Orwell extensions: Difference between revisions

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**Imported revision 451984760 - Original comment: **
Undo revision 225002 by VectorGraphics (talk). Here only 22 and 31 are used with different warts for ease of comparison between these extensions. Plus 9 isn't a reasonable tuning for orwell.
Tag: Undo
 
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Breadcrumb|Orwell}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-09-17 22:12:17 UTC</tt>.<br>
: The original revision id was <tt>451984760</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">[[Orwell]] temperament has various extensions to the 13 limit.


=Tuning Spectra=
[[Orwell]] has multiple competing [[extension]]s to the [[13-limit]]. This is evidenced by the fact that its [[support]]ing [[equal temperament]]s, [[22edo|22]] and [[31edo|31]], do less well in the 13-limit. The extensions are:
These spectra suggest possible tuning choices. For 13-limit orwell, the 5-limit minimax tuning featuring pure 6/5 eigenmonzos seems like an excellent choice, as it's right in the middle of the least squares range and very close to 13-limit least squares. Pure 13s, using the 16/13 eigenmonzo, might also please some people. For blair, pure 5/4s using the 5/4 eigenmonzo tuning is very close to 15-limit least squares and in general in the middle of the action. For winston, sticking with the 11/9 eigenmonzo minimax tuning seems reasonable.
* '''Tridecimal orwell''' ({{nowrap| 22 & 31 }}) – tempering out 99/98, 121/120, 176/175, and 275/273
* '''Blair''' ({{nowrap| 22 & 31f }}) – tempering out 65/64, 78/77, 91/90, and 99/98
* '''Winston''' ({{nowrap| 22f & 31 }}) – tempering out 66/65, 99/98, 105/104, and 121/120


==Spectrum of Orwell Tunings by Eigenmonzos==
The most important of these is tridecimal orwell, which tempers out [[352/351]] and may also be characterized by tempering out [[275/273]] instead. Supported by [[53edo|53]], it has the highest accuracy in its approximation of 13/8, but also the highest complexity. The other two extensions have lower complexity, but also lower accuracy. In winston, ~13/8 is conflated with ~18/11 and is generally tuned worse than in 31edo as a result of an improved ~18/11. In blair, ~13/8 is conflated with ~8/5 and is generally tuned worse than in 22edo as a result of an improved ~8/5.


Gencom: [2 7/6; 99/98 121/120 176/175 275/273]
Another possible path which relates a sense of compromise is to temper out [[169/168]], leading to [[doublethink]]. This has the effect of slicing the generator in two, and is supported by [[44edo|44]], 53, and [[62edo|62]].  
Gencom map: [&lt;1 0 3 1 3 8|, &lt;0 7 -3 8 2 -19|]
||~ Eigenmonzo ||~ Subminor Third ||
|| 7/6 || 266.871 ||
|| 15/11 || 268.475 ||
|| 14/11 || 269.585 ||
|| 12/11 || 270.127 ||
|| 15/14 || 270.139 ||
|| 7\31 || 270.968 ||
|| 11/9 || 271.049 ||
|| 8/7 || 271.103 ||
|| 7/5 || 271.137 ||
|| 5/4 || 271.229 ||
|| 1361367/1000000 || 271.326 (7 limit least squares) ||
|| 14/13 || 271.418 (13 and 15 limit minimax) ||
|| 19\84 || 271.429 ||
|| |0 119 -46 20 -16&gt; || 271.445 (11 limit least squares) ||
|| x^10 + 2x^3 = 8 || 271.508 (equal beating) ||
|| 16/13 || 271.551 ||
|| |0 90 -41 14&gt; || 271.561 (9 limit least squares) ||
|| 6/5 || 271.564 (5 limit minimax) ||
|| |0 -211 30 -47 -5 142&gt; || 271.567 (13 limit least squares) ||
|| |0 -236 5 -51 -3 165&gt; || 271.570 (15 limit least squares) ||
|| 1220703125/1033121304 || 271.590 (5 limit least squares) ||
|| 13/12 || 271.593 ||
|| 13/10 || 271.612 ||
|| 18/13 || 271.618 ||
|| 10/9 || 271.623 (9 limit minimax) ||
|| 15/13 || 271.641 ||
|| 12\53 || 271.698 ||
|| 4/3 || 271.708 ||
|| 13/11 || 271.942 ||
|| 16/15 || 272.067 ||
|| 9/7 || 272.514 ||
|| 5\22 || 272.727 ||
|| 11/10 || 273.001 ||
|| 11/8 || 275.659 ||


==Spectrum of Winston Tunings by Eigenmonzos==
See [[Semicomma family #Orwell]], [[Semicomma family #Blair|#Blair]], and [[Semicomma family #Winston|#Winston]] for technical data.


