Orwell extensions: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
[[Orwell|Orwell]] temperament has various extensions to the 13 limit.
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=
=Tuning Spectra=
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.
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.


==Spectrum of Orwell Tunings by Eigenmonzos==  
==Spectrum of Orwell Tunings by Eigenmonzos==


Gencom: [2 7/6; 99/98 121/120 176/175 275/273]
Gencom: [2 7/6; 99/98 121/120 176/175 275/273]
Gencom map: [&lt;1 0 3 1 3 8|, &lt;0 7 -3 8 2 -19|]
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==  
{| class="wikitable"
|-
! | 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==


Gencom: [2 7/6; 66/65 99/98 105/104 121/120]
Gencom: [2 7/6; 66/65 99/98 105/104 121/120]
Gencom map: [&lt;1 0 3 1 3 1|, &lt;0 7 -3 8 2 12|]
Gencom map: [&lt;1 0 3 1 3 1|, &lt;0 7 -3 8 2 12|]
||~ 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"
|-
! | 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==


Gencom: [2 7/6; 65/64 78/77 91/90 99/98]
Gencom: [2 7/6; 65/64 78/77 91/90 99/98]
Gencom map: [&lt;1 0 3 1 3 3|, &lt;0 7 -3 8 2 3|]
Gencom map: [&lt;1 0 3 1 3 3|, &lt;0 7 -3 8 2 3|]
||~ 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;
&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;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;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&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;14/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.418 (13 and 15 limit minimax)&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;16/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.551&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;|0 -211 30 -47 -5 142&amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.567 (13 limit least squares)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &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;
&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;13/12&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.593&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;271.612&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;271.618&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;15/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.641&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;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;
&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;
&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;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>
{| class="wikitable"
|-
! | 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
|}
[[Category:blair]]
[[Category:eigenmonzo]]
[[Category:orwell]]
[[Category:spectrum]]
[[Category:winston]]

Revision as of 00:00, 17 July 2018

Orwell temperament has various extensions to the 13 limit.

Tuning Spectra

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.

Spectrum of Orwell Tunings by Eigenmonzos

Gencom: [2 7/6; 99/98 121/120 176/175 275/273]

Gencom map: [<1 0 3 1 3 8|, <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> 271.445 (11 limit least squares)
x^10 + 2x^3 = 8 271.508 (equal beating)
16/13 271.551
|0 90 -41 14> 271.561 (9 limit least squares)
6/5 271.564 (5 limit minimax)
|0 -211 30 -47 -5 142> 271.567 (13 limit least squares)
|0 -236 5 -51 -3 165> 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

Gencom: [2 7/6; 66/65 99/98 105/104 121/120]

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

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> 270.860 (15 limit least squares)
|0 118 -61 16 -26 44> 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> 271.445 (11 limit least squares)
x^10 + 2x^3 = 8 271.508 (equal beating)
|0 90 -41 14> 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

Gencom: [2 7/6; 65/64 78/77 91/90 99/98]

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

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> 271.231 (15 limit least squares)
|0 145 -52 25 -17 -10> 271.261 (13 limit least squares)
1361367/1000000 271.326 (7 limit least squares)
19\84 271.429
|0 119 -46 20 -16> 271.445 (11 limit least squares)
x^10 + 2x^3 = 8 271.508 (equal beating)
|0 90 -41 14> 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
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