Huygens vs meanpop: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
"11-limit meantone" and "meanpop", both discussed at [[Meantone_family|Meantone family]], are two different temperaments in the 11 limit. This page compares and contrasts them in detail.
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>2012-11-04 09:32:10 UTC</tt>.<br>
: The original revision id was <tt>379021908</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">"11-limit meantone" and "meanpop", both discussed at [[Meantone family]], are two different temperaments in the 11 limit. This page compares and contrasts them in detail.


Extending meantone from the 5 limit to the 7 limit, there is one obvious mapping that is not too complex and adds hardly any additional error (so we're not talking about dominant temperament here). This is called "7-limit meantone" or "septimal meantone" and is an amazingly efficient (and beautiful) temperament. But extending it from the 7 limit to the 11 limit is not so simple. There are two mappings that are comparable in complexity and error: 11-limit meantone and meanpop.
Extending meantone from the 5 limit to the 7 limit, there is one obvious mapping that is not too complex and adds hardly any additional error (so we're not talking about dominant temperament here). This is called "7-limit meantone" or "septimal meantone" and is an amazingly efficient (and beautiful) temperament. But extending it from the 7 limit to the 11 limit is not so simple. There are two mappings that are comparable in complexity and error: 11-limit meantone and meanpop.
Line 14: Line 7:
In meanpop, 11/8 is represented by the doubly diminished fifth, for example C-Gbb. This is in the opposite direction along the circle of fifths - 13 fifths down.
In meanpop, 11/8 is represented by the doubly diminished fifth, for example C-Gbb. This is in the opposite direction along the circle of fifths - 13 fifths down.


Can meantone and meanpop be combined into a single temperament? Yes! It works wonderfully and that temperament is [[31edo]]. In 31edo the circle of fifths closes perfectly after 31 fifths, so Ex and Gbb are the same note. (In other words, the interval of the //quadruply diminished third// is tuned to 0 cents, if that makes any sense to you.) This makes everything much simpler and results in 121/120 and 243/242 being tempered out, so that 12/11~11/10 is a "neutral second" (exactly half of a minor third), and 11/9 is a "neutral third" (exactly half of a perfect fifth). Keep in mind that neither of these things are true in either meantone or meanpop.
Can meantone and meanpop be combined into a single temperament? Yes! It works wonderfully and that temperament is [[31edo|31edo]]. In 31edo the circle of fifths closes perfectly after 31 fifths, so Ex and Gbb are the same note. (In other words, the interval of the ''quadruply diminished third'' is tuned to 0 cents, if that makes any sense to you.) This makes everything much simpler and results in 121/120 and 243/242 being tempered out, so that 12/11~11/10 is a "neutral second" (exactly half of a minor third), and 11/9 is a "neutral third" (exactly half of a perfect fifth). Keep in mind that neither of these things are true in either meantone or meanpop.


||~ JI interval ||~ Meantone mapping ||~ Meanpop mapping ||
{| class="wikitable"
|| 12/11 || Doubly diminished third (A-Cbb) || Doubly augmented prime (C-Cx) ||
|-
|| 11/10 || Doubly augmented prime (C-Cx) || Doubly diminished third (A-Cbb) ||
! | JI interval
|| 11/9 || Doubly augmented second (C-Dx) || Doubly diminished fourth (C-Fbb) ||
! | Meantone mapping
|| 14/11 || Diminished fourth (C-Fb), same as 9/7 || Triply augmented second (C-Dx#) ||
! | Meanpop mapping
|| 11/8 || Doubly augmented third (C-Ex) || Doubly diminished fifth (C-Gbb) ||
|-
|| 16/11 || Doubly diminished sixth (A-Fbb) || Doubly augmented fourth (C-Fx) ||
| | 12/11
|| 11/7 || Augmented fifth (C-G#), same as 14/9 || Triply diminished seventh (A-Gbbb) ||
| | Doubly diminished third (A-Cbb)
|| 18/11 || Doubly diminished seventh (A-Gbb) || Doubly augmented fifth (C-Gx) ||
| | Doubly augmented prime (C-Cx)
|| 20/11 || Doubly diminished octave (C-Cbb) || Doubly augmented sixth (C-Ax) ||
|-
|| 11/6 || Doubly augmented sixth (C-Ax) || Double diminished octave (C-Cbb) ||
| | 11/10
| | Doubly augmented prime (C-Cx)
| | Doubly diminished third (A-Cbb)
|-
| | 11/9
| | Doubly augmented second (C-Dx)
| | Doubly diminished fourth (C-Fbb)
|-
| | 14/11
| | Diminished fourth (C-Fb), same as 9/7
| | Triply augmented second (C-Dx#)
|-
| | 11/8
| | Doubly augmented third (C-Ex)
| | Doubly diminished fifth (C-Gbb)
|-
| | 16/11
| | Doubly diminished sixth (A-Fbb)
| | Doubly augmented fourth (C-Fx)
|-
| | 11/7
| | Augmented fifth (C-G#), same as 14/9
| | Triply diminished seventh (A-Gbbb)
|-
| | 18/11
| | Doubly diminished seventh (A-Gbb)
| | Doubly augmented fifth (C-Gx)
|-
| | 20/11
| | Doubly diminished octave (C-Cbb)
| | Doubly augmented sixth (C-Ax)
|-
| | 11/6
| | Doubly augmented sixth (C-Ax)
| | Double diminished octave (C-Cbb)
|}