Gencom: [2 7/6; 66/65 99/98 105/104 121/120]
== Interval chain ==
Gencom map: [&lt;1 0 3 1 3 1|, &lt;0 7 -3 8 2 12|]
Odd harmonics 1–21 and their inverses are in '''bold'''.
||~ Eigenmonzo ||~ Subminor Third ||
|| 7/6 || 266.871 ||
|| 13/12 || 267.715 ||
|| 14/13 || 267.925 ||
|| 15/11 || 268.475 ||
|| 13/11 || 268.921 ||
|| 15/13 || 269.032 ||
|| 14/11 || 269.585 ||
|| 16/13 || 270.044 ||
|| 12/11 || 270.127 ||
|| 15/14 || 270.139 ||
|| 13/10 || 270.281 ||
|| |0 112 -67 20 -28 52&gt; || 270.860 (15 limit least squares) ||
|| |0 118 -61 16 -26 44&gt; || 270.933 (13 limit least squares) ||
|| 7\31 || 270.968 ||
|| 11/9 || 271.049 (13 and 15 limit minimax) ||
|| 8/7 || 271.103 ||
|| 7/5 || 271.137 ||
|| 5/4 || 271.229 ||
|| 1361367/1000000 || 271.326 (7 limit least squares) ||
|| 19\84 || 271.429 ||
|| |0 119 -46 20 -16&gt; || 271.445 (11 limit least squares) ||
|| x^10 + 2x^3 = 8 || 271.508 (equal beating) ||
|| |0 90 -41 14&gt; || 271.561 (9 limit least squares) ||
|| 6/5 || 271.564 (5 limit minimax) ||
|| 1220703125/1033121304 || 271.590 (5 limit least squares) ||
|| 10/9 || 271.623 (9 limit minimax) ||
|| 12\53 || 271.698 ||
|| 4/3 || 271.708 ||
|| 16/15 || 272.067 ||
|| 9/7 || 272.514 ||
|| 5\22 || 272.727 ||
|| 11/10 || 273.001 ||
|| 11/8 || 275.659 ||
|| 18/13 || 281.691 ||


==Spectrum of Blair Tunings by Eigenmonzos==  
{| class="wikitable center-1 right-2"
|-
! rowspan="3" | #
! rowspan="3" | Cents*
! colspan="4" | Approximate ratios
|-
! rowspan="2" | 11-limit
! colspan="3" | 13-limit extensions
|-
! Tridecimal orwell
! Winston
! Blair
|-
| 0
| 0.00
| '''1/1'''
|
|
|
|-
| 1
| 271.46
| 7/6
|
|
| 13/11, 15/13
|-
| 2
| 542.91
| '''11/8''', 15/11
|
| 18/13
| 35/26, 39/28
|-
| 3
| 814.37
| '''8/5'''
|
| 21/13, 52/33
| '''13/8'''
|-
| 4
| 1085.82
| '''15/8''', 28/15
|
| 13/7
| 24/13
|-
| 5
| 157.28
| 12/11, 11/10, 35/32
|
| 13/12
| 14/13
|-
| 6
| 428.73
| 14/11, 9/7, 32/25
|
|
| 13/10, 33/26
|-
| 7
| 700.19
| '''3/2'''
|
| 52/35
|
|-
| 8
| 971.64
| '''7/4'''
|
| 26/15
|
|-
| 9
| 43.10
| 49/48, 36/35, 33/32
| 40/39
| 27/26
| 26/25
|-
| 10
| 314.55
| 6/5
|
| 13/11
| 39/32
|-
| 11
| 586.01
| 7/5
|
| 39/28
| 18/13
|-
| 12
| 857.46
| 18/11
| 64/39
| '''13/8'''
| 21/13
|-
| 13
| 1128.92
| 21/11, 27/14, 48/25
| 25/13
|
| 39/20
|-
| 14
| 200.37
| '''9/8''', 28/25
|
|
|
|-
| 15
| 471.83
| '''21/16'''
|
| 13/10
|
|-
| 16
| 743.28
| 49/32, 54/35
| 20/13
|
|
|-
| 17
| 1014.74
| 9/5
|
|
|
|-
| 18
| 86.19
| 21/20
|
| 26/25
| 27/26
|-
| 19
| 357.65
| 27/22, 49/40
| '''16/13'''
| 39/32
|
|-
| 20
| 629.10
| 36/25
| 56/39
|
|
|-
| 21
| 900.56
| 27/16, 42/25
| 22/13
|
|
|-
| 22
| 1172.01
| 63/32
|
| 39/20
|
|}
<nowiki>*</nowiki> in 11-limit CWE tuning


