Orwellismic family: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Technical data page}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
The '''orwellismic family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the orwellisma, [[1728/1715]].
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-01-03 16:02:10 UTC</tt>.<br>
: The original revision id was <tt>190718138</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">The orwellismic family of rank three temperaments tempers out the orwellisma, 1728/1715.


Comma: 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.


7- and 9-limit minimax
[[Subgroup]]: 2.3.5.7
[&lt;1 0 0 0|, &lt;0 1 0 0|, &lt;3/2 3/4 3/4 -3/4|,
&lt;3/2 3/4 -1/4 1/4|]
Eigenmonzos: 2, 4/3, 7/5


Lattice basis: 7/6 0.7526 3/2 1.3931
[[Comma list]]: 1728/1715
Angle(7/6, 3/2) = 97.5081
Map to lattice: [&lt;0 0 -3 1|, &lt;0 1 0 1|]


Map: [&lt;1 0 0 2|, &lt;0 1 0 1|, &lt;0 0 3 -1|]
{{Mapping|legend=1| 1 0 0 2 | 0 1 0 1 | 0 0 3 -1 }}
Generators: 2, 3, 12/7
: mapping generators: ~2, ~3, ~12/7
Edos: 9, 14, 17, 18, 22, 26, 27, 31, 49, 53, 58, 80, 84, 111, 142
Badness: 0.000187


Scales: [[orwellismic9]]
[[Mapping to lattice]]: {{mapping| 0 0 -3 1 | 0 1 0 1 }}


==Orwellic==
Lattice basis:
Commas: 176/175, 540/539
: 7/6 length = 0.7526, 3/2 length = 1.3931
: Angle (7/6, 3/2) = 97.5081


11-limit minimax
[[Optimal tuning]]s:
[&lt;1 0 0 0 0|, &lt;-2/9 0 -5/9 5/9 4/9|, &lt;4/3 0 1/3 -1/3 1/3|,
* [[WE]]: ~2 = 1199.4176{{c}}, ~3/2 = 701.8055{{c}}, ~12/7 = 928.9445{{c}}
&lt;4/3 0 -2/3 2/3 1/3|, &lt;0 0 0 0 1|]
: [[error map]]: {{val| -0.582 -0.732 +0.520 +2.288 }}
Eigenmonzos: 2, 11/8, 7/5
* [[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 }}


Lattice basis: 7/6 0.5167 3/2 1.3055
[[Minimax tuning]]:  
Angle(7/6, 3/2) = 79.0940
* [[7-odd-limit|7-]] and [[9-odd-limit]]
Map to lattice: [&lt;0 0 -3 1 -5|, &lt;0 1 0 1 1|]
: {{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|unchanged-interval (eigenmonzo) basis]]: 2.3.7/5


Map: [&lt;1 0 0 2 -2|, &lt;0 1 0 1 1|, &lt;0 0 3 -1 5|]
{{Optimal ET sequence|legend=1| 4, 5, 9, 17, 18, 22, 27, 31, 53, 80, 84, 111, 164d, 195d, 275d, 306d }}
Generators: 2, 3, 12/7
EDOs: 9, 22, 31, 53, 58, 80, 111
Badness: 0.000411


Scales: [[orwellic22]]
[[Badness]] (Sintel): 0.826


===13-limit===
[[Projection pair]]: ~5 = 1728/343 to 2.3.7
Commas: 176/175, 351/350, 540/539


Map: [&lt;1 0 0 2 -2 3|, &lt;0 1 0 1 1 -2|, &lt;0 0 3 -1 5 5|]
Scales: [[orwellismic9]], [[orwellismic diatonic]]
EDOs: 22, 31, 53, 58, 80, 111
Badness: 0.000758


===Georgian===
== Guanyin ==
Commas: 144/143, 176/175, 196/195
{{See also|Subgroup temperaments#Guanyintet}}


Map: [&lt;1 0 0 2 -2 6|, &lt;0 1 0 1 1 1|, &lt;0 0 3 -1 5 -5|]
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]].  
EDOs: 9, 31, 58
 
Badness: 0.000904</pre></div>
[[Subgroup]]: 2.3.5.7.11
<h4>Original HTML content:</h4>
 
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Orwellismic family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The orwellismic family of rank three temperaments tempers out the orwellisma, 1728/1715.&lt;br /&gt;
[[Comma list]]: 176/175, 540/539
&lt;br /&gt;
 
Comma: 1728/1715&lt;br /&gt;
{{Mapping|legend=1| 1 0 0 2 -2 | 0 1 0 1 1 | 0 0 3 -1 5 }}
&lt;br /&gt;
 
7- and 9-limit minimax&lt;br /&gt;
Mapping to lattice: {{mapping| 0 0 -3 1 -5 | 0 1 0 1 1 }}
[&amp;lt;1 0 0 0|, &amp;lt;0 1 0 0|, &amp;lt;3/2 3/4 3/4 -3/4|, &lt;br /&gt;
 
&amp;lt;3/2 3/4 -1/4 1/4|]&lt;br /&gt;
Lattice basis:
Eigenmonzos: 2, 4/3, 7/5&lt;br /&gt;
: 7/6 length = 0.5167, 3/2 length = 1.3055
&lt;br /&gt;
: Angle (7/6, 3/2) = 79.0940
Lattice basis: 7/6 0.7526 3/2 1.3931&lt;br /&gt;
 
