Semicomma family: Difference between revisions

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__FORCETOC__
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=
 
The 5-limit parent comma for the '''semicomma family''' is the semicomma, 2109375/2097152 = |-21 3 7>. This is the amount by which three pure 3/1 twelfths exceed seven pure 8/5 minor sixths. '''Orson''', the [[5-limit|5-limit]] temperament tempering it out, has a [[generator|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|cent]]s, making it hard to treat orson as anything other than an (at least) 7-limit system, leading to orwell.
= 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.


Comma: 2109375/2097152
Comma: 2109375/2097152


[[Tuning_Ranges_of_Regular_Temperaments|valid range]]: [257.143, 276.923] (14b to 13)
[[Tuning Ranges of Regular Temperaments|valid range]]: [257.143, 276.923] (14b to 13)


nice range: [271.229, 271.708]
nice range: [271.229, 271.708]
Line 11: Line 12:
strict range: [271.229, 271.708]
strict range: [271.229, 271.708]


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


Map: [<1 0 3|, <0 7 -3|]
Map: [<1 0 3|, <0 7 -3|]
Line 19: Line 20:
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 65536/64625 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 67528125/67108864, giving the 31&243 temperament with wedgie <<28 -12 1 -84 -77 36||, or 4375/4374, giving the 53&243 temperament with wedgie <<7 -3 61 -21 77 150||.
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
* 1029/1024, leading to the 31&159 temperament with wedgie <<21 -9 -7 -63 -70 9||, or  
* 67528125/67108864, giving the 31&243 temperament with wedgie <<28 -12 1 -84 -77 36||, or  
* 4375/4374, giving the 53&243 temperament with wedgie <<7 -3 61 -21 77 150||.


=Orwell=
= Orwell =
Main article: [[Orwell|Orwell]]
{{main| Orwell }}


So called because 19\84 (as a [[fraction_of_the_octave|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|7-limit]] and naturally extends into the [[11-limit|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|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'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.


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.
Line 31: Line 35:
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|Commas]]: 225/224, 1728/1715
[[Comma]]s: 225/224, 1728/1715


7-limit
7-limit
Line 38: Line 42:
|27/11 0 3/11 -3/11>, |27/11 0 -8/11 8/11>]
|27/11 0 3/11 -3/11>, |27/11 0 -8/11 8/11>]


[[Fractional_monzos|Eigenmonzos]]: 2, 7/5
[[Eigenmonzo]]s: 2, 7/5


9-limit
9-limit
Line 45: Line 49:
|42/17 -6/17 3/17 0>, |41/17 16/17 -8/17 0>]
|42/17 -6/17 3/17 0>, |41/17 16/17 -8/17 0>]


[[Eigenmonzo|Eigenmonzos]]: 2, 10/9
[[Eigenmonzo]]s: 2, 10/9


valid range: [266.667, 272.727] (9 to 22)
valid range: [266.667, 272.727] (9 to 22)
Line 53: Line 57:
strict range: [266.871, 271.708]
strict range: [266.871, 271.708]


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


Algebraic generators: Sabra3, the real root of 12x^3-7x-48.
Algebraic generators: Sabra3, the real root of 12x^3-7x-48.
Line 61: Line 65:
Wedgie: <<7 -3 8 -21 -7 27||
Wedgie: <<7 -3 8 -21 -7 27||


EDOs: 22, 31, 53, 84, 137, 221d, 358d
{{EDOs|legend=1| 22, 31, 53, 84, 137, 221d, 358d }}


Badness: 0.0207
Badness: 0.0207


==11-limit==
==11-limit==
[[Comma|Commas]]: 99/98, 121/120, 176/175
[[Comma]]s: 99/98, 121/120, 176/175


[[Minimax_tuning|Minimax tuning]]
[[Minimax tuning]]


[|1 0 0 0 0>, |14/11 0 -7/11 7/11 0>, |27/11 0 3/11 -3/11 0>,
[|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>]
|27/11 0 -8/11 8/11 0>, |37/11 0 -2/11 2/11 0>]


[[Eigenmonzo|Eigenmonzos]]: 2, 7/5
[[Eigenmonzo]]s: 2, 7/5


valid range: [270.968, 272.727] (31 to 22)
valid range: [270.968, 272.727] (31 to 22)
Line 81: Line 85:
strict range: [270.968, 272.727]
strict range: [270.968, 272.727]