Gencom: [2 7/6; 65/64 78/77 91/90 99/98]
== Tuning spectra ==
Gencom map: [&lt;1 0 3 1 3 3|, &lt;0 7 -3 8 2 3|]
These spectra suggest possible tuning choices. For 13-limit orwell, the 5-limit minimax tuning featuring pure 5/3 eigenmonzos seems like an excellent choice, as it is right in the middle of the least squares range and very close to 13-limit least squares. Pure 13's, using the 13/8 eigenmonzo, might also please some people. For blair, pure 5/4's using the 5/4 eigenmonzo tuning is very close to 15-odd-limit least squares and in general in the middle of the action. For winston, sticking with the 11/9 eigenmonzo minimax tuning seems reasonable.
||~ Eigenmonzo ||~ Subminor Third ||
|| 15/13 || 247.741 ||
|| 13/12 || 265.357 ||
|| 14/13 || 265.660 ||
|| 7/6 || 266.871 ||
|| 15/11 || 268.475 ||
|| 18/13 || 269.398 ||
|| 14/11 || 269.585 ||
|| 12/11 || 270.127 ||
|| 15/14 || 270.139 ||
|| 7\31 || 270.968 ||
|| 11/9 || 271.049 ||
|| 8/7 || 271.103 ||
|| 7/5 || 271.137 (7, 11, 13 and 15 limit minimax) ||
|| 5/4 || 271.229 ||
|| |0 148 -49 29 -19 -11&gt; || 271.231 (15 limit least squares) ||
|| |0 145 -52 25 -17 -10&gt; || 271.261 (13 limit least squares) ||
|| 1361367/1000000 || 271.326 (7 limit least squares) ||
|| 19\84 || 271.429 ||
|| |0 119 -46 20 -16&gt; || 271.445 (11 limit least squares) ||
|| x^10 + 2x^3 = 8 || 271.508 (equal beating) ||
|| |0 90 -41 14&gt; || 271.561 (9 limit least squares) ||
|| 6/5 || 271.564 (5 limit minimax) ||
|| 1220703125/1033121304 || 271.590 (5 limit least squares) ||
|| 10/9 || 271.623 (9 limit minimax) ||
|| 12\53 || 271.698 ||
|| 4/3 || 271.708 ||
|| 16/15 || 272.067 ||
|| 9/7 || 272.514 ||
|| 5\22 || 272.727 ||
|| 11/10 || 273.001 ||
|| 11/8 || 275.659 ||
|| 13/10 || 275.702 ||
|| 16/13 || 280.176 ||
|| 13/11 || 289.210 ||</pre></div>
<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;Orwell extensions&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;a class="wiki_link" href="/Orwell"&gt;Orwell&lt;/a&gt; temperament has various extensions to the 13 limit.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Tuning Spectra"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Tuning Spectra&lt;/h1&gt;
These spectra suggest possible tuning choices. For 13-limit orwell, the 5-limit minimax tuning featuring pure 6/5 eigenmonzos seems like an excellent choice, as it's right in the middle of the least squares range and very close to 13-limit least squares. Pure 13s, using the 16/13 eigenmonzo, might also please some people. For blair, pure 5/4s using the 5/4 eigenmonzo tuning is very close to 15-limit least squares and in general in the middle of the action. For winston, sticking with the 11/9 eigenmonzo minimax tuning seems reasonable.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Tuning Spectra-Spectrum of Orwell Tunings by Eigenmonzos"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Spectrum of Orwell Tunings by Eigenmonzos&lt;/h2&gt;
&lt;br /&gt;
Gencom: [2 7/6; 99/98 121/120 176/175 275/273]&lt;br /&gt;
Gencom map: [&amp;lt;1 0 3 1 3 8|, &amp;lt;0 7 -3 8 2 -19|]&lt;br /&gt;


=== Tridecimal orwell ===
{| class="wikitable center-all left-4"
|-
! Edo<br>generators
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]
! Generator (¢)
! Comments
|-
|
| 7/6
| 266.871
|
|-
|
| 15/11
| 268.475
|
|-
|
| 11/7
| 269.585
|
|-
|
| 11/6
| 270.127
|
|-
|
| 15/14
| 270.139
|
|-
| 7\31
|
| 270.968
| Lower bound of 9- to 15-odd-limit diamond monotone
|-
|
| 11/9
| 271.049
|
|-
|
| 7/4
| 271.103
|
|-
|
| 7/5
| 271.137
|
|-
|
| 5/4
| 271.229
|
|-
|
| 1361367/1000000
| 271.326
| 7-odd-limit least squares
|-
|
| 13/7
| 271.418
| 13- and 15-odd-limit minimax
|-
| 19\84
|
| 271.429
| 84e val
|-
|
| {{monzo| 0 119 -46 20 -16 }}
| 271.445
| 11-odd-limit least squares
|-
|
| 13/8
| 271.551
|
|-
|
| {{monzo| 0 90 -41 14 }}
| 271.561
| 9-odd-limit least squares
|-
|
| 5/3
| 271.564
| 5-odd-limit minimax
|-
|
| {{monzo| 0 -211 30 -47 -5 142 }}
| 271.567
| 13-odd-limit least squares
|-
|
| {{monzo| 0 -236 5 -51 -3 165 }}
| 271.570
| 15-odd-limit least squares
|-
|
| 1220703125/1033121304
| 271.590
| 5-odd-limit least squares
|-
|
| 13/12
| 271.593
|
|-
|
| 13/10
| 271.612
|
|-
|
| 13/9
| 271.618
|
|-
|
| 9/5
| 271.623
| 9-odd-limit minimax
|-
|
| 15/13
| 271.641
|
|-
| 12\53
|
| 271.698
| Upper bound of 9- to 15-odd-limit diamond monotone
|-
|
| 3/2
| 271.708
|
|-
|
| 13/11
| 271.942
|
|-
|
| 15/8
| 272.067
|
|-
|
| 9/7
| 272.514
|
|-
| 5\22
|
| 272.727
|
|-
|
| 11/10
| 273.001
|
|-
|
| 11/8
| 275.659
|
|}


