Orwellismic family: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
The '''orwellismic family''' of rank-3 temperaments tempers out the orwellisma, [[1728/1715]].
The '''orwellismic family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the orwellisma, [[1728/1715]].


== Orwellismic ==
== Orwellismic ==
Orwellismic uses three [[~]][[7/6]] generators to reach [[8/5]] while preserving the lattice structure of the [[2.3.7 subgroup]]. Ideally, 7/6 and 5/4 are both tuned slightly sharp.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 8: Line 10:


{{Mapping|legend=1| 1 0 0 2 | 0 1 0 1 | 0 0 3 -1 }}
{{Mapping|legend=1| 1 0 0 2 | 0 1 0 1 | 0 0 3 -1 }}
: mapping generators: ~2, ~3, ~12/7
: mapping generators: ~2, ~3, ~12/7


Line 16: Line 17:
: 7/6 length = 0.7526, 3/2 length = 1.3931
: 7/6 length = 0.7526, 3/2 length = 1.3931
: Angle (7/6, 3/2) = 97.5081
: Angle (7/6, 3/2) = 97.5081
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.4176{{c}}, ~3/2 = 701.8055{{c}}, ~12/7 = 928.9445{{c}}
: [[error map]]: {{val| -0.582 -0.732 +0.520 +2.288 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.6429{{c}}, ~12/7 = 929.2316{{c}}
: error map: {{val| 0.000 -0.312 +1.381 +3.585 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]]  
* [[7-odd-limit|7-]] and [[9-odd-limit]]  
: {{Monzo list| 1 0 0 0 | 0 1 0 0 | 3/2 3/4 3/4 -3/4 | 3/2 3/4 -1/4 1/4 }}
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 3/2 3/4 3/4 -3/4 | 3/2 3/4 -1/4 1/4 }}
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.3.7/5
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7/5


{{Optimal ET sequence|legend=1| 4, 5, 9, 17, 18, 22, 27, 31, 53, 80, 84, 111, 164d, 195d, 275d, 306d }}
{{Optimal ET sequence|legend=1| 4, 5, 9, 17, 18, 22, 27, 31, 53, 80, 84, 111, 164d, 195d, 275d, 306d }}


[[Badness]]: 0.187 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.826


[[Projection pair]]: ~5 = 1728/343 to 2.3.7
[[Projection pair]]: ~5 = 1728/343 to 2.3.7
Line 31: Line 38:


== Guanyin ==
== Guanyin ==
{{See also|Subgroup temperaments#Guanyintet}}
Guanyin is the most natural [[11-limit]] extension of orwellismic, where two ~7/6 generators make [[15/11]], tempering out [[540/539]], and two sharpened ~[[5/4]]'s make [[11/7]], tempering out [[176/175]].
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Line 42: Line 53:
: 7/6 length = 0.5167, 3/2 length = 1.3055
: 7/6 length = 0.5167, 3/2 length = 1.3055
: Angle (7/6, 3/2) = 79.0940
: Angle (7/6, 3/2) = 79.0940
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.2336{{c}}, ~3/2 = 702.4159{{c}}, ~12/7 = 929.3128{{c}}
: [[error map]]: {{val| -0.7664 -0.3055 +1.6247 +1.9781 -1.5717 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2876{{c}}, ~12/7 = 929.7608{{c}}
: error map: {{val| 0.0000 +0.3326 +2.9688 +3.7009 -0.2261 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]  
* [[11-odd-limit]]  
: [{{monzo| 1 0 0 0 0 }}, {{monzo| -2/9 0 -5/9 5/9 4/9 }}, {{monzo| 4/3 0 1/3 -1/3 1/3 }}, {{monzo| 4/3 0 -2/3 2/3 1/3 }}, {{monzo| 0 0 0 0 1 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| -2/9 0 -5/9 5/9 4/9 }}, {{monzo| 4/3 0 1/3 -1/3 1/3 }}, {{monzo| 4/3 0 -2/3 2/3 1/3 }}, {{monzo| 0 0 0 0 1 }}]
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.11.7/5
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.11.7/5


{{Optimal ET sequence|legend=1| 22, 27e, 31, 53, 58, 80, 111, 280cd }}
{{Optimal ET sequence|legend=1| 22, 27e, 31, 53, 58, 80, 111, 280cd }}


[[Badness]]: 0.411 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.494


[[Projection pair]]s: ~5 = 1728/343, ~11 = 186624/16807 to 2.3.7
[[Projection pair]]s: ~5 = 1728/343, ~11 = 186624/16807 to 2.3.7
Line 57: Line 74:


; Music
; Music
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) by [[Gene Ward Smith]] – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – guanyin[22] in 111edo tuning
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) by [[Gene Ward Smith]] – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – in Guanyin[22], 111edo tuning


=== 13-limit ===
=== 13-limit ===
Line 66: Line 83:
Mapping: {{mapping| 1 0 0 2 -2 3 | 0 1 0 1 1 -2 | 0 0 3 -1 5 5 }}
Mapping: {{mapping| 1 0 0 2 -2 3 | 0 1 0 1 1 -2 | 0 0 3 -1 5 5 }}


{{Optimal ET sequence|legend=1| 22, 27e, 31, 53, 58, 80, 111, 280cdf }}
Optimal tunings:
* WE: ~2 = 1199.2015{{c}}, ~3/2 = 702.4954{{c}}, ~12/7 = 929.2881{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6821{{c}}, ~12/7 = 929.7296{{c}}


Badness: 0.758 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 22, 27e, 31, 53, 58, 80, 111, 280cdf }}
 
Badness (Sintel): 0.708


Projection pairs: ~5 = 1728/343, ~11 = 186624/16807, ~13 = 221184/16897 to 2.3.7
Projection pairs: ~5 = 1728/343, ~11 = 186624/16807, ~13 = 221184/16897 to 2.3.7
Line 81: Line 102:
Mapping: {{mapping| 1 0 0 2 -2 6 | 0 1 0 1 1 1 | 0 0 3 -1 5 -5 }}
Mapping: {{mapping| 1 0 0 2 -2 6 | 0 1 0 1 1 1 | 0 0 3 -1 5 -5 }}


{{Optimal ET sequence|legend=1| 22f, 27e, 31, 49f, 58, 138cdeff, 196cdefff, 254ccdeeffff }}
Optimal tunings:
* WE: ~2 = 1198.4320{{c}}, ~3/2 = 702.7802{{c}}, ~12/7 = 929.4834{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.5889{{c}}, ~12/7 = 930.5977{{c}}


Badness: 0.904 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 22f, 27e, 31, 49f, 58, 138cdeff, 196cdefff, 254ccdeeffff }}
 
Badness (Sintel): 0.845


Projection pairs: ~5 = 1728/343, ~11 = 186624/16807, ~13 = 16807/1296 to 2.3.7
Projection pairs: ~5 = 1728/343, ~11 = 186624/16807, ~13 = 16807/1296 to 2.3.7


== Orwellian ==
== Orwellian ==
Orwellian is another 11-limit extension of orwellismic that tempers out [[99/98]], which is supported by 11-limit [[orwell]].
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Line 93: Line 120:


{{Mapping|legend=1| 1 0 0 2 5 | 0 1 0 1 0 | 0 0 3 -1 -2 }}
{{Mapping|legend=1| 1 0 0 2 5 | 0 1 0 1 0 | 0 0 3 -1 -2 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.7839{{c}}, ~3/2 = 699.6996{{c}}, ~12/7 = 928.6481{{c}}
: [[error map]]: {{val| +0.784 -1.471 -0.369 +4.577 -4.695 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 699.6270{{c}}, ~12/7 = 928.0978{{c}}
: error map: {{val| 0.000 -2.328 -2.020 +2.703 -7.514 }}


{{Optimal ET sequence|legend=1| 4, 5, 9, 17, 22, 31, 53, 84e }}
{{Optimal ET sequence|legend=1| 4, 5, 9, 17, 22, 31, 53, 84e }}


[[Badness]]: 0.715 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.859


Scales: [[orwellian chromatic]]
Scales: [[orwellian chromatic]]

Latest revision as of 18:05, 22 January 2026

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 orwellismic family of rank-3 temperaments tempers out the orwellisma, 1728/1715.

Orwellismic

Orwellismic uses three ~7/6 generators to reach 8/5 while preserving the lattice structure of the 2.3.7 subgroup. Ideally, 7/6 and 5/4 are both tuned slightly sharp.

