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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Breadcrumb|Negri}}
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>2012-11-06 19:28:43 UTC</tt>.<br>
: The original revision id was <tt>379878890</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">[[Negri]] temperament has various extensions to the 11 limit.


=Tuning Spectra=
[[Negri]] has various competing [[extension]]s to the [[11-limit]]. This is evidenced by the fact that its [[support]]ing equal temperaments, [[10edo|10]] and [[19edo|19]], do less well in the 11-limit. The extensions are:
==Spectrum of Negri Tunings by Eigenmonzos==
* '''Undecimal negri''' (10 & 19) – tempering out 45/44, 49/48, and 56/55
* '''Negril''' (10e & 19) – tempering out 49/48, 100/99, and 225/224
* '''Negric''' (10e & 19e) – tempering out 33/32, 49/48, and 77/75
* '''Negroni''' (10 & 19e) – tempering out 49/48, 55/54, and 225/224


Gencom: [2 15/14; 45/44 49/48 56/55 78/77]
The most important of these is undecimal negri, in which the generator, representing 13/12, 14/13, 15/14, and 16/15, goes one step further to stand in for ~12/11. The poor accuracy comes with a not overly complex mapping, as you can find ~16/11 just five generator steps away. Its evil cousin, negric, finds ~11/8 where negri will find ~15/11, which is of course conflated with [[4/3]].  
Gencom map: [&lt;1 2 2 3 4 4|, &lt;0 -4 3 -2 -5 -3|]
||~ Eigenmonzo ||~ Supersecor ||
|| 16/15 || 111.731 ||
|| 8/7 || 115.587 ||
|| 11/9 || 115.803 ||
|| 15/14 || 119.443 ||
|| 16/13 || 119.824 ||
|| 7/5 || 123.498 ||
|| 15/13 || 123.871 ||
|| 13/10 || 124.298 ||
|| 4/3 || 124.511 ||
|| 10/9 || 125.673 ||
|| 6/5 || 126.337 ||
|| 18/13 || 127.324 ||
|| 9/7 || 127.486 ||
|| 14/13 || 128.298 ||
|| 5/4 || 128.771 ||
|| 11/10 || 129.374 ||
|| 11/8 || 129.736 ||
|| 7/6 || 133.435 ||
|| 15/11 || 134.238 ||
|| 13/12 || 138.573 ||
|| 14/11 || 139.169 ||
|| 13/11 || 144.605 ||
|| 12/11 || 150.637 ||


==Spectrum of Negril Tunings by Eigenmonzos==
The other pair of extensions are of higher complexity, but are well rewarded with better intonation. Negril finds ~11/8 where negroni will find ~15/11 and vice versa. They unite in [[29edo]], which can be recommended as a tuning for both.


Gencom: [2 15/14; 49/48 65/64 91/90 875/858]
Another possible path which relates a sense of compromise is to temper out [[121/120]], leading to [[wilsec]]. This has the effect of slicing the generator in two, and is supported by [[20edo|20]], [[29edo|29]], and [[38edo|38df]].  
Gencom map: [&lt;1 2 2 3 2 4|, &lt;0 -4 3 -2 14 -3|]
||~ Eigenmonzo ||~ Supersecor ||
|| 16/15 || 111.731 ||
|| 8/7 || 115.587 ||
|| 15/14 || 119.443 ||
|| 16/13 || 119.824 ||
|| 7/5 || 123.498 ||
|| 15/13 || 123.871 ||
|| 14/11 || 123.906 ||
|| 11/10 || 124.091 ||
|| 13/11 || 124.164 ||
|| 15/11 || 124.203 ||
|| 13/10 || 124.298 ||
|| 4/3 || 124.511 ||
|| 11/9 || 124.882 ||
|| 12/11 || 124.965 ||
|| 11/8 || 125.094 ||
|| 10/9 || 125.673 ||
|| 6/5 || 126.337 ||
|| 18/13 || 127.324 ||
|| 9/7 || 127.486 ||
|| 14/13 || 128.298 ||
|| 5/4 || 128.771 ||
|| 7/6 || 133.435 ||
|| 13/12 || 138.573 ||


== Interval chain ==
In the following table, odd harmonics and subharmonics 1–13 are in '''bold'''.


