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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Breadcrumb|Miracle}}
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>2014-12-26 18:27:02 UTC</tt>.<br>
: The original revision id was <tt>535923234</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 basic 7-limit [[miracle]] temperament has various extensions to the 11 and 13 limits.


=Tuning Spectra=
The [[11-limit]] [[miracle]] temperament has various [[extension]]s to the [[13-limit]]. The following temperaments are discussed in this article:
==Spectrum of Miraculous Tunings by Eigenmonzos==
* '''Miraculous''' ({{nowrap| 31 & 41 }}) – tempering out 105/104, 144/143, 196/195, and 243/242;
* '''Benediction''' ({{nowrap| 31 & 41f }}) – tempering out 225/224, 243/242, 351/350, and 385/384;
* '''Manna''' ({{nowrap| 31f & 41 }}) – tempering out 225/224, 243/242, 325/324, and 385/384;


Gencom: [2 15/14; 105/104 144/143 196/195 275/273]
In addition, we also consider the only alternative 11-limit mapping:
Gencom map: [&lt;1 1 3 3 2 4|, &lt;0 6 -7 -2 15 -3|]
* '''Revelation''' ({{nowrap| 21 & 31 }}) – tempering out 66/65, 99/98, 105/104, and 512/507.  
||~ Eigenmonzo ||~ Secor ||
|| 16/15 || 111.731 ||
|| 13/10 || 113.553 ||
|| 8/7 || 115.587 ||
|| 11/9 || 115.803 ||
|| 3\31 || 116.129 ||
|| 5/4 || 116.241 ||
|| 15/11 || 116.441 ||
|| 7/5 || 116.502 ||
|| 10\103 || 116.505 ||
|| 17\175 || 116.571 ||
|| |0 -27 25 5&gt; || 116.573 (7 limit least squares) ||
|| |0 -19 20&gt; || 116.578 (5 limit least squares) ||
|| 6/5 || 116.588 (5 and 7 limit minimax) ||
|| 11/10 || 116.591 ||
|| 12/11 || 116.596 ||
|| 14/11 || 116.617 ||
|| POTE11 || 116.633 ||
|| 7/6 || 116.641 ||
|| 7\72 || 116.667 ||
|| |0 17 -11 -6 11&gt; || 116.672 (11 limit least squares) ||
|| POTE5 || 116.673 ||
|| POTE7 || 116.675 ||
|| 10/9 || 116.716 (9, 11 and 13 limit minimax) ||
|| |0 117 -44 -19&gt; || 116.721 (9 limit least squares) ||
|| POTE13 || 116.747 ||
|| 11/8 || 116.755 ||
|| 9/7 || 116.792 ||
|| 11\113 || 116.814 ||
|| |0 127 -84 -36 100 -44&gt; || 116.820 (15 limit least squares) ||
|| |0 141 -70 -35 84 -42&gt; || 116.846 (13 limit least squares) ||
|| 4/3 || 116.993 (15 limit minimax) ||
|| 4\41 || 117.073 ||
|| 13/11 || 117.266 ||
|| 18/13 || 117.559 ||
|| 13/12 || 117.936 ||
|| 15/14 || 119.443 ||
|| 16/13 || 119.824 ||
|| 15/13 || 123.871 ||
|| 14/13 || 128.298 ||


==Spectrum of Benediction Tunings by Eigenmonzos==
As we will see in [[#Interval chain]], miraculous is the only extension whose complexity is at about the same level as the 11-limit. It is [[support]]ed by [[72edo|72f]]. The generator, representing [[15/14]], and [[16/15]], goes one step further to stand in for [[~]][[14/13]], and you can find [[11/9]]~[[16/13]] just three generator steps away. Benediction and manna are available if we want to use the more accurately tuned [[patent val]] mapping of prime [[13/1|13]] in 72edo, in which they merge into one. However, benediction benefits from a flatter tuning such as [[103edo]] whereas manna benefits from a sharper tuning such as [[113edo]].


Gencom: [2 15/14; 225/224 243/242 351/350 441/440]
Another possible path which relates a sense of compromise is to temper out [[169/168]], leading to [[semimiracle]]. This has the effect of slicing the period in two, and is supported by [[62edo|62]], 72, and [[82edo|82]]. Finally, there is [[mirage]], the [[rank-3 temperament|rank-3]] [[expansion]] of miracle with the addition of an independent generator for prime 13.  
Gencom map: [&lt;1 1 3 3 2 7|, &lt;0 6 -7 -2 15 -34|]
||~ Eigenmonzo ||~ Secor ||
|| 16/15 || 111.731 ||
|| 8/7 || 115.587 ||
|| 11/9 || 115.803 ||
|| 3\31 || 116.129 ||
|| 5/4 || 116.241 ||
|| 15/11 || 116.441 ||
|| 16/13 || 116.455 ||
|| 7/5 || 116.502 ||
|| 10\103 || 116.505 ||
|| 14/13 || 116.509 ||
|| 13/10 || 116.511 ||
|| 13/12 || 116.536 ||
|| 13/11 || 116.547 ||
|| |0 -234 39 4 -115 228&gt; || 116.56309 (13 limit least squares) ||
|| |0 -251 22 5 -131 261&gt; || 116.56348 (15 limit least squares) ||
|| 17\175 || 116.571 ||
|| |0 -27 25 5&gt; || 116.573 (7 limit least squares) ||
|| POTE13 || 116.574 ||
|| |0 -19 20&gt; || 116.578 (5 limit least squares) ||
|| 6/5 || 116.588 (5, 7 and 15 limit minimax) ||
|| 11/10 || 116.591 ||
|| 18/13 || 116.595 (13 limit minimax) ||
|| 12/11 || 116.596 ||
|| 15/13 || 116.598 ||
|| 14/11 || 116.617 ||
|| POTE11 || 116.633 ||
|| 7/6 || 116.641 ||
|| 7\72 || 116.667 ||
|| |0 17 -11 -6 11&gt; || 116.672 (11 limit least squares) ||
|| POTE5 || 116.673 ||
|| POTE7 || 116.675 ||
|| 10/9 || 116.716 (9 and 11 limit minimax) ||
|| |0 117 -44 -19&gt; || 116.721 (9 limit least squares) ||
|| 11/8 || 116.755 ||
|| 9/7 || 116.792 ||
|| 11\113 || 116.814 ||
|| 4/3 || 116.993 ||
|| 4\41 || 117.073 ||
|| 15/14 || 119.443 ||


==Spectrum of Manna Tunings by Eigenmonzos==
The 13-limit extensions can all be extended to the [[17-limit]] by recognizing [[21/16]]~[[17/13]], tempering out [[273/272]] (and many other commas such as [[715/714]] and [[833/832]]). For miraculous it implies the generator also represents [[17/16]], which is supported by 72fg. For semimiracle it implies the half-octave period represents [[17/12]]~[[24/17]].