Angle(7/6, 3/2) = 97.5081&lt;br /&gt;
[[Optimal tuning]]s:  
Map to lattice: [&amp;lt;0 0 -3 1|, &amp;lt;0 1 0 1|]&lt;br /&gt;
* [[WE]]: ~2 = 1199.2336{{c}}, ~3/2 = 702.4159{{c}}, ~12/7 = 929.3128{{c}}
&lt;br /&gt;
: [[error map]]: {{val| -0.7664 -0.3055 +1.6247 +1.9781 -1.5717 }}
Map: [&amp;lt;1 0 0 2|, &amp;lt;0 1 0 1|, &amp;lt;0 0 3 -1|]&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2876{{c}}, ~12/7 = 929.7608{{c}}
Generators: 2, 3, 12/7&lt;br /&gt;
: error map: {{val| 0.0000 +0.3326 +2.9688 +3.7009 -0.2261 }}
Edos: 9, 14, 17, 18, 22, 26, 27, 31, 49, 53, 58, 80, 84, 111, 142&lt;br /&gt;
 
Badness: 0.000187&lt;br /&gt;
[[Minimax tuning]]:  
&lt;br /&gt;
* [[11-odd-limit]]
Scales: &lt;a class="wiki_link" href="/orwellismic9"&gt;orwellismic9&lt;/a&gt;&lt;br /&gt;
: [{{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 }}]
&lt;br /&gt;
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.11.7/5
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Orwellic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Orwellic&lt;/h2&gt;
 
Commas: 176/175, 540/539&lt;br /&gt;
{{Optimal ET sequence|legend=1| 22, 27e, 31, 53, 58, 80, 111, 280cd }}
&lt;br /&gt;
 
11-limit minimax&lt;br /&gt;
[[Badness]] (Sintel): 0.494
[&amp;lt;1 0 0 0 0|, &amp;lt;-2/9 0 -5/9 5/9 4/9|, &amp;lt;4/3 0 1/3 -1/3 1/3|, &lt;br /&gt;
 
&amp;lt;4/3 0 -2/3 2/3 1/3|, &amp;lt;0 0 0 0 1|]&lt;br /&gt;
[[Projection pair]]s: ~5 = 1728/343, ~11 = 186624/16807 to 2.3.7
Eigenmonzos: 2, 11/8, 7/5&lt;br /&gt;
 
&lt;br /&gt;
Scales: [[guanyin22]]
Lattice basis: 7/6 0.5167 3/2 1.3055&lt;br /&gt;
 
Angle(7/6, 3/2) = 79.0940&lt;br /&gt;
; Music
Map to lattice: [&amp;lt;0 0 -3 1 -5|, &amp;lt;0 1 0 1 1|]&lt;br /&gt;
* ''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
&lt;br /&gt;
 
Map: [&amp;lt;1 0 0 2 -2|, &amp;lt;0 1 0 1 1|, &amp;lt;0 0 3 -1 5|]&lt;br /&gt;
=== 13-limit ===
Generators: 2, 3, 12/7&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
EDOs: 9, 22, 31, 53, 58, 80, 111&lt;br /&gt;
 
Badness: 0.000411&lt;br /&gt;
Comma list: 176/175, 351/350, 540/539
&lt;br /&gt;
 
Scales: &lt;a class="wiki_link" href="/orwellic22"&gt;orwellic22&lt;/a&gt;&lt;br /&gt;
Mapping: {{mapping| 1 0 0 2 -2 3 | 0 1 0 1 1 -2 | 0 0 3 -1 5 5 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x-Orwellic-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;13-limit&lt;/h3&gt;
Optimal tunings:
Commas: 176/175, 351/350, 540/539&lt;br /&gt;
* WE: ~2 = 1199.2015{{c}}, ~3/2 = 702.4954{{c}}, ~12/7 = 929.2881{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6821{{c}}, ~12/7 = 929.7296{{c}}
Map: [&amp;lt;1 0 0 2 -2 3|, &amp;lt;0 1 0 1 1 -2|, &amp;lt;0 0 3 -1 5 5|]&lt;br /&gt;
 
EDOs: 22, 31, 53, 58, 80, 111&lt;br /&gt;
{{Optimal ET sequence|legend=0| 22, 27e, 31, 53, 58, 80, 111, 280cdf }}
Badness: 0.000758&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 0.708
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x-Orwellic-Georgian"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Georgian&lt;/h3&gt;
 
Commas: 144/143, 176/175, 196/195&lt;br /&gt;
Projection pairs: ~5 = 1728/343, ~11 = 186624/16807, ~13 = 221184/16897 to 2.3.7
&lt;br /&gt;
 
Map: [&amp;lt;1 0 0 2 -2 6|, &amp;lt;0 1 0 1 1 1|, &amp;lt;0 0 3 -1 5 -5|]&lt;br /&gt;
=== Avalokita ===
EDOs: 9, 31, 58&lt;br /&gt;
This extension used to be known as ''georgian''.
Badness: 0.000904&lt;/body&gt;&lt;/html&gt;</pre></div>
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 144/143, 176/175, 196/195
 
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 }}
 
{{Optimal ET sequence|legend=1| 4, 5, 9, 17, 22, 31, 53, 84e }}
 
[[Badness]] (Sintel): 0.859
 
Scales: [[orwellian chromatic]]
 
[[Category:Temperament families]]
[[Category:Orwellismic family| ]] <!-- main article -->
[[Category:Rank 3]]
[[Category:Guanyin]]
[[Category:Listen]]

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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