</pre></div>
{| class="wikitable center-1 right-2"
<h4>Original HTML content:</h4>
! rowspan="3" | #
<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;Negri extensions&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;a class="wiki_link" href="/Negri"&gt;Negri&lt;/a&gt; temperament has various extensions to the 11 limit.&lt;br /&gt;
! rowspan="3" | Cents*
&lt;br /&gt;
! colspan="5" | Approximate ratios
&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;
|-
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Tuning Spectra-Spectrum of Negri Tunings by Eigenmonzos"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Spectrum of Negri Tunings by Eigenmonzos&lt;/h2&gt;
! rowspan="2" | 2.3.5.7.13 subgroup
&lt;br /&gt;
! colspan="4" | Full 13-limit extensions
Gencom: [2 15/14; 45/44 49/48 56/55 78/77]&lt;br /&gt;
|-
Gencom map: [&amp;lt;1 2 2 3 4 4|, &amp;lt;0 -4 3 -2 -5 -3|]&lt;br /&gt;
! Negri
! Negril
! Negric
! Negroni
|-
| 0
| 0.0
| '''1/1'''
|
|
|
|
|-
| 1
| 125.4
| 13/12, 14/13, 15/14, 16/15
| 12/11
|
| 11/10
|
|-
| 2
| 250.7
| 7/6, '''8/7''', 15/13
| 13/11
|
|
|
|-
| 3
| 376.1
| '''5/4''', '''16/13'''
| 11/9, 14/11
|
|
|
|-
| 4
| 501.4
| '''4/3'''
| 15/11
|
| '''11/8'''
|
|-
| 5
| 626.8
| 10/7, 13/9
| '''16/11'''
|
| 22/15
|
|-
| 6
| 752.1
| 14/9, 20/13, 32/21
|
|
| 11/7
|
|-
| 7
| 877.5
| 5/3
|
|
| 22/13
| 18/11
|-
| 8
| 1002.8
| '''16/9'''
| 20/11
|
| 11/6
|
|-
| 9
| 1128.2
| 35/18, 40/21, 52/27
| 64/33
|
| 88/45
|
|-
| 10
| 53.5
| 25/24, 28/27, 50/49, 64/63
|
| 33/32
|
| 45/44
|-
| 11
| 178.9
| 10/9
|
| 11/10
|
| 12/11
|-
| 12
| 304.3
| 25/21
|
|
| 11/9
| 13/11
|-
| 13
| 429.6
| 35/27
|
|
|
| 14/11
|-
| 14
| 555.0
| 25/18
|
| '''11/8'''
|
| 15/11
|-
| 15
| 680.3
| 40/27
|
| 22/15
|
| '''16/11'''
|-
| 16
| 805.7
| 100/63
|
| 11/7
|
|
|-
| 17
| 931.0
| 140/81
|
| 22/13
|
|
|-
| 18
| 1056.4
| 50/27
|
| 11/6
|
|
|-
| 19
| 1181.7
| 125/63, 160/81
|
| 88/45
|
| 64/33
|}
<nowiki/>* In 2.3.5.7.13-subgroup [[CWE tuning]]