=Tuning Spectra=
=Tuning Spectra=
==Spectrum of Undecimal Meantone Tunings by Eigenmonzos==
||~ Eigenmonzo ||~ Fifth ||
|| 10/9 || 691.202 ||
|| 6/5 || 694.786 ||
|| 9/7 || 695.614 ||
|| 7/6 || 696.319 ||
|| 5/4 || 696.578 ||
|| 11/9 || 696.713 (minimax tuning) ||
|| 8/7 || 696.883 ||
|| 12/11 || 697.021 ||
|| 7/5 || 697.085 ||
|| 11/8 || 697.295 ||
|| 11/10 || 697.500 ||
|| 14/11 || 697.812 ||
|| 4/3 || 701.955 ||


==Spectrum of Meanpop Tunings by Eigenmonzos==  
==Spectrum of Undecimal Meantone Tunings by Eigenmonzos==
||~ Eigenmonzo ||~ Fifth ||
|| 10/9 || 691.202 ||
|| 6/5 || 694.786 ||
|| 9/7 || 695.614 ||
|| 11/8 || 696.052 ||
|| 11/10 || 696.176 ||
|| 7/6 || 696.319 ||
|| 14/11 || 696.413 ||
|| 12/11 || 696.474 ||
|| 5/4 || 696.578 (minimax tuning) ||
|| 11/9 || 696.839 ||
|| 8/7 || 696.883 ||
|| 7/5 || 697.085 ||
|| 4/3 || 701.955 ||


</pre></div>
{| class="wikitable"
<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;Meantone vs meanpop&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&amp;quot;11-limit meantone&amp;quot; and &amp;quot;meanpop&amp;quot;, both discussed at &lt;a class="wiki_link" href="/Meantone%20family"&gt;Meantone family&lt;/a&gt;, are two different temperaments in the 11 limit. This page compares and contrasts them in detail.&lt;br /&gt;
! | Eigenmonzo
&lt;br /&gt;
! | Fifth
Extending meantone from the 5 limit to the 7 limit, there is one obvious mapping that is not too complex and adds hardly any additional error (so we're not talking about dominant temperament here). This is called &amp;quot;7-limit meantone&amp;quot; or &amp;quot;septimal meantone&amp;quot; and is an amazingly efficient (and beautiful) temperament. But extending it from the 7 limit to the 11 limit is not so simple. There are two mappings that are comparable in complexity and error: 11-limit meantone and meanpop.&lt;br /&gt;
|-
&lt;br /&gt;
| | 10/9
In 11-limit meantone, 11/8 is represented by the doubly augmented third, for example C-Ex (where &amp;quot;x&amp;quot; represents the standard double sharp symbol, equivalent in meaning to &amp;quot;##&amp;quot;). This is 18 fifths along the circle of fifths; Ex is 18 fifths up from C.&lt;br /&gt;
| | 691.202
&lt;br /&gt;
|-
In meanpop, 11/8 is represented by the doubly diminished fifth, for example C-Gbb. This is in the opposite direction along the circle of fifths - 13 fifths down.&lt;br /&gt;
| | 6/5
&lt;br /&gt;
| | 694.786
Can meantone and meanpop be combined into a single temperament? Yes! It works wonderfully and that temperament is &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt;. In 31edo the circle of fifths closes perfectly after 31 fifths, so Ex and Gbb are the same note. (In other words, the interval of the &lt;em&gt;quadruply diminished third&lt;/em&gt; is tuned to 0 cents, if that makes any sense to you.) This makes everything much simpler and results in 121/120 and 243/242 being tempered out, so that 12/11~11/10 is a &amp;quot;neutral second&amp;quot; (exactly half of a minor third), and 11/9 is a &amp;quot;neutral third&amp;quot; (exactly half of a perfect fifth). Keep in mind that neither of these things are true in either meantone or meanpop.&lt;br /&gt;
|-
&lt;br /&gt;
| | 9/7
| | 695.614
|-
| | 7/6
| | 696.319
|-
| | 5/4
| | 696.578
|-
| | 11/9
| | 696.713 (minimax tuning)
|-
| | 8/7
| | 696.883
|-
| | 12/11
| | 697.021
|-
| | 7/5
| | 697.085
|-
| | 11/8
| | 697.295
|-
| | 11/10
| | 697.500
|-
| | 14/11
| | 697.812
|-
| | 4/3
| | 701.955
|}


