering out the semicomma, 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.
Subgroup: 2.3.5


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


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


[[POTE generator]]: ~75/64 = 271.627
[[POTE generator]]: ~75/64 = 271.627


[[Tuning Ranges]]:  
[[Tuning ranges]]:  
* valid range: [257.143, 276.923] (3\14 to 3\13)
* valid range: [257.143, 276.923] (3\14 to 3\13)
* nice range: [271.229, 271.708]
* nice range: [271.229, 271.708]
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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 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
[[Comma list]]: 225/224, 1728/1715


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


{{Multival|legend=1| 7 -3 8 -21 -7 27 }}
{{Multival|legend=1| 7 -3 8 -21 -7 27 }}
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[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]]
* [[7-odd-limit]]
: [|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>]
: [{{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 }}]
: [[Eigenmonzo]]s: 2, 7/5
: [[Eigenmonzo]]s: 2, 7/5
* 9-odd-limit
* 9-odd-limit
: [|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>]
: [{{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 }}]
: [[Eigenmonzo]]s: 2, 10/9
: [[Eigenmonzo]]s: 2, 10/9


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* valid range: [266.667, 272.727] (9 to 22)
* valid range: [266.667, 272.727] (2\9 to 5\22)
* nice range: [266.871, 271.708]
* nice range: [266.871, 271.708]
* strict range: [266.871, 271.708]
* strict range: [266.871, 271.708]
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{{Val list|legend=1| 22, 31, 53, 84, 137, 221d, 358d }}
{{Val list|legend=1| 22, 31, 53, 84, 137, 221d, 358d }}


Badness: 0.0207
[[Badness]]: 0.0207


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Comma list: 99/98, 121/120, 176/175
Comma list: 99/98, 121/120, 176/175


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


[[POTE generator]]: ~7/6 = 271.426
POTE generator: ~7/6 = 271.426


Minimax tuning:  
Minimax tuning:  
* 11-odd-limit
* 11-odd-limit
: [|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>]
: [{{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 }}]
: [[Eigenmonzo]]s: 2, 7/5
: Eigenmonzos: 2, 7/5


Tuning ranges:  
Tuning ranges:  
* valid range: [270.968, 272.727] (31 to 22)
* valid range: [270.968, 272.727] (7\31 to 5\22)
* nice range: [266.871, 275.659]
* nice range: [266.871, 275.659]
* strict range: [270.968, 272.727]
* strict range: [270.968, 272.727]
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=== 13-limit ===
=== 13-limit ===
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: [<1 0 3 1 3 8|, <0 7 -3 8 2 -19|]
Mapping: [{{val| 1 0 3 1 3 8 }}, {{val| 0 7 -3 8 2 -19 }}]


POTE generator: ~7/6 = 271.546
POTE generator: ~7/6 = 271.546


Tuning ranges:  
Tuning ranges:  
* valid range: [270.968, 271.698] (31 to 53)
* valid range: [270.968, 271.698] (7\31 to 12\53)
* nice range: [266.871, 275.659]
* nice range: [266.871, 275.659]
* strict range: [270.968, 271.698]
* strict range: [270.968, 271.698]
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=== Blair ===
=== Blair ===
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: [<1 0 3 1 3 3|, <0 7 -3 8 2 3|]
Mapping: [{{val| 1 0 3 1 3 3 }}, {{val| 0 7 -3 8 2 3 }}]


POTE generator: ~7/6 = 271.301
POTE generator: ~7/6 = 271.301
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=== Winston ===
=== Winston ===
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: [<1 0 3 1 3 1|, <0 7 -3 8 2 12|]
Mapping: [{{val| 1 0 3 1 3 1 }}, {{val| 0 7 -3 8 2 12 }}]


[[POTE generator]]: ~7/6 = 271.088
POTE generator: ~7/6 = 271.088


Tuning ranges:  
Tuning ranges:  
* valid range: [270.968, 272.727] (31 to 22f)
* valid range: [270.968, 272.727] (7\31 to 5\22)
* nice range: [266.871, 281.691]
* nice range: [266.871, 281.691]
* strict range: [270.968, 272.727]
* strict range: [270.968, 272.727]
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=== Doublethink ===
=== Doublethink ===
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: [<1 0 3 1 3 2|, <0 14 -6 16 4 15|]
Mapping: [{{val| 1 0 3 1 3 2 }}, {{val| 0 14 -6 16 4 15 }}]


POTE generator: ~13/12 = 135.723
POTE generator: ~13/12 = 135.723


Tuning ranges:  
Tuning ranges:  
* valid range: [135.484, 136.364] (62 to 44)
* valid range: [135.484, 136.364] (7\62 to 5\44)
* nice range: [128.298, 138.573]
* nice range: [128.298, 138.573]
* strict range: [135.484, 136.364]
* strict range: [135.484, 136.364]
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== Newspeak ==
== Newspeak ==
Subgroup: 2.3.5.7.11
Comma list: 225/224, 441/440, 1728/1715
Comma list: 225/224, 441/440, 1728/1715


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


POTE generator: ~7/6 = 271.288
POTE generator: ~7/6 = 271.288


Tuning ranges:  
Tuning ranges:  
* valid range: [270.968, 271.698] (31 to 53)
* valid range: [270.968, 271.698] (7\31 to 12\53)
* nice range: [266.871, 272.514]
* nice range: [266.871, 272.514]
* strict range: [270.968, 271.698]
* strict range: [270.968, 271.698]
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== Borwell ==
== Borwell ==
Subgroup: 2.3.5.7.11
Comma list: 225/224, 243/242, 1728/1715
Comma list: 225/224, 243/242, 1728/1715


Mapping: [<1 7 0 9 17|, <0 -14 6 -16 -35|]
Mapping: [{{val| 1 7 0 9 17 }}, {{val| 0 -14 6 -16 -35 }}]


POTE generator: ~55/36 = 735.752
POTE generator: ~55/36 = 735.752
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= Triwell =
= Triwell =
Subgroup: 2.3.5.7
[[Comma list]]: 1029/1024, 235298/234375
[[Comma list]]: 1029/1024, 235298/234375


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


{{Multival|legend=1| 21 -9 -7 -63 -70 9 }}
{{Multival|legend=1| 21 -9 -7 -63 -70 9 }}
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== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Comma list: 385/384, 441/440, 456533/455625
Comma list: 385/384, 441/440, 456533/455625


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


POTE generator: ~448/375 = 309.471
POTE generator: ~448/375 = 309.471
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