&lt;table class="wiki_table"&gt;
=== Winston ===
    &lt;tr&gt;
{| class="wikitable center-all left-4"
        &lt;th&gt;Eigenmonzo&lt;br /&gt;
|-
&lt;/th&gt;
! Edo<br>generators
        &lt;th&gt;Subminor Third&lt;br /&gt;
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]
&lt;/th&gt;
! Generator (¢)
    &lt;/tr&gt;
! Comments
    &lt;tr&gt;
|-
        &lt;td&gt;7/6&lt;br /&gt;
|
&lt;/td&gt;
| 7/6
        &lt;td&gt;266.871&lt;br /&gt;
| 266.871
&lt;/td&gt;
|
    &lt;/tr&gt;
|-
    &lt;tr&gt;
|
        &lt;td&gt;15/11&lt;br /&gt;
| 13/12
&lt;/td&gt;
| 267.715
        &lt;td&gt;268.475&lt;br /&gt;
|
&lt;/td&gt;
|-
    &lt;/tr&gt;
|
    &lt;tr&gt;
| 13/7
        &lt;td&gt;14/11&lt;br /&gt;
| 267.925
&lt;/td&gt;
|
        &lt;td&gt;269.585&lt;br /&gt;
|-
&lt;/td&gt;
|
    &lt;/tr&gt;
| 15/11
    &lt;tr&gt;
| 268.475
        &lt;td&gt;12/11&lt;br /&gt;
|
&lt;/td&gt;
|-
        &lt;td&gt;270.127&lt;br /&gt;
|
&lt;/td&gt;
| 13/11
    &lt;/tr&gt;
| 268.921
    &lt;tr&gt;
|
        &lt;td&gt;15/14&lt;br /&gt;
|-
&lt;/td&gt;
|
        &lt;td&gt;270.139&lt;br /&gt;
| 15/13
&lt;/td&gt;
| 269.032
    &lt;/tr&gt;
|
    &lt;tr&gt;
|-
        &lt;td&gt;7\31&lt;br /&gt;
|
&lt;/td&gt;
| 11/7
        &lt;td&gt;270.968&lt;br /&gt;
| 269.585
&lt;/td&gt;
|
    &lt;/tr&gt;
|-
    &lt;tr&gt;
|
        &lt;td&gt;11/9&lt;br /&gt;
| 13/8
&lt;/td&gt;
| 270.044
        &lt;td&gt;271.049&lt;br /&gt;
|
&lt;/td&gt;
|-
    &lt;/tr&gt;
|
    &lt;tr&gt;
| 11/6
        &lt;td&gt;8/7&lt;br /&gt;
| 270.127
&lt;/td&gt;
|
        &lt;td&gt;271.103&lt;br /&gt;
|-
&lt;/td&gt;
|
    &lt;/tr&gt;
| 15/14
    &lt;tr&gt;
| 270.139
        &lt;td&gt;7/5&lt;br /&gt;
|
&lt;/td&gt;
|-
        &lt;td&gt;271.137&lt;br /&gt;
|
&lt;/td&gt;
| 13/10
    &lt;/tr&gt;
| 270.281
    &lt;tr&gt;
|
        &lt;td&gt;5/4&lt;br /&gt;
|-
&lt;/td&gt;
|
        &lt;td&gt;271.229&lt;br /&gt;
| {{monzo| 0 112 -67 20 -28 52 }}
&lt;/td&gt;
| 270.860
    &lt;/tr&gt;
| 15-odd-limit least squares
    &lt;tr&gt;
|-
        &lt;td&gt;1361367/1000000&lt;br /&gt;
|
&lt;/td&gt;
| {{monzo| 0 118 -61 16 -26 44 }}
        &lt;td&gt;271.326 (7 limit least squares)&lt;br /&gt;
| 270.933
&lt;/td&gt;
| 13-odd-limit least squares
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 7\31
        &lt;td&gt;14/13&lt;br /&gt;
|
&lt;/td&gt;
| 270.968
        &lt;td&gt;271.418 (13 and 15 limit minimax)&lt;br /&gt;
| Lower bound of 9- to 15-odd-limit diamond monotone
&lt;/td&gt;
|-
    &lt;/tr&gt;
|
    &lt;tr&gt;
| 11/9
        &lt;td&gt;19\84&lt;br /&gt;
| 271.049
&lt;/td&gt;
| 13- and 15-odd-limit minimax
        &lt;td&gt;271.429&lt;br /&gt;
|-
&lt;/td&gt;
|
    &lt;/tr&gt;
| 7/4
    &lt;tr&gt;
| 271.103
        &lt;td&gt;|0 119 -46 20 -16&amp;gt;&lt;br /&gt;
|
&lt;/td&gt;
|-
        &lt;td&gt;271.445 (11 limit least squares)&lt;br /&gt;
|
&lt;/td&gt;
| 7/5
    &lt;/tr&gt;
| 271.137
    &lt;tr&gt;
|
        &lt;td&gt;x^10 + 2x^3 = 8&lt;br /&gt;
|-
&lt;/td&gt;
|
        &lt;td&gt;271.508 (equal beating)&lt;br /&gt;
| 5/4
&lt;/td&gt;
| 271.229
    &lt;/tr&gt;
|
    &lt;tr&gt;
|-
        &lt;td&gt;16/13&lt;br /&gt;
|
&lt;/td&gt;
| 1361367/1000000
        &lt;td&gt;271.551&lt;br /&gt;
| 271.326
&lt;/td&gt;
| 7-odd-limit least squares
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 19\84
        &lt;td&gt;|0 90 -41 14&amp;gt;&lt;br /&gt;
|
&lt;/td&gt;
| 271.429
        &lt;td&gt;271.561 (9 limit least squares)&lt;br /&gt;
| 84eff val
&lt;/td&gt;
|-
    &lt;/tr&gt;
|
    &lt;tr&gt;
| {{monzo| 0 119 -46 20 -16 }}
        &lt;td&gt;6/5&lt;br /&gt;
| 271.445
&lt;/td&gt;
| 11-odd-limit least squares
        &lt;td&gt;271.564 (5 limit minimax)&lt;br /&gt;
|-
&lt;/td&gt;
|
    &lt;/tr&gt;
| {{monzo| 0 90 -41 14 }}
    &lt;tr&gt;
| 271.561
        &lt;td&gt;|0 -211 30 -47 -5 142&amp;gt;&lt;br /&gt;
| 9-odd-limit least squares
&lt;/td&gt;
|-
        &lt;td&gt;271.567 (13 limit least squares)&lt;br /&gt;
|
&lt;/td&gt;
| 5/3
    &lt;/tr&gt;
| 271.564
    &lt;tr&gt;
| 5-odd-limit minimax
        &lt;td&gt;|0 -236 5 -51 -3 165&amp;gt;&lt;br /&gt;
|-
&lt;/td&gt;
|
        &lt;td&gt;271.570 (15 limit least squares)&lt;br /&gt;
| 1220703125/1033121304
&lt;/td&gt;
| 271.590
    &lt;/tr&gt;
| 5-odd-limit least squares
    &lt;tr&gt;
|-
        &lt;td&gt;1220703125/1033121304&lt;br /&gt;
|
&lt;/td&gt;
| 9/5
        &lt;td&gt;271.590 (5 limit least squares)&lt;br /&gt;
| 271.623
&lt;/td&gt;
| 9-odd-limit minimax
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 12\53
        &lt;td&gt;13/12&lt;br /&gt;
|
&lt;/td&gt;
| 271.698
        &lt;td&gt;271.593&lt;br /&gt;
| 53f val
&lt;/td&gt;
|-
    &lt;/tr&gt;
|
    &lt;tr&gt;
| 3/2
        &lt;td&gt;13/10&lt;br /&gt;
| 271.708
&lt;/td&gt;
|
        &lt;td&gt;271.612&lt;br /&gt;
|-
&lt;/td&gt;
|
    &lt;/tr&gt;
| 15/8
    &lt;tr&gt;
| 272.067
        &lt;td&gt;18/13&lt;br /&gt;
|
&lt;/td&gt;
|-
        &lt;td&gt;271.618&lt;br /&gt;
|
&lt;/td&gt;
| 9/7
    &lt;/tr&gt;
| 272.514
    &lt;tr&gt;
|
        &lt;td&gt;10/9&lt;br /&gt;
|-
&lt;/td&gt;
| 5\22
        &lt;td&gt;271.623 (9 limit minimax)&lt;br /&gt;
|
&lt;/td&gt;
| 272.727
    &lt;/tr&gt;
| 22f val, upper bound of 9- to 15-odd-limit diamond monotone
    &lt;tr&gt;
|-
        &lt;td&gt;15/13&lt;br /&gt;
|
&lt;/td&gt;
| 11/10
        &lt;td&gt;271.641&lt;br /&gt;
| 273.001
&lt;/td&gt;
|
    &lt;/tr&gt;
|-
    &lt;tr&gt;
|
        &lt;td&gt;12\53&lt;br /&gt;
| 11/8
&lt;/td&gt;
| 275.659
        &lt;td&gt;271.698&lt;br /&gt;
|
&lt;/td&gt;
|-
    &lt;/tr&gt;
|
    &lt;tr&gt;
| 13/9
        &lt;td&gt;4/3&lt;br /&gt;
| 281.691
&lt;/td&gt;
|
        &lt;td&gt;271.708&lt;br /&gt;
|}
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;13/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.942&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;16/15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;272.067&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;9/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;272.514&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;5\22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;272.727&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11/10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;273.001&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11/8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;275.659&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
=== Blair ===
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Tuning Spectra-Spectrum of Winston Tunings by Eigenmonzos"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Spectrum of Winston Tunings by Eigenmonzos&lt;/h2&gt;
&lt;br /&gt;
Gencom: [2 7/6; 66/65 99/98 105/104 121/120]&lt;br /&gt;
Gencom map: [&amp;lt;1 0 3 1 3 1|, &amp;lt;0 7 -3 8 2 12|]&lt;br /&gt;