==Spectrum of Meanpop Tunings by Eigenmonzos==


&lt;table class="wiki_table"&gt;
{| class="wikitable"
    &lt;tr&gt;
|-
        &lt;th&gt;JI interval&lt;br /&gt;
! | Eigenmonzo
&lt;/th&gt;
! | Fifth
        &lt;th&gt;Meantone mapping&lt;br /&gt;
|-
&lt;/th&gt;
| | 10/9
        &lt;th&gt;Meanpop mapping&lt;br /&gt;
| | 691.202
&lt;/th&gt;
|-
    &lt;/tr&gt;
| | 6/5
    &lt;tr&gt;
| | 694.786
        &lt;td&gt;12/11&lt;br /&gt;
|-
&lt;/td&gt;
| | 9/7
        &lt;td&gt;Doubly diminished third (A-Cbb)&lt;br /&gt;
| | 695.614
&lt;/td&gt;
|-
        &lt;td&gt;Doubly augmented prime (C-Cx)&lt;br /&gt;
| | 11/8
&lt;/td&gt;
| | 696.052
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| | 11/10
        &lt;td&gt;11/10&lt;br /&gt;
| | 696.176
&lt;/td&gt;
|-
        &lt;td&gt;Doubly augmented prime (C-Cx)&lt;br /&gt;
| | 7/6
&lt;/td&gt;
| | 696.319
        &lt;td&gt;Doubly diminished third (A-Cbb)&lt;br /&gt;
|-
&lt;/td&gt;
| | 14/11
    &lt;/tr&gt;
| | 696.413
    &lt;tr&gt;
|-
        &lt;td&gt;11/9&lt;br /&gt;
| | 12/11
&lt;/td&gt;
| | 696.474
        &lt;td&gt;Doubly augmented second (C-Dx)&lt;br /&gt;
|-
&lt;/td&gt;
| | 5/4
        &lt;td&gt;Doubly diminished fourth (C-Fbb)&lt;br /&gt;
| | 696.578 (minimax tuning)
&lt;/td&gt;
|-
    &lt;/tr&gt;
| | 11/9
    &lt;tr&gt;
| | 696.839
        &lt;td&gt;14/11&lt;br /&gt;
|-
&lt;/td&gt;
| | 8/7
        &lt;td&gt;Diminished fourth (C-Fb), same as 9/7&lt;br /&gt;
| | 696.883
&lt;/td&gt;
|-
        &lt;td&gt;Triply augmented second (C-Dx#)&lt;br /&gt;
| | 7/5
&lt;/td&gt;
| | 697.085
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| | 4/3
        &lt;td&gt;11/8&lt;br /&gt;
| | 701.955
&lt;/td&gt;
|}
        &lt;td&gt;Doubly augmented third (C-Ex)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Doubly diminished fifth (C-Gbb)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;16/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Doubly diminished sixth (A-Fbb)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Doubly augmented fourth (C-Fx)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Augmented fifth (C-G#), same as 14/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Triply diminished seventh (A-Gbbb)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;18/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Doubly diminished seventh (A-Gbb)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Doubly augmented fifth (C-Gx)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;20/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Doubly diminished octave (C-Cbb)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Doubly augmented sixth (C-Ax)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11/6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Doubly augmented sixth (C-Ax)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Double diminished octave (C-Cbb)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&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 Undecimal Meantone Tunings by Eigenmonzos"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Spectrum of Undecimal Meantone Tunings by Eigenmonzos&lt;/h2&gt;
 