== Tuning spectra ==
=== Undecimal negri ===


&lt;table class="wiki_table"&gt;
{| class="wikitable center-all left-4"
    &lt;tr&gt;
|-
        &lt;th&gt;Eigenmonzo&lt;br /&gt;
! Edo<br>generator
&lt;/th&gt;
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]
        &lt;th&gt;Supersecor&lt;br /&gt;
! Generator (¢)
&lt;/th&gt;
! Comments
    &lt;/tr&gt;
|-
    &lt;tr&gt;
|
        &lt;td&gt;16/15&lt;br /&gt;
| 15/8
&lt;/td&gt;
| 111.731
        &lt;td&gt;111.731&lt;br /&gt;
|
&lt;/td&gt;
|-
    &lt;/tr&gt;
|
    &lt;tr&gt;
| 7/4
        &lt;td&gt;8/7&lt;br /&gt;
| 115.587
&lt;/td&gt;
|
        &lt;td&gt;115.587&lt;br /&gt;
|-
&lt;/td&gt;
|
    &lt;/tr&gt;
| 11/9
    &lt;tr&gt;
| 115.803
        &lt;td&gt;11/9&lt;br /&gt;
|
&lt;/td&gt;
|-
        &lt;td&gt;115.803&lt;br /&gt;
|
&lt;/td&gt;
| 15/14
    &lt;/tr&gt;
| 119.443
    &lt;tr&gt;
|
        &lt;td&gt;15/14&lt;br /&gt;
|-
&lt;/td&gt;
|
        &lt;td&gt;119.443&lt;br /&gt;
| 13/8
&lt;/td&gt;
| 119.824
    &lt;/tr&gt;
|
    &lt;tr&gt;
|-
        &lt;td&gt;16/13&lt;br /&gt;
| 1\10
&lt;/td&gt;
|
        &lt;td&gt;119.824&lt;br /&gt;
| 120.000
&lt;/td&gt;
|
    &lt;/tr&gt;
|-
    &lt;tr&gt;
|
        &lt;td&gt;7/5&lt;br /&gt;
| 7/5
&lt;/td&gt;
| 123.498
        &lt;td&gt;123.498&lt;br /&gt;
|
&lt;/td&gt;
|-
    &lt;/tr&gt;
|
    &lt;tr&gt;
| 15/13
        &lt;td&gt;15/13&lt;br /&gt;
| 123.871
&lt;/td&gt;
|
        &lt;td&gt;123.871&lt;br /&gt;
|-
&lt;/td&gt;
| 3\29
    &lt;/tr&gt;
|
    &lt;tr&gt;
| 124.138
        &lt;td&gt;13/10&lt;br /&gt;
|
&lt;/td&gt;
|-
        &lt;td&gt;124.298&lt;br /&gt;
|
&lt;/td&gt;
| 13/10
    &lt;/tr&gt;
| 124.298
    &lt;tr&gt;
|
        &lt;td&gt;4/3&lt;br /&gt;
|-
&lt;/td&gt;
|
        &lt;td&gt;124.511&lt;br /&gt;
| 3/2
&lt;/td&gt;
| 124.511
    &lt;/tr&gt;
| 7- and 9-odd-limit minimax
    &lt;tr&gt;
|-
        &lt;td&gt;10/9&lt;br /&gt;
| 5\48
&lt;/td&gt;
|
        &lt;td&gt;125.673&lt;br /&gt;
| 125.000
&lt;/td&gt;
|
    &lt;/tr&gt;
|-
    &lt;tr&gt;
|
        &lt;td&gt;6/5&lt;br /&gt;
| {{monzo| 0 -13 15 -5 }}
&lt;/td&gt;
| 125.469
        &lt;td&gt;126.337&lt;br /&gt;
| 7-odd-limit least squares
&lt;/td&gt;
|-
    &lt;/tr&gt;
|
    &lt;tr&gt;
| {{monzo| 0 63 -26 -1 }}
        &lt;td&gt;18/13&lt;br /&gt;
| 125.579
&lt;/td&gt;
| 9-odd-limit least squares