Gencom: [2 15/14; 225/224 243/242 325/324 385/384]
[[225/224]] factors into ([[400/399]])·([[513/512]]) in the [[19-limit]], suggesting that miracle can be extended to include prime 19 by tempering out both commas. However, this means 31edo is no longer in the valid [[diamond monotone]] range. Alternatively, 19 can be reached by tempering out [[324/323]] and [[495/494]], which is called prism. The two merge in manna.
Gencom map: [&lt;1 1 3 3 2 0|, &lt;0 6 -7 -2 15 38|]
||~ Eigenmonzo ||~ Secor ||
|| 16/15 || 111.731 ||
|| 8/7 || 115.587 ||
|| 11/9 || 115.803 ||
|| 3\31 || 116.129 ||
|| 5/4 || 116.241 ||
|| 15/11 || 116.441 ||
|| 7/5 || 116.502 ||
|| 10\103 || 116.505 ||
|| 17\175 || 116.571 ||
|| |0 -27 25 5&gt; || 116.573 (7 limit least squares) ||
|| |0 -19 20&gt; || 116.578 (5 limit least squares) ||
|| 6/5 || 116.588 (5 and 7 limit minimax) ||
|| 11/10 || 116.591 ||
|| 12/11 || 116.596 ||
|| 14/11 || 116.617 ||
|| POTE11 || 116.633 ||
|| 7/6 || 116.641 ||
|| 7\72 || 116.667 ||
|| |0 17 -11 -6 11&gt; || 116.672 (11 limit least squares) ||
|| POTE5 || 116.673 ||
|| POTE7 || 116.675 ||
|| 10/9 || 116.716 (9 and 11 limit minimax) ||
|| |0 117 -44 -19&gt; || 116.721 (9 limit least squares) ||
|| 15/13 || 116.725 (15 limit minimax) ||
|| POTE13 || 116.739 ||
|| 11/8 || 116.755 ||
|| 13/10 || 116.760 (13 limit minimax) ||
|| |0 -37 -166 -77 59 243&gt; || 116.764 (15 limit least squares) ||
|| |0 18 -111 -76 43 204&gt; || 116.780 (13 limit least squares) ||
|| 9/7 || 116.792 ||
|| 14/13 || 116.79254 ||
|| 18/13 || 116.79299 ||
|| 11\113 || 116.814 ||
|| 13/12 || 116.830 ||
|| 16/13 || 116.856 ||
|| 13/11 || 116.922 ||
|| 4/3 || 116.993 ||
|| 4\41 || 117.073 ||
|| 15/14 || 119.443 ||


==Spectrum of Revelation Tunings by Eigenmonzos==
There is also a natural extension to the [[23-limit]] that tempers out [[300/299]] and [[392/391]]. For prism, this also tempers out [[760/759]], which conflates [[23/19]] with [[40/33]].


Gencom: [2 15/14; 66/65 99/98 105/104 1001/1000]
For technical information see [[Gamelismic clan #Miracle]].
Gencom map: [&lt;1 1 3 3 5 4|, &lt;0 6 -7 -2 -16 -3|]
||~ Eigenmonzo ||~ Secor ||
|| 16/15 || 111.731 ||
|| 13/10 || 113.553 ||
|| 13/11 || 114.555 ||
|| 11/10 || 115.000 ||
|| 14/11 || 115.536 ||
|| 11/8 || 115.543 ||
|| 8/7 || 115.587 ||
|| 15/11 || 115.797 ||
|| 12/11 || 115.938 ||
|| 3\31 || 116.129 ||
|| 11/9 || 116.164 (11, 13 and 15 limit minimax) ||
|| |0 -195 35 5 89&gt; || 116.198 (11 limit least squares) ||
|| |0 -251 22 5 117 13&gt; || 116.229 (15 limit least squares) ||
|| 5/4 || 116.241 ||
|| |0 -234 39 4 102 11&gt; || 116.249 (13 limit least squares) ||
|| POTE13 || 116.268 ||
|| POTE11 || 116.277 ||
|| 7/5 || 116.502 ||
|| 10\103 || 116.505 ||
|| 17\175 || 116.571 ||
|| |0 -27 25 5&gt; || 116.573 (7 limit least squares) ||
|| |0 -19 20&gt; || 116.578 (5 limit least squares) ||
|| 6/5 || 116.588 (5 and 7 limit minimax) ||
|| 7/6 || 116.641 ||
|| 7\72 || 116.667 ||
|| POTE5 || 116.673 ||
|| POTE7 || 116.675 ||
|| 10/9 || 116.716 (9 limit minimax) ||
|| |0 117 -44 -19&gt; || 116.721 (9 limit least squares) ||
|| 9/7 || 116.792 ||
|| 11\113 || 116.814 ||
|| 4/3 || 116.993 ||
|| 4\41 || 117.073 ||
|| 18/13 || 117.559 ||
|| 13/12 || 117.936 ||
|| 15/14 || 119.443 ||
|| 16/13 || 119.824 ||
|| 15/13 || 123.871 ||
|| 14/13 || 128.298 ||</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;Miracle extensions&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The basic 7-limit &lt;a class="wiki_link" href="/miracle"&gt;miracle&lt;/a&gt; temperament has various extensions to the 11 and 13 limits.&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;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Tuning Spectra-Spectrum of Miraculous Tunings by Eigenmonzos"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Spectrum of Miraculous Tunings by Eigenmonzos&lt;/h2&gt;
&lt;br /&gt;
Gencom: [2 15/14; 105/104 144/143 196/195 275/273]&lt;br /&gt;
Gencom map: [&amp;lt;1 1 3 3 2 4|, &amp;lt;0 6 -7 -2 15 -3|]&lt;br /&gt;


== Interval chain ==
In the following table, odd harmonics and subharmonics 1–21 are labeled in '''bold'''.