{| class="wikitable center-all left-4"
|-
! Edo<br>generators
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]
! Generator (¢)
! Comments
|-
|
| 15/13
| 247.741
|
|-
|
| 13/12
| 265.357
|
|-
|
| 13/7
| 265.660
|
|-
|
| 7/6
| 266.871
|
|-
|
| 15/11
| 268.475
|
|-
|
| 13/9
| 269.398
|
|-
|
| 11/7
| 269.585
|
|-
|
| 11/6
| 270.127
|
|-
|
| 15/14
| 270.139
|
|-
| 7\31
|
| 270.968
| 31f val
|-
|
| 11/9
| 271.049
|
|-
|
| 7/4
| 271.103
|
|-
|
| 7/5
| 271.137
| 7-, 11-, 13- and 15-odd-limit minimax
|-
|
| 5/4
| 271.229
|
|-
|
| {{monzo| 0 148 -49 29 -19 -11 }}
| 271.231
| 15-odd-limit least squares
|-
|
| {{monzo| 0 145 -52 25 -17 -10 }}
| 271.261
| 13-odd-limit least squares
|-
|
| 1361367/1000000
| 271.326
| 7-odd-limit least squares
|-
| 19\84
|
| 271.429
| 84efff val
|-
|
| {{monzo| 0 119 -46 20 -16 }}
| 271.445
| 11-odd-limit least squares
|-
|
| {{monzo| 0 90 -41 14 }}
| 271.561
| 9-odd-limit least squares
|-
|
| 5/3
| 271.564
| 5-odd-limit minimax
|-
|
| 1220703125/1033121304
| 271.590
| 5-odd-limit least squares
|-
|
| 9/5
| 271.623
| 9-odd-limit minimax
|-
| 12\53
|
| 271.698
| 53ff val
|-
|
| 3/2
| 271.708
|
|-
|
| 15/8
| 272.067
|
|-
|
| 9/7
| 272.514
|
|-
| 5\22
|
| 272.727
|
|-
|
| 11/10
| 273.001
|
|-
|
| 11/8
| 275.659
|
|-
|
| 13/10
| 275.702
|
|-
|
| 13/8
| 280.176
|
|-
|
| 13/11
| 289.210
|
|}