        &lt;td&gt;127.324&lt;br /&gt;
|-
&lt;/td&gt;
|
    &lt;/tr&gt;
| 9/5
    &lt;tr&gt;
| 125.673
        &lt;td&gt;9/7&lt;br /&gt;
|
&lt;/td&gt;
|-
        &lt;td&gt;127.486&lt;br /&gt;
|
&lt;/td&gt;
| {{monzo| 6 11 -10 }}
    &lt;/tr&gt;
| 126.238
    &lt;tr&gt;
| 5-odd-limit least squares
        &lt;td&gt;14/13&lt;br /&gt;
|-
&lt;/td&gt;
| 2\19
        &lt;td&gt;128.298&lt;br /&gt;
|
&lt;/td&gt;
| 126.316
    &lt;/tr&gt;
|
    &lt;tr&gt;
|-
        &lt;td&gt;5/4&lt;br /&gt;
|
&lt;/td&gt;
| 5/3
        &lt;td&gt;128.771&lt;br /&gt;
| 126.337
&lt;/td&gt;
| 5-odd-limit minimax
    &lt;/tr&gt;
|-
    &lt;tr&gt;
|
        &lt;td&gt;11/10&lt;br /&gt;
| {{monzo| 0 79 -40 -5 16 2 }}
&lt;/td&gt;
| 126.445
        &lt;td&gt;129.374&lt;br /&gt;
| 13-odd-limit least squares
&lt;/td&gt;
|-
    &lt;/tr&gt;
|
    &lt;tr&gt;
| {{monzo| 0 34 -17 -2 7 }}
        &lt;td&gt;11/8&lt;br /&gt;
| 126.511
&lt;/td&gt;
| 11-odd-limit least squares
        &lt;td&gt;129.736&lt;br /&gt;
|-
&lt;/td&gt;
|
    &lt;/tr&gt;
| {{monzo| 0 73 -46 -4 20 4 }}
    &lt;tr&gt;
| 126.619
        &lt;td&gt;7/6&lt;br /&gt;
| 15-odd-limit least squares
&lt;/td&gt;
|-
        &lt;td&gt;133.435&lt;br /&gt;
|
&lt;/td&gt;
| 13/9
    &lt;/tr&gt;
| 127.324
    &lt;tr&gt;
|
        &lt;td&gt;15/11&lt;br /&gt;
|-
&lt;/td&gt;
|
        &lt;td&gt;134.238&lt;br /&gt;
| 9/7
&lt;/td&gt;
| 127.486
    &lt;/tr&gt;
| 11-, 13- and 15-odd-limit minimax
    &lt;tr&gt;
|-
        &lt;td&gt;13/12&lt;br /&gt;
| 5\47
&lt;/td&gt;
|
        &lt;td&gt;138.573&lt;br /&gt;
| 127.660
&lt;/td&gt;
|
    &lt;/tr&gt;
|-
    &lt;tr&gt;
|
        &lt;td&gt;14/11&lt;br /&gt;
| 13/7
&lt;/td&gt;
| 128.298
        &lt;td&gt;139.169&lt;br /&gt;
|
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 3\28
    &lt;tr&gt;
|
        &lt;td&gt;13/11&lt;br /&gt;
| 128.571
&lt;/td&gt;
|
        &lt;td&gt;144.605&lt;br /&gt;
|-
&lt;/td&gt;
|
    &lt;/tr&gt;
| 5/4
    &lt;tr&gt;
| 128.771
        &lt;td&gt;12/11&lt;br /&gt;
|
&lt;/td&gt;
|-
        &lt;td&gt;150.637&lt;br /&gt;
|
&lt;/td&gt;
| 11/10
    &lt;/tr&gt;
| 129.374
&lt;/table&gt;
|
|-
|
| 11/8
| 129.736
|
|-
| 1\9
|
| 133.333
|
|-
|
| 7/6
| 133.435
|
|-
|
| 15/11
| 134.238
|
|-
|
| 13/12
| 138.573
|
|-
|
| 11/7
| 139.169
|
|-
|
| 13/11
| 144.605
|
|-
|
| 11/6
| 150.637
|
|}