&lt;table class="wiki_table"&gt;
{| class="wikitable center-1 right-2"
    &lt;tr&gt;
|-
        &lt;th&gt;Eigenmonzo&lt;br /&gt;
! rowspan="3" | #
&lt;/th&gt;
! rowspan="3" | Cents*
        &lt;th&gt;Secor&lt;br /&gt;
! colspan="4" | Approximate ratios
&lt;/th&gt;
|-
    &lt;/tr&gt;
! rowspan="2" | 11-limit
    &lt;tr&gt;
! colspan="3" | 17-limit extensions
        &lt;td&gt;16/15&lt;br /&gt;
|-
&lt;/td&gt;
! Miraculous
        &lt;td&gt;111.731&lt;br /&gt;
! Benediction
&lt;/td&gt;
! Manna
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 0
        &lt;td&gt;13/10&lt;br /&gt;
| 0.0
&lt;/td&gt;
| '''1/1'''
        &lt;td&gt;113.553&lt;br /&gt;
|
&lt;/td&gt;
|
    &lt;/tr&gt;
|
    &lt;tr&gt;
|-
        &lt;td&gt;8/7&lt;br /&gt;
| 1
&lt;/td&gt;
| 116.6
        &lt;td&gt;115.587&lt;br /&gt;
| 15/14, '''16/15'''
&lt;/td&gt;
| 14/13, '''17/16'''
    &lt;/tr&gt;
|
    &lt;tr&gt;
|
        &lt;td&gt;11/9&lt;br /&gt;
|-
&lt;/td&gt;
| 2
        &lt;td&gt;115.803&lt;br /&gt;
| 233.3
&lt;/td&gt;
| '''8/7'''
    &lt;/tr&gt;
| 15/13, 17/15
    &lt;tr&gt;
|
        &lt;td&gt;3\31&lt;br /&gt;
|
&lt;/td&gt;
|-
        &lt;td&gt;116.129&lt;br /&gt;
| 3
&lt;/td&gt;
| 349.9
    &lt;/tr&gt;
| 11/9
    &lt;tr&gt;
| '''16/13''', 17/14, 21/17
        &lt;td&gt;5/4&lt;br /&gt;
|
&lt;/td&gt;
|
        &lt;td&gt;116.241&lt;br /&gt;
|-
&lt;/td&gt;
| 4
    &lt;/tr&gt;
| 466.6
    &lt;tr&gt;
| '''21/16'''
        &lt;td&gt;15/11&lt;br /&gt;
| 13/10, 17/13
&lt;/td&gt;
| 17/13
        &lt;td&gt;116.441&lt;br /&gt;
| 17/13
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 5
    &lt;tr&gt;
| 583.2
        &lt;td&gt;7/5&lt;br /&gt;
| 7/5
&lt;/td&gt;
| 24/17
        &lt;td&gt;116.502&lt;br /&gt;
|
&lt;/td&gt;
|
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 6
        &lt;td&gt;10\103&lt;br /&gt;
| 699.9
&lt;/td&gt;
| '''3/2'''
        &lt;td&gt;116.505&lt;br /&gt;
|
&lt;/td&gt;
|
    &lt;/tr&gt;
|
    &lt;tr&gt;
|-
        &lt;td&gt;17\175&lt;br /&gt;
| 7
&lt;/td&gt;
| 816.5
        &lt;td&gt;116.571&lt;br /&gt;
| '''8/5'''
&lt;/td&gt;
| 21/13
    &lt;/tr&gt;
|
    &lt;tr&gt;
|
        &lt;td&gt;|0 -27 25 5&amp;gt;&lt;br /&gt;
|-
&lt;/td&gt;
| 8
        &lt;td&gt;116.573 (7 limit least squares)&lt;br /&gt;
| 933.2
&lt;/td&gt;
| 12/7
    &lt;/tr&gt;
| 17/10
    &lt;tr&gt;
|
        &lt;td&gt;|0 -19 20&amp;gt;&lt;br /&gt;
|
&lt;/td&gt;
|-
        &lt;td&gt;116.578 (5 limit least squares)&lt;br /&gt;
| 9
&lt;/td&gt;
| 1049.8
    &lt;/tr&gt;
| 11/6
    &lt;tr&gt;
| 24/13
        &lt;td&gt;6/5&lt;br /&gt;
|
&lt;/td&gt;
|
        &lt;td&gt;116.588 (5 and 7 limit minimax)&lt;br /&gt;
|-
&lt;/td&gt;
| 10
    &lt;/tr&gt;
| 1166.5
    &lt;tr&gt;
| 49/25, 55/28, 63/32, <br>88/45, 96/49, 108/55
        &lt;td&gt;11/10&lt;br /&gt;
| 39/20, 51/26, 77/39, <br>128/65, 135/68, 168/85
&lt;/td&gt;
| 51/26, 100/51
        &lt;td&gt;116.591&lt;br /&gt;
| 51/26, 65/33
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 11
    &lt;tr&gt;
| 83.1
        &lt;td&gt;12/11&lt;br /&gt;
| 21/20, 22/21
&lt;/td&gt;
| 18/17, 26/25
        &lt;td&gt;116.596&lt;br /&gt;
|
&lt;/td&gt;
|
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 12
        &lt;td&gt;14/11&lt;br /&gt;
| 199.8
&lt;/td&gt;
| '''9/8'''
        &lt;td&gt;116.617&lt;br /&gt;
|
&lt;/td&gt;
|
    &lt;/tr&gt;
|
    &lt;tr&gt;
|-
        &lt;td&gt;POTE11&lt;br /&gt;
| 13
&lt;/td&gt;
| 316.4
        &lt;td&gt;116.633&lt;br /&gt;
| 6/5
&lt;/td&gt;
|
    &lt;/tr&gt;
|
    &lt;tr&gt;
|
        &lt;td&gt;7/6&lt;br /&gt;
|-
&lt;/td&gt;
| 14
        &lt;td&gt;116.641&lt;br /&gt;
| 433.1
&lt;/td&gt;
| 9/7
    &lt;/tr&gt;
| 22/17
    &lt;tr&gt;
|
        &lt;td&gt;7\72&lt;br /&gt;
|
&lt;/td&gt;
|-
        &lt;td&gt;116.667&lt;br /&gt;
| 15
&lt;/td&gt;
| 549.7
    &lt;/tr&gt;
| '''11/8'''
    &lt;tr&gt;
| 18/13
        &lt;td&gt;|0 17 -11 -6 11&amp;gt;&lt;br /&gt;
|
&lt;/td&gt;
|
        &lt;td&gt;116.672 (11 limit least squares)&lt;br /&gt;
|-
&lt;/td&gt;
| 16
    &lt;/tr&gt;
| 666.3
    &lt;tr&gt;
| 22/15