Subgroup: 2.3.5.7

Comma list: 1728/1715

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

mapping generators: ~2, ~3, ~12/7

Mapping to lattice: [⟨0 0 -3 1], ⟨0 1 0 1]]

Lattice basis:

7/6 length = 0.7526, 3/2 length = 1.3931
Angle (7/6, 3/2) = 97.5081

Optimal tunings:

  • WE: ~2 = 1199.4176 ¢, ~3/2 = 701.8055 ¢, ~12/7 = 928.9445 ¢
error map: ⟨-0.582 -0.732 +0.520 +2.288]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.6429 ¢, ~12/7 = 929.2316 ¢
error map: ⟨0.000 -0.312 +1.381 +3.585]

Minimax tuning:

[[1 0 0 0⟩, [0 1 0 0⟩, [3/2 3/4 3/4 -3/4⟩, [3/2 3/4 -1/4 1/4⟩]
unchanged-interval (eigenmonzo) basis: 2.3.7/5

Optimal ET sequence: 4, 5, 9, 17, 18, 22, 27, 31, 53, 80, 84, 111, 164d, 195d, 275d, 306d

Badness (Sintel): 0.826

Projection pair: ~5 = 1728/343 to 2.3.7

Scales: orwellismic9, orwellismic diatonic

Guanyin

See also: Subgroup temperaments § Guanyintet

Guanyin is the most natural 11-limit extension of orwellismic, where two ~7/6 generators make 15/11, tempering out 540/539, and two sharpened ~5/4's make 11/7, tempering out 176/175.

Subgroup: 2.3.5.7.11

Comma list: 176/175, 540/539

Mapping: [⟨1 0 0 2 -2], ⟨0 1 0 1 1], ⟨0 0 3 -1 5]]

Mapping to lattice: [⟨0 0 -3 1 -5], ⟨0 1 0 1 1]]

Lattice basis:

7/6 length = 0.5167, 3/2 length = 1.3055
Angle (7/6, 3/2) = 79.0940

Optimal tunings:

  • WE: ~2 = 1199.2336 ¢, ~3/2 = 702.4159 ¢, ~12/7 = 929.3128 ¢
error map: ⟨-0.7664 -0.3055 +1.6247 +1.9781 -1.5717]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2876 ¢, ~12/7 = 929.7608 ¢
error map: ⟨0.0000 +0.3326 +2.9688 +3.7009 -0.2261]

Minimax tuning:

[[1 0 0 0 0⟩, [-2/9 0 -5/9 5/9 4/9⟩, [4/3 0 1/3 -1/3 1/3⟩, [4/3 0 -2/3 2/3 1/3⟩, [0 0 0 0 1⟩]
unchanged-interval (eigenmonzo) basis: 2.11.7/5

Optimal ET sequence: 22, 27e, 31, 53, 58, 80, 111, 280cd

Badness (Sintel): 0.494

Projection pairs: ~5 = 1728/343, ~11 = 186624/16807 to 2.3.7

Scales: guanyin22

Music

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 540/539

Mapping: [⟨1 0 0 2 -2 3], ⟨0 1 0 1 1 -2], ⟨0 0 3 -1 5 5]]

Optimal tunings:

  • WE: ~2 = 1199.2015 ¢, ~3/2 = 702.4954 ¢, ~12/7 = 929.2881 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6821 ¢, ~12/7 = 929.7296 ¢

Optimal ET sequence: 22, 27e, 31, 53, 58, 80, 111, 280cdf

Badness (Sintel): 0.708

Projection pairs: ~5 = 1728/343, ~11 = 186624/16807, ~13 = 221184/16897 to 2.3.7

Avalokita

This extension used to be known as georgian.

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 176/175, 196/195

Mapping: [⟨1 0 0 2 -2 6], ⟨0 1 0 1 1 1], ⟨0 0 3 -1 5 -5]]

Optimal tunings:

  • WE: ~2 = 1198.4320 ¢, ~3/2 = 702.7802 ¢, ~12/7 = 929.4834 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.5889 ¢, ~12/7 = 930.5977 ¢

Optimal ET sequence: 22f, 27e, 31, 49f, 58, 138cdeff, 196cdefff, 254ccdeeffff

Badness (Sintel): 0.845

Projection pairs: ~5 = 1728/343, ~11 = 186624/16807, ~13 = 16807/1296 to 2.3.7

Orwellian

Orwellian is another 11-limit extension of orwellismic that tempers out 99/98, which is supported by 11-limit orwell.

Subgroup: 2.3.5.7.11

Comma list: 99/98, 385/384

Mapping: [⟨1 0 0 2 5], ⟨0 1 0 1 0], ⟨0 0 3 -1 -2]]

Optimal tunings:

  • WE: ~2 = 1200.7839 ¢, ~3/2 = 699.6996 ¢, ~12/7 = 928.6481 ¢
error map: ⟨+0.784 -1.471 -0.369 +4.577 -4.695]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 699.6270 ¢, ~12/7 = 928.0978 ¢
error map: ⟨0.000 -2.328 -2.020 +2.703 -7.514]

Optimal ET sequence: 4, 5, 9, 17, 22, 31, 53, 84e

Badness (Sintel): 0.859

Scales: orwellian chromatic

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