&lt;br /&gt;
=== Negril ===
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Tuning Spectra-Spectrum of Negril Tunings by Eigenmonzos"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Spectrum of Negril Tunings by Eigenmonzos&lt;/h2&gt;
&lt;br /&gt;
Gencom: [2 15/14; 49/48 65/64 91/90 875/858]&lt;br /&gt;
Gencom map: [&amp;lt;1 2 2 3 2 4|, &amp;lt;0 -4 3 -2 14 -3|]&lt;br /&gt;


{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! Unchanged interval<br>(eigenmonzo)
! Generator (¢)
! Comments
|-
|
| 15/8
| 111.731
|
|-
|
| 7/4
| 115.587
|
|-
|
| 15/14
| 119.443
|
|-
|
| 13/8
| 119.824
|
|-
| 1\10
|
| 120.000
|
|-
|
| 7/5
| 123.498
|
|-
|
| 15/13
| 123.871
|
|-
|
| 11/7
| 123.906
|
|-
|
| 11/10
| 124.091
|
|-
| 3\29
|
| 124.138
|
|-
|
| 13/11
| 124.164
|
|-
|
| 15/11
| 124.203
|
|-
|
| 13/10
| 124.298
|
|-
|
| 3/2
| 124.511
| 7-, 9-, 11-, 13- and 15-odd-limit minimax
|-
|
| {{monzo| 0 -149 8 -15 113 -23 }}
| 124.700
| 15-odd-limit least squares
|-
|
| {{monzo| 0 -136 21 -14 98 -21 }}
| 124.764
| 13-odd-limit least squares
|-
|
| {{monzo| 0 -125 15 -15 81 }}
| 124.854
| 11-odd-limit least squares
|-
|
| 11/9
| 124.882
|
|-
|
| 11/6
| 124.965
|
|-
| 5\48
|
| 125.000
|
|-
|
| 11/8
| 125.094
|
|-
|
| {{monzo| 0 -13 15 -5 }}
| 125.469
| 7-odd-limit least squares
|-
|
| {{monzo| 0 63 -26 -1 }}
| 125.579
| 9-odd-limit least squares
|-
|
| 9/5
| 125.673
|
|-
|
| {{monzo| 6 11 -10 }}
| 126.238
| 5-odd-limit least squares
|-
| 2\19
|
| 126.316
|
|-
|
| 5/3
| 126.337
| 5-odd-limit minimax
|-
|
| 13/9
| 127.324
|
|-
|
| 9/7
| 127.486
|
|-
| 5\47
|
| 127.660
|
|-
|
| 13/7
| 128.298
|
|-
| 3\28
|
| 128.571
|
|-
|
| 5/4
| 128.771
|
|-
| 1\9
|
| 133.333
|
|-
|
| 7/6
| 133.435
|
|-
|
| 13/12
| 138.573
|
|}