        &lt;td&gt;POTE5&lt;br /&gt;
|
&lt;/td&gt;
|
        &lt;td&gt;116.673&lt;br /&gt;
|
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 17
    &lt;tr&gt;
| 783.0
        &lt;td&gt;POTE7&lt;br /&gt;
| 11/7
&lt;/td&gt;
|
        &lt;td&gt;116.675&lt;br /&gt;
|
&lt;/td&gt;
|
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 18
        &lt;td&gt;10/9&lt;br /&gt;
| 899.6
&lt;/td&gt;
| 27/16, 42/25
        &lt;td&gt;116.716 (9, 11 and 13 limit minimax)&lt;br /&gt;
| 22/13
&lt;/td&gt;
|
    &lt;/tr&gt;
|
    &lt;tr&gt;
|-
        &lt;td&gt;|0 117 -44 -19&amp;gt;&lt;br /&gt;
| 19
&lt;/td&gt;
| 1016.3
        &lt;td&gt;116.721 (9 limit least squares)&lt;br /&gt;
| 9/5
&lt;/td&gt;
|
    &lt;/tr&gt;
|
    &lt;tr&gt;
|
        &lt;td&gt;POTE13&lt;br /&gt;
|-
&lt;/td&gt;
| 20
        &lt;td&gt;116.747&lt;br /&gt;
| 1132.9
&lt;/td&gt;
| 27/14, 48/25
    &lt;/tr&gt;
|
    &lt;tr&gt;
|
        &lt;td&gt;11/8&lt;br /&gt;
| 52/27
&lt;/td&gt;
|-
        &lt;td&gt;116.755&lt;br /&gt;
| 21
&lt;/td&gt;
| 49.6
    &lt;/tr&gt;
| 33/32, 36/35
    &lt;tr&gt;
| 27/26
        &lt;td&gt;9/7&lt;br /&gt;
| 35/34, 40/39
&lt;/td&gt;
| 34/33
        &lt;td&gt;116.792&lt;br /&gt;
|-
&lt;/td&gt;
| 22
    &lt;/tr&gt;
| 166.2
    &lt;tr&gt;
| 11/10
        &lt;td&gt;11\113&lt;br /&gt;
|
&lt;/td&gt;
|
        &lt;td&gt;116.814&lt;br /&gt;
|
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 23
    &lt;tr&gt;
| 282.9
        &lt;td&gt;|0 127 -84 -36 100 -44&amp;gt;&lt;br /&gt;
| 33/28
&lt;/td&gt;
|
        &lt;td&gt;116.820 (15 limit least squares)&lt;br /&gt;
| 20/17
&lt;/td&gt;
| 13/11
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 24
        &lt;td&gt;|0 141 -70 -35 84 -42&amp;gt;&lt;br /&gt;
| 399.5
&lt;/td&gt;
| 44/35
        &lt;td&gt;116.846 (13 limit least squares)&lt;br /&gt;
|
&lt;/td&gt;
|
    &lt;/tr&gt;
|
    &lt;tr&gt;
|-
        &lt;td&gt;4/3&lt;br /&gt;
| 25
&lt;/td&gt;
| 516.2
        &lt;td&gt;116.993 (15 limit minimax)&lt;br /&gt;
| 27/20
&lt;/td&gt;
|
    &lt;/tr&gt;
|
    &lt;tr&gt;
|
        &lt;td&gt;4\41&lt;br /&gt;
|-
&lt;/td&gt;
| 26
        &lt;td&gt;117.073&lt;br /&gt;
| 632.8
&lt;/td&gt;
| 36/25
    &lt;/tr&gt;
|
    &lt;tr&gt;
|
        &lt;td&gt;13/11&lt;br /&gt;
| 13/9
&lt;/td&gt;
|-
        &lt;td&gt;117.266&lt;br /&gt;
| 27
&lt;/td&gt;
| 749.5
    &lt;/tr&gt;
| 54/35, 77/50
    &lt;tr&gt;
|
        &lt;td&gt;18/13&lt;br /&gt;
| 20/13
&lt;/td&gt;
| 17/11
        &lt;td&gt;117.559&lt;br /&gt;
|-
&lt;/td&gt;
| 28
    &lt;/tr&gt;
| 866.1
    &lt;tr&gt;
| 33/20
        &lt;td&gt;13/12&lt;br /&gt;
|
&lt;/td&gt;
| 28/17
        &lt;td&gt;117.936&lt;br /&gt;
|
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 29
    &lt;tr&gt;
| 982.8
        &lt;td&gt;15/14&lt;br /&gt;
| 44/25
&lt;/td&gt;
|
        &lt;td&gt;119.443&lt;br /&gt;
| 30/17
&lt;/td&gt;
|
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 30
        &lt;td&gt;16/13&lt;br /&gt;
| 1099.4
&lt;/td&gt;
| 66/35
        &lt;td&gt;119.824&lt;br /&gt;
|
&lt;/td&gt;
| '''32/17'''
    &lt;/tr&gt;
| 17/9
    &lt;tr&gt;
|-
        &lt;td&gt;15/13&lt;br /&gt;
| 31
&lt;/td&gt;
| 16.1
        &lt;td&gt;123.871&lt;br /&gt;
| 81/80, 99/98, 121/120
&lt;/td&gt;
| 66/65
    &lt;/tr&gt;
| 105/104, 120/119, 136/135, <br>144/143, 154/153, 170/169
    &lt;tr&gt;
| 65/64, 78/77, <br>85/84, 91/90
        &lt;td&gt;14/13&lt;br /&gt;
|-
&lt;/td&gt;
| 32
        &lt;td&gt;128.298&lt;br /&gt;
| 132.7
&lt;/td&gt;
| 27/25
    &lt;/tr&gt;
|
&lt;/table&gt;
| 14/13
| 13/12
|-
| 33
| 249.3
| 81/70
|
| 15/13
| 52/45
|-
| 34
| 366.0
| 99/80
|
| '''16/13''', 21/17
| 26/21
|-
| 35
| 482.6
| 33/25
|
|
|
|-
| 36
| 599.3
| 99/70
|
| 24/17
| 17/12
|-
| 37
| 715.9
| 121/80
|
|
|
|-
| 38
| 832.6
| 121/75
|
| 21/13
| '''13/8''', 34/21
|-
| 39
| 949.2
| 121/70
|
| 45/26
| 26/15
|-
| 40
| 1065.9
| 231/125
|
| 24/13
| 13/7
|-
| 41
| 1182.5
| 99/50
|
| 77/39, 128/65, <br>168/85, 180/91
| 119/60, 143/72, 135/68, <br>153/77, 169/85, 195/98
|}
<nowiki/>* In 11-limit [[CWE tuning]], octave reduced