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


Map: [<1 0 3 1 3|, <0 7 -3 8 2|]
Map: [<1 0 3 1 3|, <0 7 -3 8 2|]


[[EDO|Edos]]: [[22edo|22]], [[31edo|31]], [[53edo|53]], [[84edo|84e]]
{{EDOs|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
Commas: 99/98, 121/120, 176/175, 275/273


Line 98: Line 102:
strict range: [270.968, 271.698]
strict range: [270.968, 271.698]


[[POTE_tuning|POTE generator]]: ~7/6 = 271.546
[[POTE generator]]: ~7/6 = 271.546


Map: [<1 0 3 1 3 8|, <0 7 -3 8 2 -19|]
Map: [<1 0 3 1 3 8|, <0 7 -3 8 2 -19|]


EDOs: 22, 31, 53, 84e, 137e
{{EDOs|legend=1| 22, 31, 53, 84e, 137e }}


Badness: 0.0197
Badness: 0.0197


[[Orwell#Music|Music in Orwell]]
[[Orwell #Music|Music in Orwell]]


==Blair==
=== Blair ===
Commas: 65/64, 78/77, 91/90, 99/98
Commas: 65/64, 78/77, 91/90, 99/98


Line 121: Line 125:
Map: [<1 0 3 1 3 3|, <0 7 -3 8 2 3|]
Map: [<1 0 3 1 3 3|, <0 7 -3 8 2 3|]


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


Badness: 0.0231
Badness: 0.0231


==Newspeak==
=== Winston ===
Commas: 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]
 
POTE tuning: ~7/6 = 271.288
 
Map: [<1 0 3 1 -4|, <0 7 -3 8 33|]
 
EDOs: 31, 84, 115, 376b, 491bd, 606bde
 
Badness: 0.0314
 
==Winston==
Commas: 66/65, 99/98, 105/104, 121/120
Commas: 66/65, 99/98, 105/104, 121/120


Line 151: Line 138:
strict range: [270.968, 272.727]
strict range: [270.968, 272.727]


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


Map: [<1 0 3 1 3 1|, <0 7 -3 8 2 12|]
Map: [<1 0 3 1 3 1|, <0 7 -3 8 2 12|]


EDOs: 22f, 31
{{EDOs|legend=1| 22f, 31 }}


Badness: 0.0199
Badness: 0.0199


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


Line 172: Line 159:
Map: [<1 0 3 1 3 2|, <0 14 -6 16 4 15|]
Map: [<1 0 3 1 3 2|, <0 14 -6 16 4 15|]


EDOs: 9, 35, 44, 53, 62, 115ef, 168ef
{{EDOs|legend=1| 9, 35, 44, 53, 62, 115ef, 168ef }}


Badness: 0.0271
Badness: 0.0271


=Borwell=
== Newspeak ==
Commas: 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]
 
POTE tuning: ~7/6 = 271.288
 
Map: [<1 0 3 1 -4|, <0 7 -3 8 33|]
 
{{EDOs|legend=1| 31, 84, 115, 376b, 491bd, 606bde }}
 
Badness: 0.0314
 
== Borwell ==
Commas: 225/224, 243/242, 1728/1715
Commas: 225/224, 243/242, 1728/1715


Line 183: Line 187:
Map: [<1 7 0 9 17|, <0 -14 6 -16 -35|]
Map: [<1 7 0 9 17|, <0 -14 6 -16 -35|]


EDOs: 31, 106, 137, 442bd
{{EDOs|legend=1| 31, 106, 137, 442bd }}


Badness: 0.0384
Badness: 0.0384


=Triwell=
= Triwell =
Commas: 1029/1024, 235298/234375
Commas: 1029/1024, 235298/234375


Line 196: Line 200:
Wedgie: <<21 -9 -7 -63 -70 9||
Wedgie: <<21 -9 -7 -63 -70 9||


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


Badness: 0.0806
Badness: 0.0806


==11-limit==
== 11-limit ==
Commas: 385/384, 441/440, 456533/455625
Commas: 385/384, 441/440, 456533/455625


Line 207: Line 211:
Map: [<1 7 0 1 13|, <0 -21 9 7 -37]]
Map: [<1 7 0 1 13|, <0 -21 9 7 -37]]


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


Badness: 0.0298
Badness: 0.0298
Line 214: Line 218:
[[Category:Temperament family]]
[[Category:Temperament family]]
[[Category:Semicomma]]
[[Category:Semicomma]]
[[Category:Rank 2]]
[[Category:Orwell]]
[[Category:Listen]]
[[Category:Listen]]
[[Category:Orwell]]
[[Category:Todo:add_definition]]
[[Category:Todo:intro]]

Revision as of 06:17, 14 January 2021

The 5-limit parent comma for the semicomma family is the semicomma, 2109375/2097152 = [-21 3 7⟩. This is the amount by which three pure 3/1 twelfths exceed seven pure 8/5 minor sixths.