&lt;table class="wiki_table"&gt;
=== Negric ===
    &lt;tr&gt;
        &lt;th&gt;Eigenmonzo&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Supersecor&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;16/15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;111.731&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;115.587&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;119.443&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;119.824&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;123.498&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;123.871&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;123.906&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;124.091&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;124.164&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;124.203&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;124.298&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;124.511&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;124.882&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;124.965&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;125.094&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;125.673&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;126.337&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;127.324&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;127.486&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;128.298&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;128.771&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;133.435&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;138.573&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;/body&gt;&lt;/html&gt;</pre></div>
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! Unchanged interval<br>(eigenmonzo)
! Generator (¢)
! Comments
|-
|
| 15/8
| 111.731
|
|-
|
| 7/4
| 115.587
|
|-
|
| 15/14
| 119.443
|
|-
|
| 13/8
| 119.824
|
|-
| 1\10
|
| 120.000
|
|-
|
| 7/5
| 123.498
|
|-
|
| 15/13
| 123.871
|
|-
| 3\29
|
| 124.138
|
|-
|
| 13/10
| 124.298
|
|-
|
| 3/2
| 124.511
| 7- and 9-odd-limit minimax
|-
| 5\48
|
| 125.000
|
|-
|
| {{monzo| 0 -13 15 -5 }}
| 125.469
| 7-odd-limit least squares
|-
|
| {{monzo| 0 63 -26 -1 }}
| 125.579
| 9-odd-limit least squares
|-
|
| 9/5
| 125.673
|
|-
|
| {{monzo| 6 11 -10 }}
| 126.238
| 5-odd-limit least squares
|-
| 2\19
|
| 126.316
|
|-
|
| 5/3
| 126.337
| 5-odd-limit minimax
|-
|
| 13/9
| 127.324
|
|-
|
| 9/7
| 127.486
|
|-
|
| {{monzo| 0 106 -31 4 -38 11 }}
| 127.602
| 13-odd-limit least squares
|-
| 5\47
|
| 127.660
|
|-
|
| {{monzo| 0 -95 25 -5 31 }}
| 127.706
| 11 limit least squares
|-
|
| {{monzo| 0 109 -28 5 -43 13 }}
| 127.718
| 15-odd-limit least squares
|-
|
| 13/7
| 128.298
|
|-
| 3\28
|
| 128.571
|
|-
|
| 5/4
| 128.771
|
|-
|
| 11/9
| 128.951
| 11-, 13- and 15-odd-limit minimax
|-
|
| 13/11
| 130.113
|
|-
|
| 11/7
| 130.415
|
|-
|
| 11/6
| 131.170
|
|-
|
| 15/11
| 132.610
|
|-
| 1\9
|
| 133.333
|
|-
|
| 7/6
| 133.435
|
|-
|
| 11/8
| 137.829
|
|-
|
| 13/12
| 138.573
|
|-
|
| 11/10
| 165.004
|
|}
 
=== Negroni ===
 
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! Unchanged interval<br>(eigenmonzo)
! Generator (¢)
! Comments
|-
|
| 15/8
| 111.731
|
|-
|
| 7/4
| 115.587
|
|-
|
| 15/14
| 119.443
|
|-
|
| 13/8
| 119.824
|
|-
| 1\10
|
| 120.000
|
|-
|
| 11/9
| 121.799
| 11-, 13- and 15-odd-limit minimax
|-
|
| 11/6
| 122.785
|
|-
|
| 11/8
| 123.245
|
|-
|
| 7/5
| 123.498
|
|-
|
| 15/13
| 123.871
|
|-
|
| 15/11
| 124.068
|
|-
|
| 13/11
| 124.101
|
|-
| 3\29
|
| 124.138
|
|-
|
| 11/10
| 124.166
|
|-
|
| {{monzo| 0 19 -22 -7 32 }}
| 124.180
| 11-odd-limit least squares
|-
|
| {{monzo| 0 33 -66 -14 90 -6 }}
| 124.186
| 15-odd-limit least squares
|-
|
| {{monzo| 0 49 -50 -15 76 -8 }}
| 124.215
| 13-odd-limit least squares
|-
|
| 13/10
| 124.298
|
|-
|
| 11/7
| 124.424
|
|-
|
| 3/2
| 124.511
| 7- and 9-odd-limit minimax
|-
| 5\48
|
| 125.000
|
|-
|
| {{monzo| 0 -13 15 -5 }}
| 125.469
| 7-odd-limit least squares
|-
|
| {{monzo| 0 63 -26 -1 }}
| 125.579
| 9-odd-limit least squares
|-
|
| 9/5
| 125.673
|
|-
|
| {{monzo| 6 11 -10 }}
| 126.238
| 5-odd-limit least squares
|-
|
| 5/3
| 126.337
| 5-odd-limit minimax
|-
|
| 13/9
| 127.324
|
|-
|
| 9/7
| 127.486
|
|-
| 5\47
|
| 127.660
|
|-
|
| 13/7
| 128.298
|
|-
| 3\28
|
| 128.571
|
|-
|
| 5/4
| 128.771
|
|-
| 1\9
|
| 133.333
|
|-
|
| 7/6
| 133.435
|
|-
|
| 13/12
| 138.573
|
|}
 