&lt;br /&gt;
== Tunings ==
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Tuning Spectra-Spectrum of Benediction Tunings by Eigenmonzos"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Spectrum of Benediction Tunings by Eigenmonzos&lt;/h2&gt;
=== Norm-based tunings ===
&lt;br /&gt;
* 5-limit POTE: ~16/15 = 116.673{{c}}
Gencom: [2 15/14; 225/224 243/242 351/350 441/440]&lt;br /&gt;
* 7-limit POTE: ~15/14 = 116.675{{c}}
Gencom map: [&amp;lt;1 1 3 3 2 7|, &amp;lt;0 6 -7 -2 15 -34|]&lt;br /&gt;
* 11-limit POTE
** Miracle: ~15/14 = 116.633{{c}}
** Revelation: ~15/14 = 116.277{{c}}
* 13-limit POTE
** Miraculous: ~15/14 = 116.747{{c}}
** Benediction: ~15/14 = 116.574{{c}}
** Manna: ~15/14 = 116.739{{c}}
** Revelation: ~15/14 = 116.268{{c}}


=== Target tunings ===
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Target tunings (miracle)
|-
! rowspan="2" | Target
! colspan="2" | Minimax
! colspan="2" | Least squares
|-
! Generator
! Eigenmonzo*
! Generator
! Eigenmonzo*
|-
| 5-odd-limit
| ~16/15 = 116.588{{c}}
| 5/3
| ~16/15 = 116.578{{c}}
| {{Monzo| 0 -19 20 }}
|-
| 7-odd-limit
| ~15/14 = 116.588{{c}}
| 5/3
| ~15/14 = 116.573{{c}}
| {{Monzo| 0 -27 25 5 }}
|-
| 9-odd-limit
| ~15/14 = 116.716{{c}}
| 9/5
| ~15/14 = 116.721{{c}}
| {{Monzo| 0 117 -44 -19 }}
|-
| 11-odd-limit
| ~15/14 = 116.716{{c}}
| 9/5
| ~15/14 = 116.672{{c}}
| {{Monzo| 0 17 -11 -6 11 }}
|}


&lt;table class="wiki_table"&gt;
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
    &lt;tr&gt;
|+ style="white-space: nowrap;" | Target tunings (miraculous)
        &lt;th&gt;Eigenmonzo&lt;br /&gt;
|-
&lt;/th&gt;
! rowspan="2" | Target
        &lt;th&gt;Secor&lt;br /&gt;
! colspan="2" | Minimax
&lt;/th&gt;
! colspan="2" | Least squares
    &lt;/tr&gt;
|-
    &lt;tr&gt;
! Generator
        &lt;td&gt;16/15&lt;br /&gt;
! Eigenmonzo*
&lt;/td&gt;
! Generator
        &lt;td&gt;111.731&lt;br /&gt;
! Eigenmonzo*
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 13-odd-limit
    &lt;tr&gt;
| ~15/14 = 116.716{{c}}
        &lt;td&gt;8/7&lt;br /&gt;
| 9/5
&lt;/td&gt;
| ~15/14 = 116.846{{c}}
        &lt;td&gt;115.587&lt;br /&gt;
| {{Monzo| 0 141 -70 -35 84 -42 }}
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 15-odd-limit
    &lt;tr&gt;
| ~15/14 = 116.993{{c}}
        &lt;td&gt;11/9&lt;br /&gt;
| 3/2
&lt;/td&gt;
| ~15/14 = 116.820{{c}}
        &lt;td&gt;115.803&lt;br /&gt;
| {{Monzo| 0 127 -84 -36 100 -44 }}
&lt;/td&gt;
|}
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;3\31&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;116.129&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;116.241&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;116.441&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;116.455&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;116.502&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;10\103&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;116.505&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;116.509&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;116.511&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;116.536&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;116.547&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;|0 -234 39 4 -115 228&amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;116.56309 (13 limit least squares)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;|0 -251 22 5 -131 261&amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;116.56348 (15 limit least squares)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;17\175&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;116.571&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;|0 -27 25 5&amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;116.573 (7 limit least squares)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;POTE13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;116.574&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;|0 -19 20&amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;116.578 (5 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;116.588 (5, 7 and 15 limit minimax)&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;116.591&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;116.595 (13 limit minimax)&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;116.596&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;116.598&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;116.617&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;POTE11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;116.633&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;116.641&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7\72&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;116.667&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;|0 17 -11 -6 11&amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;116.672 (11 limit least squares)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;POTE5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;116.673&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;POTE7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;116.675&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;116.716 (9 and 11 limit minimax)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;|0 117 -44 -19&amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;116.721 (9 limit least squares)&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;116.755&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;116.792&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11\113&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;116.814&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;116.993&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4\41&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;117.073&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;/table&gt;


&lt;br /&gt;
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Tuning Spectra-Spectrum of Manna Tunings by Eigenmonzos"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Spectrum of Manna Tunings by Eigenmonzos&lt;/h2&gt;
|+ style="white-space: nowrap;" | Target tunings (benediction)
&lt;br /&gt;
|-
Gencom: [2 15/14; 225/224 243/242 325/324 385/384]&lt;br /&gt;
! rowspan="2" | Target
Gencom map: [&amp;lt;1 1 3 3 2 0|, &amp;lt;0 6 -7 -2 15 38|]&lt;br /&gt;
! colspan="2" | Minimax
! colspan="2" | Least squares
|-
! Generator
! Eigenmonzo*
! Generator
! Eigenmonzo*
|-
| 13-odd-limit
| ~15/14 = 116.595{{c}}
| 13/9
| ~15/14 = 116.56309{{c}}
| {{Monzo| 0 -234 39 4 -115 228 }}
|-
| 15-odd-limit
| ~15/14 = 116.588{{c}}
| 5/3
| ~15/14 = 116.56348{{c}}
| {{Monzo| 0 -251 22 5 -131 261 }}
|}


{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Target tunings (manna)
|-
! rowspan="2" | Target
! colspan="2" | Minimax
! colspan="2" | Least squares
|-
! Generator
! Eigenmonzo*
! Generator
! Eigenmonzo*
|-
| 13-odd-limit
| ~15/14 = 116.760{{c}}
| 13/10
| ~15/14 = 116.780{{c}}
| {{Monzo| 0 18 -111 -76 43 204 }}
|-
| 15-odd-limit
| ~15/14 = 116.725{{c}}
| 15/13
| ~15/14 = 116.764{{c}}
| {{Monzo| 0 -37 -166 -77 59 243 }}
|}