&lt;table class="wiki_table"&gt;
[[Category:Orwell]]
    &lt;tr&gt;
[[Category:Temperament extensions]]
        &lt;th&gt;Eigenmonzo&lt;br /&gt;
[[Category:Rank-2 temperaments]]
&lt;/th&gt;
        &lt;th&gt;Subminor Third&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7/6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;266.871&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;13/12&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;267.715&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;14/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;267.925&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;15/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;268.475&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;13/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;268.921&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;15/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;269.032&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;14/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;269.585&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;16/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;270.044&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;12/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;270.127&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;15/14&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;270.139&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;13/10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;270.281&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;|0 112 -67 20 -28 52&amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;270.860 (15 limit least squares)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;|0 118 -61 16 -26 44&amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;270.933 (13 limit least squares)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7\31&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;270.968&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.049 (13 and 15 limit minimax)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;8/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.103&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.137&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;5/4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.229&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1361367/1000000&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.326 (7 limit least squares)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;19\84&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.429&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;|0 119 -46 20 -16&amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.445 (11 limit least squares)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;x^10 + 2x^3 = 8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.508 (equal beating)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;|0 90 -41 14&amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.561 (9 limit least squares)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;6/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.564 (5 limit minimax)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1220703125/1033121304&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.590 (5 limit least squares)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;10/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.623 (9 limit minimax)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;12\53&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.698&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.708&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;16/15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;272.067&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;9/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;272.514&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;5\22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;272.727&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11/10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;273.001&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11/8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;275.659&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;18/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;281.691&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Tuning Spectra-Spectrum of Blair Tunings by Eigenmonzos"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Spectrum of Blair Tunings by Eigenmonzos&lt;/h2&gt;
&lt;br /&gt;
Gencom: [2 7/6; 65/64 78/77 91/90 99/98]&lt;br /&gt;
Gencom map: [&amp;lt;1 0 3 1 3 3|, &amp;lt;0 7 -3 8 2 3|]&lt;br /&gt;
 
