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Orwellismic family: Difference between revisions

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


== Orwellismic ==
== Orwellismic ==
Subgroup: 2.3.5.7
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
[[Comma list]]: 1728/1715


[[Mapping]]: [{{val| 1 0 0 2 }}, {{val| 0 1 0 1 }}, {{val| 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
[[Mapping to lattice]]: {{mapping| 0 0 -3 1 | 0 1 0 1 }}
 
[[Mapping to lattice]]: [{{val| 0 0 -3 1 }}, {{val| 0 1 0 1 }}]


Lattice basis:  
Lattice basis:  
: 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| 1 0 0 0 }}, {{monzo| 0 1 0 0 }}, {{monzo| 3/2 3/4 3/4 -3/4 }}, {{monzo| 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]]s (unchanged intervals): 2, 4/3, 7/5
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7/5
 
{{Val list|legend=1| 9, 14, 17, 18, 22, 26, 27, 31, 49, 53, 58, 80, 84, 111, 142, 164d, 306d, 417cd }}
 
[[Badness]]: 0.187 × 10<sup>-3</sup>


[[Projection pair]]: 5 1728/343 to 2.3.7
{{Optimal ET sequence|legend=1| 4, 5, 9, 17, 18, 22, 27, 31, 53, 80, 84, 111, 164d, 195d, 275d, 306d }}


Scales: [[orwellismic9]]
[[Badness]] (Sintel): 0.826


== Orwellian  ==
[[Projection pair]]: ~5 = 1728/343 to 2.3.7
Subgroup: 2.3.5.7.11


[[Comma list]]: 99/98, 385/384
Scales: [[orwellismic9]], [[orwellismic diatonic]]


[[Mapping]]: [{{val| 1 0 0 2 5 }}, {{val| 0 1 0 1 0 }}, {{val| 0 0 3 -1 -2 }}]
== Guanyin ==
{{See also|Subgroup temperaments#Guanyintet}}


{{Val list|legend=1| 22, 31, 53, 84e, 163e, 194e, 247cde, 278bdee }}
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]].


[[Badness]]: 0.715 × 10<sup>-3</sup>
[[Subgroup]]: 2.3.5.7.11


== Guanyin  ==
[[Comma list]]: 176/175, 540/539
[[Comma list]]: 176/175, 540/539


[[Mapping]]: [{{val| 1 0 0 2 -2 }}, {{val| 0 1 0 1 1 }}, {{val| 0 0 3 -1 5 }}]
{{Mapping|legend=1| 1 0 0 2 -2 | 0 1 0 1 1 | 0 0 3 -1 5 }}


Mapping to lattice: [{{val| 0 0 -3 1 -5 }}, {{val| 0 1 0 1 1 }}]
Mapping to lattice: {{mapping| 0 0 -3 1 -5 | 0 1 0 1 1 }}


Lattice basis:  
Lattice basis:  
: 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]]s (unchanged intervals): 2, 11/8, 7/5
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.11.7/5


{{Val list|legend=1| 9, 22, 31, 53, 58, 80, 111, 280cd, 391cd }}
{{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


Scales: [[guanyin22]]
Scales: [[guanyin22]]


; Music:
; Music
* [http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene Ward Smith]]
* ''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 ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


Mapping: [{{val| 1 0 0 2 -2 3 }}, {{val| 0 1 0 1 1 -2 }}, {{val| 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 }}


Vals: {{Val list| 22, 31, 53, 58, 80, 111, 280cdf, 391cdf }}
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 }}


Projection pairs: 5 1728/343 11 186624/16807 221184/16897 to 2.3.7
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''.


=== Georgian  ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


Mapping: [{{val| 1 0 0 2 -2 6 }}, {{val| 0 1 0 1 1 1 }}, {{val| 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 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}}
 
{{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
 
== 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|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 }}


Vals: {{Val list| 9, 31, 46, 48, 58 }}
{{Optimal ET sequence|legend=1| 4, 5, 9, 17, 22, 31, 53, 84e }}


Badness: 0.904 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.859


Projection pairs: 5 1728/343 11 186624/16807 16807/1296 to 2.3.7
Scales: [[orwellian chromatic]]


[[Category:Regular temperament theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Orwellismic family| ]] <!-- main article -->
[[Category:Orwellismic family| ]] <!-- main article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
[[Category:Guanyin]]
[[Category:Guanyin]]
[[Category:Listen]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/Orwellismic_family"