&lt;table class="wiki_table"&gt;
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
    &lt;tr&gt;
|+ style="white-space: nowrap;" | Target tunings (revelation)
        &lt;th&gt;Eigenmonzo&lt;br /&gt;
|-
&lt;/th&gt;
! rowspan="2" | Target
        &lt;th&gt;Secor&lt;br /&gt;
! colspan="2" | Minimax
&lt;/th&gt;
! colspan="2" | Least squares
    &lt;/tr&gt;
|-
    &lt;tr&gt;
! Generator
        &lt;td&gt;16/15&lt;br /&gt;
! Eigenmonzo*
&lt;/td&gt;
! Generator
        &lt;td&gt;111.731&lt;br /&gt;
! Eigenmonzo*
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 11-odd-limit
    &lt;tr&gt;
| ~15/14 = 116.164{{c}}
        &lt;td&gt;8/7&lt;br /&gt;
| 11/9
&lt;/td&gt;
| ~15/14 = 116.198{{c}}
        &lt;td&gt;115.587&lt;br /&gt;
| {{Monzo| 0 -195 35 5 89 }}
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 13-odd-limit
    &lt;tr&gt;
| ~15/14 = 116.164{{c}}
        &lt;td&gt;11/9&lt;br /&gt;
| 11/9
&lt;/td&gt;
| ~15/14 = 116.249{{c}}
        &lt;td&gt;115.803&lt;br /&gt;
| {{Monzo| 0 -234 39 4 102 11 }}
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 15-odd-limit
    &lt;tr&gt;
| ~15/14 = 116.164{{c}}
        &lt;td&gt;3\31&lt;br /&gt;
| 11/9
&lt;/td&gt;
| ~15/14 = 116.229{{c}}
        &lt;td&gt;116.129&lt;br /&gt;
| {{Monzo| 0 -251 22 5 117 13 }}
&lt;/td&gt;
|}
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;5/4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;116.241&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;116.441&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;116.502&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;10\103&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;116.505&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;17\175&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;116.571&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;|0 -27 25 5&amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;116.573 (7 limit least squares)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;|0 -19 20&amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;116.578 (5 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;116.588 (5 and 7 limit minimax)&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;116.591&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;116.596&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;116.617&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;POTE11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;116.633&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;116.641&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7\72&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;116.667&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;|0 17 -11 -6 11&amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;116.672 (11 limit least squares)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;POTE5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;116.673&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;POTE7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;116.675&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;116.716 (9 and 11 limit minimax)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;|0 117 -44 -19&amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;116.721 (9 limit least squares)&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;116.725 (15 limit minimax)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;POTE13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;116.739&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;116.755&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;116.760 (13 limit minimax)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;|0 -37 -166 -77 59 243&amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;116.764 (15 limit least squares)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;|0 18 -111 -76 43 204&amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;116.780 (13 limit least squares)&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;116.792&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;116.79254&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;116.79299&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11\113&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;116.814&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;116.830&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;116.856&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;116.922&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;116.993&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4\41&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;117.073&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;/table&gt;


&lt;br /&gt;
=== Tuning spectra ===
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Tuning Spectra-Spectrum of Revelation Tunings by Eigenmonzos"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Spectrum of Revelation Tunings by Eigenmonzos&lt;/h2&gt;
==== Miraculous ====
&lt;br /&gt;
{| class="wikitable center-all left-4"
Gencom: [2 15/14; 66/65 99/98 105/104 1001/1000]&lt;br /&gt;
|-
Gencom map: [&amp;lt;1 1 3 3 5 4|, &amp;lt;0 6 -7 -2 -16 -3|]&lt;br /&gt;
! Edo<br>generator
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]
! Generator (¢)
! Comments
|-
|
| 17/16
| 104.955
|
|-
|
| 17/15
| 108.343
|
|-
|
| 15/8
| 111.731
|
|-
|
| 17/14
| 112.043
|
|-
|
| 13/10
| 113.553
|
|-
|
| 17/10
| 114.830
|
|-
|
| 7/4
| 115.587
|
|-
|
| 11/9
| 115.803
|
|-
|
| 17/13
| 116.107
|
|-
| 3\31
|
| 116.129
| Lower bound of 11- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
|-
|
| 5/4
| 116.241
|
|-
|
| 15/11
| 116.441
|
|-
|
| 7/5
| 116.502
|
|-
| 10\103
|
| 116.505
| 103fg val
|-
| 17\175
|
| 116.571
| 175ffggg val
|-
|
| 5/3
| 116.588
| 5- and 7-odd-limit minimax
|-
|
| 11/10
| 116.591
|
|-
|
| 11/6
| 116.596
|
|-
|
| 11/7
| 116.617
|
|-
|
| 7/6
| 116.641
|
|-
| 7\72
|
| 116.667
| 72fg val
|-
|
| 9/5
| 116.716
| 9-, 11- and 13-odd-limit minimax
|-
|
| 11/8
| 116.755
|
|-
| 18\185
|
| 116.757
| 185cffggg val
|-
|
| 9/7
| 116.792
|
|-
| 11\113
|
| 116.814
| 113fgg val
|-
|
| 3/2
| 116.993
| 15-odd-limit minimax
|-
| 4\41
|
| 117.073
| Upper bound of 11- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
|-
|
| 13/11
| 117.266
|
|-
|
| 13/9
| 117.559
|
|-
|
| 17/11
| 117.597
|
|-
|
| 13/12
| 117.936
|
|-
|
| 17/9
| 118.087
|
|-
|
| 17/12
| 119.400
|
|-
|
| 15/14
| 119.443
|
|-
|
| 13/8
| 119.824
|
|-
|
| 21/17
| 121.942
|
|-
|
| 15/13
| 123.871
|
|-
|
| 13/7
| 128.298
|
|}


==== Benediction ====
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! Unchanged interval<br>(eigenmonzo)
! Generator (¢)
! Comments
|-
|
| 15/8
| 111.731
|
|-
|
| 7/4
| 115.587
|
|-
|
| 11/9
| 115.803
|
|-
|
| 17/13
| 116.107
|
|-
| 3\31
|
| 116.129
| Lower bound of 11- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
|-
|
| 5/4
| 116.241
|
|-
|
| 15/11
| 116.441
|
|-
|
| 13/8
| 116.455
|
|-
|
| 17/16
| 116.501
|
|-
|
| 7/5
| 116.502
|
|-
| 10\103
|
| 116.505
|
|-
|
| 13/7
| 116.509
|
|-
|
| 13/10
| 116.511
|
|-
|
| 13/12
| 116.536
|
|-
|
| 13/11
| 116.547
|
|-
|
| 17/14
| 116.567
|
|-
| 17\175
|
| 116.571
| 175f val
|-
|
| 17/10
| 116.581
|
|-
|
| 17/12
| 116.583
|
|-
|
| 17/11
| 116.586
|
|-
|
| 5/3
| 116.588
| 5-, 7- and 15-odd-limit minimax
|-
|
| 11/10
| 116.591
|
|-
|
| 13/9
| 116.595
| 13-odd-limit minimax
|-
|
| 11/6
| 116.596
|
|-
|
| 15/13
| 116.598
|
|-
|
| 11/7
| 116.617
|
|-
|
| 7/6
| 116.641
|
|-
|
| 17/9
| 116.642
|
|-
|
| 21/17
| 116.642
|
|-
|
| 17/15
| 116.666
|
|-
| 7\72
|
| 116.667
| Upper bound of 13- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
|-
|
| 9/5
| 116.716
| 9- and 11-odd-limit minimax
|-
|
| 11/8
| 116.755
|
|-
| 18\185
|
| 116.757
| 185cfffgg val
|-
|
| 9/7
| 116.792
|
|-
| 11\113
|
| 116.814
| 113ffg val
|-
|
| 3/2
| 116.993
|
|-
| 4\41
|
| 117.073
| 41fg val, upper bound of 11-odd-limit diamond monotone
|-
|
| 15/14
| 119.443
|
|}


