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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.  
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.  


= 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.


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


[[Tuning Ranges of Regular Temperaments|valid range]]: [257.143, 276.923] (14b to 13)
[[Mapping]]: [<1 0 3|, <0 7 -3|]
 
nice range: [271.229, 271.708]
 
strict range: [271.229, 271.708]


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


Map: [<1 0 3|, <0 7 -3|]
[[Tuning Ranges]]:
* valid range: [257.143, 276.923] (3\14 to 3\13)
* nice range: [271.229, 271.708]
* strict range: [271.229, 271.708]


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


Badness: 0.0408
[[Badness]]: 0.0408


== Seven limit children ==
== Seven limit children ==
The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. Adding 65536/65625 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 65536/65625 leads to orwell, but we could also add
* 1029/1024, leading to the 31&159 temperament with wedgie <<21 -9 -7 -63 -70 9||, or  
* 1029/1024, leading to the 31&159 temperament with wedgie {{multival| 21 -9 -7 -63 -70 9 }}, or  
* 67528125/67108864, giving the 31&243 temperament with wedgie <<28 -12 1 -84 -77 36||, or  
* 67528125/67108864, giving the 31&243 temperament with wedgie {{multival| 28 -12 1 -84 -77 36 }}, or  
* 4375/4374, giving the 53&243 temperament with wedgie <<7 -3 61 -21 77 150||.
* 4375/4374, giving the 53&243 temperament with wedgie {{multival| 7 -3 61 -21 77 150 }}.


= 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 <<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'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.


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.
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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.


[[Comma]]s: 225/224, 1728/1715
[[Comma list]]: 225/224, 1728/1715
 
7-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>]
 
[[Eigenmonzo]]s: 2, 7/5
 
9-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>]
 
[[Eigenmonzo]]s: 2, 10/9
 
valid range: [266.667, 272.727] (9 to 22)


nice range: [266.871, 271.708]
Mapping: [<1 0 3 1|, <0 7 -3 8|]


strict range: [266.871, 271.708]
{{Multival|legend=1| 7 -3 8 -21 -7 27 }}


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


Algebraic generators: Sabra3, the real root of 12x^3-7x-48.
[[Minimax tuning]]:  
* [[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>]
: [[Eigenmonzo]]s: 2, 7/5
* 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>]
: [[Eigenmonzo]]s: 2, 10/9


Map: [<1 0 3 1|, <0 7 -3 8|]
[[Tuning ranges]]:
* valid range: [266.667, 272.727] (9 to 22)
* nice range: [266.871, 271.708]
* strict range: [266.871, 271.708]


Wedgie: <<7 -3 8 -21 -7 27||
[[Algebraic generator]]: Sabra3, the real root of 12''x<sup>3</sup> - 7''x'' - 48.


{{EDOs|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 ==
[[Comma]]s: 99/98, 121/120, 176/175
Comma list: 99/98, 121/120, 176/175


[[Minimax tuning]]
Mapping: [&lt;1 0 3 1 3|, &lt;0 7 -3 8 2|]


[|1 0 0 0 0&gt;, |14/11 0 -7/11 7/11 0&gt;, |27/11 0 3/11 -3/11 0&gt;,
[[POTE generator]]: ~7/6 = 271.426
|27/11 0 -8/11 8/11 0&gt;, |37/11 0 -2/11 2/11 0&gt;]
 
[[Eigenmonzo]]s: 2, 7/5
 
valid range: [270.968, 272.727] (31 to 22)
 
nice range: [266.871, 275.659]
 
strict range: [270.968, 272.727]


[[POTE generator]]: ~7/6 = 271.426
Minimax tuning:
* 11-odd-limit
: [|1 0 0 0 0&gt;, |14/11 0 -7/11 7/11 0&gt;, |27/11 0 3/11 -3/11 0&gt;, |27/11 0 -8/11 8/11 0&gt;, |37/11 0 -2/11 2/11 0&gt;]
: [[Eigenmonzo]]s: 2, 7/5


Map: [&lt;1 0 3 1 3|, &lt;0 7 -3 8 2|]
Tuning ranges:
* valid range: [270.968, 272.727] (31 to 22)
* nice range: [266.871, 275.659]
* strict range: [270.968, 272.727]


{{EDOs|legend=1| 22, 31, 53, 84e }}
{{Val list|legend=1| 22, 31, 53, 84e }}


Badness: 0.0152
Badness: 0.0152


=== 13-limit ===
=== 13-limit ===
Commas: 99/98, 121/120, 176/175, 275/273
Comma list: 99/98, 121/120, 176/175, 275/273


valid range: [270.968, 271.698] (31 to 53)
Mapping: [&lt;1 0 3 1 3 8|, &lt;0 7 -3 8 2 -19|]


nice range: [266.871, 275.659]
POTE generator: ~7/6 = 271.546


strict range: [270.968, 271.698]
Tuning ranges:
* valid range: [270.968, 271.698] (31 to 53)
* nice range: [266.871, 275.659]
* strict range: [270.968, 271.698]


[[POTE generator]]: ~7/6 = 271.546
{{Val list|legend=1| 22, 31, 53, 84e, 137e }}
 
Map: [&lt;1 0 3 1 3 8|, &lt;0 7 -3 8 2 -19|]
 
{{EDOs|legend=1| 22, 31, 53, 84e, 137e }}


Badness: 0.0197
Badness: 0.0197
[[Orwell #Music|Music in Orwell]]