 
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;th&gt;Eigenmonzo&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Subminor Third&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;15/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;247.741&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;13/12&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;265.357&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;14/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;265.660&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7/6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;266.871&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;15/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;268.475&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;18/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;269.398&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;14/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;269.585&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;12/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;270.127&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;15/14&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;270.139&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7\31&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;270.968&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.049&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;8/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.103&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.137 (7, 11, 13 and 15 limit minimax)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;5/4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.229&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;|0 148 -49 29 -19 -11&amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.231 (15 limit least squares)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;|0 145 -52 25 -17 -10&amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.261 (13 limit least squares)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1361367/1000000&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.326 (7 limit least squares)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;19\84&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.429&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;|0 119 -46 20 -16&amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.445 (11 limit least squares)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;x^10 + 2x^3 = 8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.508 (equal beating)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;|0 90 -41 14&amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.561 (9 limit least squares)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;6/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.564 (5 limit minimax)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1220703125/1033121304&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.590 (5 limit least squares)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;10/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.623 (9 limit minimax)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;12\53&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.698&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.708&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;16/15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;272.067&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;9/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;272.514&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;5\22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;272.727&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11/10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;273.001&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11/8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;275.659&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;13/10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;275.702&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;16/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;280.176&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;13/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;289.210&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;/body&gt;&lt;/html&gt;</pre></div>

Latest revision as of 13:20, 2 March 2026

< Orwell

Orwell has multiple competing extensions to the 13-limit. This is evidenced by the fact that its supporting equal temperaments, 22 and 31, do less well in the 13-limit. The extensions are:

  • Tridecimal orwell (22 & 31) – tempering out 99/98, 121/120, 176/175, and 275/273
  • Blair (22 & 31f) – tempering out 65/64, 78/77, 91/90, and 99/98
  • Winston (22f & 31) – tempering out 66/65, 99/98, 105/104, and 121/120

The most important of these is tridecimal orwell, which tempers out 352/351 and may also be characterized by tempering out 275/273 instead. Supported by 53, it has the highest accuracy in its approximation of 13/8, but also the highest complexity. The other two extensions have lower complexity, but also lower accuracy. In winston, ~13/8 is conflated with ~18/11 and is generally tuned worse than in 31edo as a result of an improved ~18/11. In blair, ~13/8 is conflated with ~8/5 and is generally tuned worse than in 22edo as a result of an improved ~8/5.

Another possible path which relates a sense of compromise is to temper out 169/168, leading to doublethink. This has the effect of slicing the generator in two, and is supported by 44, 53, and 62.

See Semicomma family #Orwell, #Blair, and #Winston for technical data.

Interval chain

Odd harmonics 1–21 and their inverses are in bold.

# Cents* Approximate ratios
11-limit 13-limit extensions
Tridecimal orwell Winston Blair
0 0.00 1/1
1 271.46 7/6 13/11, 15/13
2 542.91 11/8, 15/11 18/13 35/26, 39/28
3 814.37 8/5 21/13, 52/33 13/8
4 1085.82 15/8, 28/15 13/7 24/13
5 157.28 12/11, 11/10, 35/32 13/12 14/13
6 428.73 14/11, 9/7, 32/25 13/10, 33/26
7 700.19 3/2 52/35
8 971.64 7/4 26/15
9 43.10 49/48, 36/35, 33/32 40/39 27/26 26/25
10 314.55 6/5 13/11 39/32
11 586.01 7/5 39/28 18/13
12 857.46 18/11 64/39 13/8 21/13
13 1128.92 21/11, 27/14, 48/25 25/13 39/20
14 200.37 9/8, 28/25
15 471.83 21/16 13/10
16 743.28 49/32, 54/35 20/13
17 1014.74 9/5
18 86.19 21/20 26/25 27/26
19 357.65 27/22, 49/40 16/13 39/32
20 629.10 36/25 56/39
21 900.56 27/16, 42/25 22/13
22 1172.01 63/32 39/20