&lt;table class="wiki_table"&gt;
==== Manna ====
    &lt;tr&gt;
{| class="wikitable center-all left-4"
        &lt;th&gt;Eigenmonzo&lt;br /&gt;
|-
&lt;/th&gt;
! Edo<br>generator
        &lt;th&gt;Secor&lt;br /&gt;
! Unchanged interval<br>(eigenmonzo)
&lt;/th&gt;
! Generator (¢)
    &lt;/tr&gt;
! Comments
    &lt;tr&gt;
|-
        &lt;td&gt;16/15&lt;br /&gt;
|
&lt;/td&gt;
| 15/8
        &lt;td&gt;111.731&lt;br /&gt;
| 111.731
&lt;/td&gt;
|
    &lt;/tr&gt;
|-
    &lt;tr&gt;
|
        &lt;td&gt;13/10&lt;br /&gt;
| 7/4
&lt;/td&gt;
| 115.587
        &lt;td&gt;113.553&lt;br /&gt;
|
&lt;/td&gt;
|-
    &lt;/tr&gt;
|
    &lt;tr&gt;
| 11/9
        &lt;td&gt;13/11&lt;br /&gt;
| 115.803
&lt;/td&gt;
|
        &lt;td&gt;114.555&lt;br /&gt;
|-
&lt;/td&gt;
|
    &lt;/tr&gt;
| 17/13
    &lt;tr&gt;
| 116.107
        &lt;td&gt;11/10&lt;br /&gt;
|
&lt;/td&gt;
|-
        &lt;td&gt;115.000&lt;br /&gt;
| 3\31
&lt;/td&gt;
|
    &lt;/tr&gt;
| 116.129
    &lt;tr&gt;
| 31fg val, lower bound of 11-odd-limit diamond monotone
        &lt;td&gt;14/11&lt;br /&gt;
|-
&lt;/td&gt;
|
        &lt;td&gt;115.536&lt;br /&gt;
| 5/4
&lt;/td&gt;
| 116.241
    &lt;/tr&gt;
|
    &lt;tr&gt;
|-
        &lt;td&gt;11/8&lt;br /&gt;
|
&lt;/td&gt;
| 15/11
        &lt;td&gt;115.543&lt;br /&gt;
| 116.441
&lt;/td&gt;
|
    &lt;/tr&gt;
|-
    &lt;tr&gt;
|
        &lt;td&gt;8/7&lt;br /&gt;
| 7/5
&lt;/td&gt;
| 116.502
        &lt;td&gt;115.587&lt;br /&gt;
|
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 10\103
    &lt;tr&gt;
|
        &lt;td&gt;15/11&lt;br /&gt;
| 116.505
&lt;/td&gt;
| 103ffgg val
        &lt;td&gt;115.797&lt;br /&gt;
|-
&lt;/td&gt;
| 17\175
    &lt;/tr&gt;
|
    &lt;tr&gt;
| 116.571
        &lt;td&gt;12/11&lt;br /&gt;
| 175fffgg val
&lt;/td&gt;
|-
        &lt;td&gt;115.938&lt;br /&gt;
|
&lt;/td&gt;
| 5/3
    &lt;/tr&gt;
| 116.588
    &lt;tr&gt;
| 5- and 7-odd-limit minimax
        &lt;td&gt;3\31&lt;br /&gt;
|-
&lt;/td&gt;
|
        &lt;td&gt;116.129&lt;br /&gt;
| 11/10
&lt;/td&gt;
| 116.591
    &lt;/tr&gt;
|
    &lt;tr&gt;
|-
        &lt;td&gt;11/9&lt;br /&gt;
|
&lt;/td&gt;
| 11/6
        &lt;td&gt;116.164 (11, 13 and 15 limit minimax)&lt;br /&gt;
| 116.596
&lt;/td&gt;
|
    &lt;/tr&gt;
|-
    &lt;tr&gt;
|
        &lt;td&gt;|0 -195 35 5 89&amp;gt;&lt;br /&gt;
| 11/7
&lt;/td&gt;
| 116.617
        &lt;td&gt;116.198 (11 limit least squares)&lt;br /&gt;
|
&lt;/td&gt;
|-
    &lt;/tr&gt;
|
    &lt;tr&gt;
| 7/6
        &lt;td&gt;|0 -251 22 5 117 13&amp;gt;&lt;br /&gt;
| 116.641
&lt;/td&gt;
|
        &lt;td&gt;116.229 (15 limit least squares)&lt;br /&gt;
|-
&lt;/td&gt;
| 7\72
    &lt;/tr&gt;
|
    &lt;tr&gt;
| 116.667
        &lt;td&gt;5/4&lt;br /&gt;
| Lower bound of 13- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
&lt;/td&gt;
|-
        &lt;td&gt;116.241&lt;br /&gt;
|
&lt;/td&gt;
| 17/15
    &lt;/tr&gt;
| 116.667
    &lt;tr&gt;
|
        &lt;td&gt;|0 -234 39 4 102 11&amp;gt;&lt;br /&gt;
|-
&lt;/td&gt;
|
        &lt;td&gt;116.249 (13 limit least squares)&lt;br /&gt;
| 21/17
&lt;/td&gt;
| 116.689
    &lt;/tr&gt;
|
    &lt;tr&gt;
|-
        &lt;td&gt;POTE13&lt;br /&gt;
|
&lt;/td&gt;
| 17/9
        &lt;td&gt;116.268&lt;br /&gt;
| 116.702
&lt;/td&gt;
|
    &lt;/tr&gt;
|-
    &lt;tr&gt;
|
        &lt;td&gt;POTE11&lt;br /&gt;
| 17/10
&lt;/td&gt;
| 116.707
        &lt;td&gt;116.277&lt;br /&gt;
|
&lt;/td&gt;
|-
    &lt;/tr&gt;
|
    &lt;tr&gt;
| 9/5
        &lt;td&gt;7/5&lt;br /&gt;
| 116.716
&lt;/td&gt;
| 9- and 11-odd-limit minimax
        &lt;td&gt;116.502&lt;br /&gt;
|-
&lt;/td&gt;
|
    &lt;/tr&gt;
| 15/13
    &lt;tr&gt;
| 116.725
        &lt;td&gt;10\103&lt;br /&gt;
| 15-odd-limit minimax
&lt;/td&gt;
|-
        &lt;td&gt;116.505&lt;br /&gt;
|
&lt;/td&gt;
| 17/14
    &lt;/tr&gt;
| 116.730
    &lt;tr&gt;
|
        &lt;td&gt;17\175&lt;br /&gt;
|-
&lt;/td&gt;
|
        &lt;td&gt;116.571&lt;br /&gt;
| 17/12
&lt;/td&gt;
| 116.750