=== Blair ===
=== Blair ===
Commas: 65/64, 78/77, 91/90, 99/98
Comma list: 65/64, 78/77, 91/90, 99/98
 
valid range: []
 
nice range: [265.357, 289.210]


strict range: []
Mapping: [&lt;1 0 3 1 3 3|, &lt;0 7 -3 8 2 3|]


POTE generator: ~7/6 = 271.301
POTE generator: ~7/6 = 271.301


Map: [&lt;1 0 3 1 3 3|, &lt;0 7 -3 8 2 3|]
Tuning ranges:
* valid range: []
* nice range: [265.357, 289.210]
* strict range: []


{{EDOs|legend=1| 9, 22, 31f }}
{{Val list|legend=1| 9, 22, 31f }}


Badness: 0.0231
Badness: 0.0231


=== Winston ===
=== Winston ===
Commas: 66/65, 99/98, 105/104, 121/120
Comma list: 66/65, 99/98, 105/104, 121/120
 
valid range: [270.968, 272.727] (31 to 22f)
 
nice range: [266.871, 281.691]


strict range: [270.968, 272.727]
Mapping: [&lt;1 0 3 1 3 1|, &lt;0 7 -3 8 2 12|]


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


Map: [&lt;1 0 3 1 3 1|, &lt;0 7 -3 8 2 12|]
Tuning ranges:
* valid range: [270.968, 272.727] (31 to 22f)
* nice range: [266.871, 281.691]
* strict range: [270.968, 272.727]


{{EDOs|legend=1| 22f, 31 }}
{{Val list|legend=1| 22f, 31 }}


Badness: 0.0199
Badness: 0.0199


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


valid range: [135.484, 136.364] (62 to 44)
Mapping: [&lt;1 0 3 1 3 2|, &lt;0 14 -6 16 4 15|]


nice range: [128.298, 138.573]
POTE generator: ~13/12 = 135.723


strict range: [135.484, 136.364]
Tuning ranges:
* valid range: [135.484, 136.364] (62 to 44)
* nice range: [128.298, 138.573]
* strict range: [135.484, 136.364]


POTE tuning: ~13/12 = 135.723
{{Val list|legend=1| 9, 35, 44, 53, 62, 115ef, 168ef }}
 
Map: [&lt;1 0 3 1 3 2|, &lt;0 14 -6 16 4 15|]
 
{{EDOs|legend=1| 9, 35, 44, 53, 62, 115ef, 168ef }}


Badness: 0.0271
Badness: 0.0271


== Newspeak ==
== Newspeak ==
Commas: 225/224, 441/440, 1728/1715
Comma list: 225/224, 441/440, 1728/1715
 
valid range: [270.968, 271.698] (31 to 53)
 
nice range: [266.871, 272.514]


strict range: [270.968, 271.698]
Mapping: [&lt;1 0 3 1 -4|, &lt;0 7 -3 8 33|]


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


Map: [&lt;1 0 3 1 -4|, &lt;0 7 -3 8 33|]
Tuning ranges:
* valid range: [270.968, 271.698] (31 to 53)
* nice range: [266.871, 272.514]
* strict range: [270.968, 271.698]


{{EDOs|legend=1| 31, 84, 115, 376b, 491bd, 606bde }}
{{Val list|legend=1| 31, 84, 115, 376b, 491bd, 606bde }}


Badness: 0.0314
Badness: 0.0314


== Borwell ==
== Borwell ==
Commas: 225/224, 243/242, 1728/1715
Comma list: 225/224, 243/242, 1728/1715
 
Mapping: [&lt;1 7 0 9 17|, &lt;0 -14 6 -16 -35|]


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


Map: [&lt;1 7 0 9 17|, &lt;0 -14 6 -16 -35|]
{{Val list|legend=1| 31, 106, 137, 442bd }}
 
{{EDOs|legend=1| 31, 106, 137, 442bd }}


Badness: 0.0384
Badness: 0.0384


= Triwell =
= Triwell =
Commas: 1029/1024, 235298/234375
[[Comma list]]: 1029/1024, 235298/234375


POTE generator: ~448/375 = 309.472
[[Mapping]]: [&lt;1 7 0 1|, &lt;0 -21 9 7]]


Map: [&lt;1 7 0 1|, &lt;0 -21 9 7]]
{{Multival|legend=1| 21 -9 -7 -63 -70 9 }}


Wedgie: &lt;&lt;21 -9 -7 -63 -70 9||
[[POTE generator]]: ~448/375 = 309.472


{{EDOs|legend=1| 31, 97, 128, 159, 190 }}
{{Val list|legend=1| 31, 97, 128, 159, 190 }}


Badness: 0.0806
[[Badness]]: 0.0806


== 11-limit ==
== 11-limit ==
Commas: 385/384, 441/440, 456533/455625
Comma list: 385/384, 441/440, 456533/455625
 
Mapping: [&lt;1 7 0 1 13|, &lt;0 -21 9 7 -37]]


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


Map: [&lt;1 7 0 1 13|, &lt;0 -21 9 7 -37]]
{{Val list|legend=1| 31, 97, 128, 159, 190 }}
 
{{EDOs|legend=1| 31, 97, 128, 159, 190 }}


Badness: 0.0298
Badness: 0.0298


[[Category:Theory]]
[[Category:Regular temperament theory]]
[[Category:Temperament family]]
[[Category:Temperament family]]
[[Category:Semicomma]]
[[Category:Semicomma]]
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Orson]]
[[Category:Orwell]]
[[Category:Orwell]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/Semicomma_family"