* in 11-limit CWE tuning

Tuning spectra

These spectra suggest possible tuning choices. For 13-limit orwell, the 5-limit minimax tuning featuring pure 5/3 eigenmonzos seems like an excellent choice, as it is right in the middle of the least squares range and very close to 13-limit least squares. Pure 13's, using the 13/8 eigenmonzo, might also please some people. For blair, pure 5/4's using the 5/4 eigenmonzo tuning is very close to 15-odd-limit least squares and in general in the middle of the action. For winston, sticking with the 11/9 eigenmonzo minimax tuning seems reasonable.

Tridecimal orwell

Edo
generators 		Unchanged interval
(eigenmonzo) 		Generator (¢) Comments
7/6 266.871
15/11 268.475
11/7 269.585
11/6 270.127
15/14 270.139
7\31 270.968 Lower bound of 9- to 15-odd-limit diamond monotone
11/9 271.049
7/4 271.103
7/5 271.137
5/4 271.229
1361367/1000000 271.326 7-odd-limit least squares
13/7 271.418 13- and 15-odd-limit minimax
19\84 271.429 84e val
[0 119 -46 20 -16⟩ 271.445 11-odd-limit least squares
13/8 271.551
[0 90 -41 14⟩ 271.561 9-odd-limit least squares
5/3 271.564 5-odd-limit minimax
[0 -211 30 -47 -5 142⟩ 271.567 13-odd-limit least squares
[0 -236 5 -51 -3 165⟩ 271.570 15-odd-limit least squares
1220703125/1033121304 271.590 5-odd-limit least squares
13/12 271.593
13/10 271.612
13/9 271.618
9/5 271.623 9-odd-limit minimax
15/13 271.641
12\53 271.698 Upper bound of 9- to 15-odd-limit diamond monotone
3/2 271.708
13/11 271.942
15/8 272.067
9/7 272.514
5\22 272.727
11/10 273.001
11/8 275.659

Winston

Edo
generators 		Unchanged interval
(eigenmonzo) 		Generator (¢) Comments
7/6 266.871
13/12 267.715
13/7 267.925
15/11 268.475
13/11 268.921
15/13 269.032
11/7 269.585
13/8 270.044
11/6 270.127
15/14 270.139
13/10 270.281
[0 112 -67 20 -28 52⟩ 270.860 15-odd-limit least squares
[0 118 -61 16 -26 44⟩ 270.933 13-odd-limit least squares
7\31 270.968 Lower bound of 9- to 15-odd-limit diamond monotone
11/9 271.049 13- and 15-odd-limit minimax
7/4 271.103
7/5 271.137
5/4 271.229
1361367/1000000 271.326 7-odd-limit least squares
19\84 271.429 84eff val
[0 119 -46 20 -16⟩ 271.445 11-odd-limit least squares
[0 90 -41 14⟩ 271.561 9-odd-limit least squares
5/3 271.564 5-odd-limit minimax
1220703125/1033121304 271.590 5-odd-limit least squares
9/5 271.623 9-odd-limit minimax
12\53 271.698 53f val
3/2 271.708
15/8 272.067
9/7 272.514
5\22 272.727 22f val, upper bound of 9- to 15-odd-limit diamond monotone
11/10 273.001
11/8 275.659
13/9 281.691

Blair

Edo
generators 		Unchanged interval
(eigenmonzo) 		Generator (¢) Comments
15/13 247.741
13/12 265.357
13/7 265.660
7/6 266.871
15/11 268.475
13/9 269.398
11/7 269.585
11/6 270.127
15/14 270.139
7\31 270.968 31f val
11/9 271.049
7/4 271.103
7/5 271.137 7-, 11-, 13- and 15-odd-limit minimax
5/4 271.229
[0 148 -49 29 -19 -11⟩ 271.231 15-odd-limit least squares
[0 145 -52 25 -17 -10⟩ 271.261 13-odd-limit least squares
1361367/1000000 271.326 7-odd-limit least squares
19\84 271.429 84efff val
[0 119 -46 20 -16⟩ 271.445 11-odd-limit least squares
[0 90 -41 14⟩ 271.561 9-odd-limit least squares
5/3 271.564 5-odd-limit minimax
1220703125/1033121304 271.590 5-odd-limit least squares
9/5 271.623 9-odd-limit minimax
12\53 271.698 53ff val
3/2 271.708
15/8 272.067
9/7 272.514
5\22 272.727
11/10 273.001
11/8 275.659
13/10 275.702
13/8 280.176
13/11 289.210
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