    &lt;/tr&gt;
|
    &lt;tr&gt;
|-
        &lt;td&gt;|0 -27 25 5&amp;gt;&lt;br /&gt;
|
&lt;/td&gt;
| 11/8
        &lt;td&gt;116.573 (7 limit least squares)&lt;br /&gt;
| 116.755
&lt;/td&gt;
|
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 18\185
        &lt;td&gt;|0 -19 20&amp;gt;&lt;br /&gt;
|
&lt;/td&gt;
| 116.757
        &lt;td&gt;116.578 (5 limit least squares)&lt;br /&gt;
| 185cf val
&lt;/td&gt;
|-
    &lt;/tr&gt;
|
    &lt;tr&gt;
| 13/10
        &lt;td&gt;6/5&lt;br /&gt;
| 116.760
&lt;/td&gt;
| 13-odd-limit minimax
        &lt;td&gt;116.588 (5 and 7 limit minimax)&lt;br /&gt;
|-
&lt;/td&gt;
|
    &lt;/tr&gt;
| 17/16
    &lt;tr&gt;
| 116.785
        &lt;td&gt;7/6&lt;br /&gt;
|
&lt;/td&gt;
|-
        &lt;td&gt;116.641&lt;br /&gt;
|
&lt;/td&gt;
| 9/7
    &lt;/tr&gt;
| 116.792
    &lt;tr&gt;
|
        &lt;td&gt;7\72&lt;br /&gt;
|-
&lt;/td&gt;
|
        &lt;td&gt;116.667&lt;br /&gt;
| 13/7
&lt;/td&gt;
| 116.79254
    &lt;/tr&gt;
|
    &lt;tr&gt;
|-
        &lt;td&gt;POTE5&lt;br /&gt;
|
&lt;/td&gt;
| 13/9
        &lt;td&gt;116.673&lt;br /&gt;
| 116.79299
&lt;/td&gt;
|
    &lt;/tr&gt;
|-
    &lt;tr&gt;
|
        &lt;td&gt;POTE7&lt;br /&gt;
| 17/11
&lt;/td&gt;
| 116.801
        &lt;td&gt;116.675&lt;br /&gt;
|
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 11\113
    &lt;tr&gt;
|
        &lt;td&gt;10/9&lt;br /&gt;
| 116.814
&lt;/td&gt;
|
        &lt;td&gt;116.716 (9 limit minimax)&lt;br /&gt;
|-
&lt;/td&gt;
|
    &lt;/tr&gt;
| 13/12
    &lt;tr&gt;
| 116.830
        &lt;td&gt;|0 117 -44 -19&amp;gt;&lt;br /&gt;
|
&lt;/td&gt;
|-
        &lt;td&gt;116.721 (9 limit least squares)&lt;br /&gt;
|
&lt;/td&gt;
| 13/8
    &lt;/tr&gt;
| 116.856
    &lt;tr&gt;
|
        &lt;td&gt;9/7&lt;br /&gt;
|-
&lt;/td&gt;
|
        &lt;td&gt;116.792&lt;br /&gt;
| 13/11
&lt;/td&gt;
| 116.922
    &lt;/tr&gt;
|
    &lt;tr&gt;
|-
        &lt;td&gt;11\113&lt;br /&gt;
|
&lt;/td&gt;
| 3/2
        &lt;td&gt;116.814&lt;br /&gt;
| 116.993
&lt;/td&gt;
|
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 4\41
        &lt;td&gt;4/3&lt;br /&gt;
|
&lt;/td&gt;
| 117.073
        &lt;td&gt;116.993&lt;br /&gt;
| Upper bound of 11- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
&lt;/td&gt;
|-
    &lt;/tr&gt;
|
    &lt;tr&gt;
| 15/14
        &lt;td&gt;4\41&lt;br /&gt;
| 119.443
&lt;/td&gt;
|
        &lt;td&gt;117.073&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;117.559&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;117.936&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;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;14/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;128.298&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;/body&gt;&lt;/html&gt;</pre></div>
==== Revelation ====
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! Unchanged interval<br>(eigenmonzo)
! Generator (¢)
! Comments
|-
|
| 15/8
| 111.731
|
|-
|
| 13/10
| 113.553
|
|-
| 2\21
|
| 114.286
|
|-
|
| 13/11
| 114.555
|
|-
|
| 11/10
| 115.000
|
|-
| 5\52
|
| 115.385
| 52f val
|-
|
| 11/7
| 115.536
|
|-
|
| 11/8
| 115.543
|
|-
|
| 7/4
| 115.587
|
|-
|
| 15/11
| 115.797
|
|-
|
| 11/6
| 115.938
|
|-
| 3\31
|
| 116.129
| 11- to 15-odd-limit, <br>and 13-limit 21-odd-limit diamond monotone (singleton)
|-
|
| 11/9
| 116.164
| 11-, 13- and 15-odd-limit minimax
|-
|
| 5/4
| 116.241
|
|-
|
| 7/5
| 116.502
|
|-
|
| 5/3
| 116.588
| 5- and 7-odd-limit minimax
|-
|
| 7/6
| 116.641
|
|-
| 7\72
|
| 116.667
| 72ee val
|-
|
| 9/5
| 116.716
| 9-odd-limit minimax
|-
|
| 9/7
| 116.792
|
|-
|
| 3/2
| 116.993
|
|-
| 4\41
|
| 117.073
| 41ef val
|-
|
| 13/9
| 117.559
|
|-
|
| 13/12
| 117.936
|
|-
|
| 15/14
| 119.443
|
|-
|
| 13/8
| 119.824
|
|-
|
| 15/13
| 123.871
|
|-
|
| 13/7
| 128.298
|
|}
 
[[Category:Miracle]]
[[Category:Temperament extensions]]
[[Category:Rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/Miracle_extensions"