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 cents, making it hard to treat orson as anything other than an (at least) 7-limit system, leading to orwell.

Comma: 2109375/2097152

valid range: [257.143, 276.923] (14b to 13)

nice range: [271.229, 271.708]

strict range: [271.229, 271.708]

POTE generator: ~75/64 = 271.627

Map: [<1 0 3|, <0 7 -3|]

EDOs: 22, 31, 53, 190, 243, 296, 645c

Badness: 0.0408

Seven limit children

The second comma of the 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
  • 67528125/67108864, giving the 31&243 temperament with wedgie <<28 -12 1 -84 -77 36||, or
  • 4375/4374, giving the 53&243 temperament with wedgie <<7 -3 61 -21 77 150||.

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's compatible with 22, 31, 53 and 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.

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 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.

Commas: 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>]

Eigenmonzos: 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>]

Eigenmonzos: 2, 10/9

valid range: [266.667, 272.727] (9 to 22)

nice range: [266.871, 271.708]

strict range: [266.871, 271.708]

POTE generator: ~7/6 = 271.509

Algebraic generators: Sabra3, the real root of 12x^3-7x-48.

Map: [<1 0 3 1|, <0 7 -3 8|]

Wedgie: <<7 -3 8 -21 -7 27||

EDOs: 22, 31, 53, 84, 137, 221d, 358d

Badness: 0.0207

11-limit

Commas: 99/98, 121/120, 176/175

Minimax tuning

[|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>]

Eigenmonzos: 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

Map: [<1 0 3 1 3|, <0 7 -3 8 2|]

EDOs: 22, 31, 53, 84e

Badness: 0.0152

13-limit

Commas: 99/98, 121/120, 176/175, 275/273

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

Map: [<1 0 3 1 3 8|, <0 7 -3 8 2 -19|]

EDOs: 22, 31, 53, 84e, 137e

Badness: 0.0197

Music in Orwell

Blair

Commas: 65/64, 78/77, 91/90, 99/98

valid range: []

nice range: [265.357, 289.210]

strict range: []

POTE generator: ~7/6 = 271.301

Map: [<1 0 3 1 3 3|, <0 7 -3 8 2 3|]

EDOs: 9, 22, 31f

Badness: 0.0231

Winston

Commas: 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]

POTE generator: ~7/6 = 271.088

Map: [<1 0 3 1 3 1|, <0 7 -3 8 2 12|]

EDOs: 22f, 31

Badness: 0.0199

Doublethink

Commas: 99/98, 121/120, 169/168, 176/175

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

Map: [<1 0 3 1 3 2|, <0 14 -6 16 4 15|]

EDOs: 9, 35, 44, 53, 62, 115ef, 168ef

Badness: 0.0271

Newspeak

Commas: 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]

POTE tuning: ~7/6 = 271.288

Map: [<1 0 3 1 -4|, <0 7 -3 8 33|]

EDOs: 31, 84, 115, 376b, 491bd, 606bde

Badness: 0.0314

Borwell

Commas: 225/224, 243/242, 1728/1715

POTE generator: ~55/36 = 735.752

Map: [<1 7 0 9 17|, <0 -14 6 -16 -35|]

EDOs: 31, 106, 137, 442bd

Badness: 0.0384

Triwell

Commas: 1029/1024, 235298/234375

POTE generator: ~448/375 = 309.472

Map: [<1 7 0 1|, <0 -21 9 7]]

Wedgie: <<21 -9 -7 -63 -70 9||

EDOs: 31, 97, 128, 159, 190

Badness: 0.0806

11-limit

Commas: 385/384, 441/440, 456533/455625

POTE generator: ~448/375 = 309.471

Map: [<1 7 0 1 13|, <0 -21 9 7 -37]]

EDOs: 31, 97, 128, 159, 190

Badness: 0.0298

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