[[Category:Negri]]
[[Category:Temperament extensions]]
[[Category:Rank-2 temperaments]]

Latest revision as of 11:57, 6 August 2025

< Negri

Negri has various competing extensions to the 11-limit. This is evidenced by the fact that its supporting equal temperaments, 10 and 19, do less well in the 11-limit. The extensions are:

  • Undecimal negri (10 & 19) – tempering out 45/44, 49/48, and 56/55
  • Negril (10e & 19) – tempering out 49/48, 100/99, and 225/224
  • Negric (10e & 19e) – tempering out 33/32, 49/48, and 77/75
  • Negroni (10 & 19e) – tempering out 49/48, 55/54, and 225/224

The most important of these is undecimal negri, in which the generator, representing 13/12, 14/13, 15/14, and 16/15, goes one step further to stand in for ~12/11. The poor accuracy comes with a not overly complex mapping, as you can find ~16/11 just five generator steps away. Its evil cousin, negric, finds ~11/8 where negri will find ~15/11, which is of course conflated with 4/3.

The other pair of extensions are of higher complexity, but are well rewarded with better intonation. Negril finds ~11/8 where negroni will find ~15/11 and vice versa. They unite in 29edo, which can be recommended as a tuning for both.

Another possible path which relates a sense of compromise is to temper out 121/120, leading to wilsec. This has the effect of slicing the generator in two, and is supported by 20, 29, and 38df.

Interval chain

In the following table, odd harmonics and subharmonics 1–13 are in bold.

# Cents* Approximate ratios
2.3.5.7.13 subgroup Full 13-limit extensions
Negri Negril Negric Negroni
0 0.0 1/1
1 125.4 13/12, 14/13, 15/14, 16/15 12/11 11/10
2 250.7 7/6, 8/7, 15/13 13/11
3 376.1 5/4, 16/13 11/9, 14/11
4 501.4 4/3 15/11 11/8
5 626.8 10/7, 13/9 16/11 22/15
6 752.1 14/9, 20/13, 32/21 11/7
7 877.5 5/3 22/13 18/11
8 1002.8 16/9 20/11 11/6
9 1128.2 35/18, 40/21, 52/27 64/33 88/45
10 53.5 25/24, 28/27, 50/49, 64/63 33/32 45/44
11 178.9 10/9 11/10 12/11
12 304.3 25/21 11/9 13/11
13 429.6 35/27 14/11
14 555.0 25/18 11/8 15/11
15 680.3 40/27 22/15 16/11
16 805.7 100/63 11/7
17 931.0 140/81 22/13
18 1056.4 50/27 11/6
19 1181.7 125/63, 160/81 88/45 64/33

* In 2.3.5.7.13-subgroup CWE tuning

Tuning spectra

Undecimal negri

Edo
generator 		Unchanged interval
(eigenmonzo) 		Generator (¢) Comments
15/8 111.731
7/4 115.587
11/9 115.803
15/14 119.443
13/8 119.824
1\10 120.000
7/5 123.498
15/13 123.871
3\29 124.138
13/10 124.298
3/2 124.511 7- and 9-odd-limit minimax
5\48 125.000
[0 -13 15 -5⟩ 125.469 7-odd-limit least squares
[0 63 -26 -1⟩ 125.579 9-odd-limit least squares
9/5 125.673
[6 11 -10⟩ 126.238 5-odd-limit least squares
2\19 126.316
5/3 126.337 5-odd-limit minimax
[0 79 -40 -5 16 2⟩ 126.445 13-odd-limit least squares
[0 34 -17 -2 7⟩ 126.511 11-odd-limit least squares
[0 73 -46 -4 20 4⟩ 126.619 15-odd-limit least squares
13/9 127.324
9/7 127.486 11-, 13- and 15-odd-limit minimax
5\47 127.660
13/7 128.298
3\28 128.571
5/4 128.771
11/10 129.374
11/8 129.736
1\9 133.333
7/6 133.435
15/11 134.238
13/12 138.573
11/7 139.169
13/11 144.605
11/6 150.637