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;th&gt;Eigenmonzo&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Fifth&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;10/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;691.202&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;694.786&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;695.614&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;696.319&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;696.578&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;696.713 (minimax tuning)&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;696.883&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;697.021&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;697.085&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;697.295&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;697.500&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;697.812&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;701.955&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 Meanpop Tunings by Eigenmonzos"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Spectrum of Meanpop Tunings by Eigenmonzos&lt;/h2&gt;
 
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;th&gt;Eigenmonzo&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Fifth&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;10/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;691.202&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;694.786&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;695.614&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;696.052&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;696.176&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;696.319&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;696.413&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;696.474&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;696.578 (minimax tuning)&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;696.839&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;696.883&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;697.085&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;701.955&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;/body&gt;&lt;/html&gt;</pre></div>

Revision as of 00:00, 17 July 2018

"11-limit meantone" and "meanpop", both discussed at Meantone family, are two different temperaments in the 11 limit. This page compares and contrasts them in detail.

Extending meantone from the 5 limit to the 7 limit, there is one obvious mapping that is not too complex and adds hardly any additional error (so we're not talking about dominant temperament here). This is called "7-limit meantone" or "septimal meantone" and is an amazingly efficient (and beautiful) temperament. But extending it from the 7 limit to the 11 limit is not so simple. There are two mappings that are comparable in complexity and error: 11-limit meantone and meanpop.

In 11-limit meantone, 11/8 is represented by the doubly augmented third, for example C-Ex (where "x" represents the standard double sharp symbol, equivalent in meaning to "##"). This is 18 fifths along the circle of fifths; Ex is 18 fifths up from C.

In meanpop, 11/8 is represented by the doubly diminished fifth, for example C-Gbb. This is in the opposite direction along the circle of fifths - 13 fifths down.

Can meantone and meanpop be combined into a single temperament? Yes! It works wonderfully and that temperament is 31edo. In 31edo the circle of fifths closes perfectly after 31 fifths, so Ex and Gbb are the same note. (In other words, the interval of the quadruply diminished third is tuned to 0 cents, if that makes any sense to you.) This makes everything much simpler and results in 121/120 and 243/242 being tempered out, so that 12/11~11/10 is a "neutral second" (exactly half of a minor third), and 11/9 is a "neutral third" (exactly half of a perfect fifth). Keep in mind that neither of these things are true in either meantone or meanpop.

JI interval Meantone mapping Meanpop mapping
12/11 Doubly diminished third (A-Cbb) Doubly augmented prime (C-Cx)
11/10 Doubly augmented prime (C-Cx) Doubly diminished third (A-Cbb)
11/9 Doubly augmented second (C-Dx) Doubly diminished fourth (C-Fbb)
14/11 Diminished fourth (C-Fb), same as 9/7 Triply augmented second (C-Dx#)
11/8 Doubly augmented third (C-Ex) Doubly diminished fifth (C-Gbb)
16/11 Doubly diminished sixth (A-Fbb) Doubly augmented fourth (C-Fx)
11/7 Augmented fifth (C-G#), same as 14/9 Triply diminished seventh (A-Gbbb)
18/11 Doubly diminished seventh (A-Gbb) Doubly augmented fifth (C-Gx)
20/11 Doubly diminished octave (C-Cbb) Doubly augmented sixth (C-Ax)
11/6 Doubly augmented sixth (C-Ax) Double diminished octave (C-Cbb)

Tuning Spectra

Spectrum of Undecimal Meantone Tunings by Eigenmonzos

Eigenmonzo Fifth
10/9 691.202
6/5 694.786
9/7 695.614
7/6 696.319
5/4 696.578
11/9 696.713 (minimax tuning)
8/7 696.883
12/11 697.021
7/5 697.085
11/8 697.295
11/10 697.500
14/11 697.812
4/3 701.955

Spectrum of Meanpop Tunings by Eigenmonzos

Eigenmonzo Fifth
10/9 691.202
6/5 694.786
9/7 695.614
11/8 696.052
11/10 696.176
7/6 696.319
14/11 696.413
12/11 696.474
5/4 696.578 (minimax tuning)
11/9 696.839
8/7 696.883
7/5 697.085
4/3 701.955
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