Negril

Edo
generator 		Unchanged interval
(eigenmonzo) 		Generator (¢) Comments
15/8 111.731
7/4 115.587
15/14 119.443
13/8 119.824
1\10 120.000
7/5 123.498
15/13 123.871
11/7 123.906
11/10 124.091
3\29 124.138
13/11 124.164
15/11 124.203
13/10 124.298
3/2 124.511 7-, 9-, 11-, 13- and 15-odd-limit minimax
[0 -149 8 -15 113 -23⟩ 124.700 15-odd-limit least squares
[0 -136 21 -14 98 -21⟩ 124.764 13-odd-limit least squares
[0 -125 15 -15 81⟩ 124.854 11-odd-limit least squares
11/9 124.882
11/6 124.965
5\48 125.000
11/8 125.094
[0 -13 15 -5⟩ 125.469 7-odd-limit least squares
[0 63 -26 -1⟩ 125.579 9-odd-limit least squares
9/5 125.673
[6 11 -10⟩ 126.238 5-odd-limit least squares
2\19 126.316
5/3 126.337 5-odd-limit minimax
13/9 127.324
9/7 127.486
5\47 127.660
13/7 128.298
3\28 128.571
5/4 128.771
1\9 133.333
7/6 133.435
13/12 138.573

Negric

Edo
generator 		Unchanged interval
(eigenmonzo) 		Generator (¢) Comments
15/8 111.731
7/4 115.587
15/14 119.443
13/8 119.824
1\10 120.000
7/5 123.498
15/13 123.871
3\29 124.138
13/10 124.298
3/2 124.511 7- and 9-odd-limit minimax
5\48 125.000
[0 -13 15 -5⟩ 125.469 7-odd-limit least squares
[0 63 -26 -1⟩ 125.579 9-odd-limit least squares
9/5 125.673
[6 11 -10⟩ 126.238 5-odd-limit least squares
2\19 126.316
5/3 126.337 5-odd-limit minimax
13/9 127.324
9/7 127.486
[0 106 -31 4 -38 11⟩ 127.602 13-odd-limit least squares
5\47 127.660
[0 -95 25 -5 31⟩ 127.706 11 limit least squares
[0 109 -28 5 -43 13⟩ 127.718 15-odd-limit least squares
13/7 128.298
3\28 128.571
5/4 128.771
11/9 128.951 11-, 13- and 15-odd-limit minimax
13/11 130.113
11/7 130.415
11/6 131.170
15/11 132.610
1\9 133.333
7/6 133.435
11/8 137.829
13/12 138.573
11/10 165.004

Negroni

Edo
generator 		Unchanged interval
(eigenmonzo) 		Generator (¢) Comments
15/8 111.731
7/4 115.587
15/14 119.443
13/8 119.824
1\10 120.000
11/9 121.799 11-, 13- and 15-odd-limit minimax
11/6 122.785
11/8 123.245
7/5 123.498
15/13 123.871
15/11 124.068
13/11 124.101
3\29 124.138
11/10 124.166
[0 19 -22 -7 32⟩ 124.180 11-odd-limit least squares
[0 33 -66 -14 90 -6⟩ 124.186 15-odd-limit least squares
[0 49 -50 -15 76 -8⟩ 124.215 13-odd-limit least squares
13/10 124.298
11/7 124.424
3/2 124.511 7- and 9-odd-limit minimax
5\48 125.000
[0 -13 15 -5⟩ 125.469 7-odd-limit least squares
[0 63 -26 -1⟩ 125.579 9-odd-limit least squares
9/5 125.673
[6 11 -10⟩ 126.238 5-odd-limit least squares
5/3 126.337 5-odd-limit minimax
13/9 127.324
9/7 127.486
5\47 127.660
13/7 128.298
3\28 128.571
5/4 128.771
1\9 133.333
7/6 133.435
13/12 138.573
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