Huygens vs meanpop: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Breadcrumb|Meantone}}
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.
{{Wikipedia|Septimal meantone temperament #11-limit meantone}}


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.
'''Undecimal meantone''' (also known as '''huygens''') and '''meanpop''', both discussed at [[meantone family]], are two different temperaments in the [[11-limit]]. This page compares and contrasts them in detail.


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.
Extending meantone from the [[5-limit]] to the [[7-limit]], there is one obvious mapping (for standard meantone tunings) which does not split the fifth that is not too complex and adds hardly any additional error (so we are not talking about [[dominant (temperament)|dominant]] 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: huygens (12 & 31) and meanpop (19 & 31).


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.
In 11-limit huygens, 11/8 is represented by the doubly augmented third, for example C–E𝄪. This is 18 fifths along the [[chain of fifths]]; E𝄪 is 18 fifths up from C. Huygens is tuned best sharp of 31edo, around 697 cents.


||~ JI interval ||~ Meantone mapping ||~ Meanpop mapping ||
In meanpop, 11/8 is represented by the doubly diminished fifth, for example C–G𝄫. This is in the opposite direction along the circle of fifths – 13 fifths down. Meanpop is tuned best flat of 31edo, around 696 cents.
|| 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=
In the [[13-limit]], meanpop extends by [[105/104]], whereas meantone forks into fokkertone, grosstone, and meridetone.  
==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==
Can huygens 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 E𝄪 and G𝄫 are the same note. (In other words, the interval of the ''quadruply diminished third'' is tuned to 0 cents, setting a minor third equal to four chromatic semitones. Expressed in tempered fifths and octave-reduced, this interval is the [[31-comma]] {{monzo| -49 31 }}, which is the 3-limit comma tempered out in 31edo.) This makes everything much simpler and results in [[121/120]] and [[243/242]] being tempered out, so that 12/11~11/10 is a true neutral second (exactly half of a minor third), and 11/9 is a true neutral third (exactly half of a perfect fifth). Keep in mind that neither of these things are true in either huygens or meanpop.
||~ 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>
== Interval chain ==
<h4>Original HTML content:</h4>
{| class="wikitable center-1 right-2"
<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;
|-
&lt;br /&gt;
! rowspan="3" | #
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;
! rowspan="3" | Cents*
&lt;br /&gt;
! colspan="3" | Approximate ratios
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;
|-
&lt;br /&gt;
! rowspan="2" | 7-limit
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;
! colspan="2" | 11-limit extensions
&lt;br /&gt;
|-
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;
! Meantone
&lt;br /&gt;
! Meanpop
|-
| 0
| 0.0
| '''1/1'''
|
|
|-
| 1
| 696.7
| '''3/2'''
|
|
|-
| 2
| 193.3
| '''9/8''', 10/9, 28/25
|
|
|-
| 3
| 890.0
| 5/3
|
|
|-
| 4
| 386.6
| '''5/4'''
|
|
|-
| 5
| 1083.3
| '''15/8''', 28/15
|
|
|-
| 6
| 579.9
| 7/5, 25/18
|
|
|-
| 7
| 76.6
| 21/20, 25/24, 28/27
| 22/21
|
|-
| 8
| 773.2
| 14/9, 25/16
| 11/7
|
|-
| 9
| 269.9
| 7/6
|
|
|-
| 10
| 966.6
| '''7/4'''
|
|
|-
| 11
| 463.2
| 21/16
|
|
|-
| 12
| 1159.9
| 35/18, 49/25, 63/32
| 55/28, 88/45
| 64/33
|-
| 13
| 656.5
| 35/24
| 22/15
| '''16/11'''
|-
| 14
| 153.2
| 35/32
| 11/10
| 12/11
|-
| 15
| 849.8
| 49/30
| 33/20, 44/27
| 18/11
|-
| 16
| 346.5
| 49/40
| 11/9
| 27/22, 40/33
|-
| 17
| 1043.2
| 49/27
| 11/6
| 20/11
|-
| 18
| 539.8
| 49/36
| '''11/8'''
| 15/11
|-
| 19
| 36.5
| 49/48
| 33/32
| 45/44, 56/55
|}
<nowiki/>* In 7-limit [[CWE]] tuning, octave reduced


== Selected intervals ==
{| class="wikitable center-3 center-5"
! rowspan="2" | JI interval
! colspan="2" | Huygens mapping
! colspan="2" | Meanpop mapping
|-
! Nominals
! Fifth steps
! Nominals
! Fifth steps
|-
| 33/32
| Doubly augmented seventh minus an octave (C–B𝄪)
| +19
| Diminished second (C–D𝄫)
| -12
|-
| 22/21
| Augmented unison (C–C♯), same as 25/24
| +7
| Triply diminished third (C–E𝄫𝄫)
| -24
|-
| 12/11
| Doubly diminished third (C–E𝄫♭)
| -17
| Doubly augmented unison (C–C𝄪)
| +14
|-
| 11/10
| Doubly augmented unison (C–C𝄪)
| +14
| Doubly diminished third (C–E𝄫♭)
| -17
|-
| 112/99
| Diminished third (C–E𝄫), same as 8/7
| -10
| Triply augmented unison (C–C𝄪♯)
| +21
|-
| 33/28
| Augmented second (C–D♯), same as 7/6
| +9
| Triply diminished fourth (C–F𝄫♭)
| -22
|-
| 27/22, 40/33
| Doubly diminished fourth (C–F𝄫)
| -15
| Doubly augmented second (C–D𝄪)
| +16
|-
| 11/9
| Doubly augmented second (C–D𝄪)
| +16
| Doubly diminished fourth (C–F𝄫)
| -15
|-
| 14/11
| Diminished fourth (C–F♭), same as 9/7
| -8
| Triply augmented second (C–D𝄪♯)
| +23
|-
| 15/11
| Doubly diminished fifth (C–G𝄫)
| -13
| Doubly augmented third (C–E𝄪)
| +18
|-
| 11/8
| Doubly augmented third (C–E𝄪)
| +18
| Doubly diminished fifth (C–G𝄫)
| -13
|-
| 16/11
| Doubly diminished sixth (C–A𝄫♭)
| -18
| Doubly augmented fourth (C–F𝄪)
| +13
|-
| 22/15
| Doubly augmented fourth (C–F𝄪)
| +13
| Doubly diminished sixth (C–A𝄫♭)
| -18
|-
| 11/7
| Augmented fifth (C–G♯), same as 14/9
| +8
| Triply diminished seventh (C–B𝄫𝄫)
| -23
|-
| 18/11
| Doubly diminished seventh (C–B𝄫♭)
| -16
| Doubly augmented fifth (C–G𝄪)
| +15
|-
| 33/20, 44/27
| Doubly augmented fifth (C–G𝄪)
| +15
| Doubly diminished seventh (C–B𝄫♭)
| -16
|-
| 56/33
| Diminished seventh (C–B𝄫), same as 12/7
| -9
| Triply augmented fifth (C–G𝄪♯)
|  +22
|-
| 99/56
| Augmented sixth (C–A♯), same as 7/4
| +10
| Triply diminished octave (C–C𝄫♭)
| -21
|-
| 20/11
| Doubly diminished octave (C–C𝄫)
| -14
| Doubly augmented sixth (C–A𝄪)
| +17
|-
| 11/6
| Doubly augmented sixth (C–A𝄪)
| +17
| Doubly diminished octave (C–C𝄫)
| -14
|-
| 21/11
| Diminished octave (C–C♭), same as 48/25
| -7
| Triply augmented sixth (C–A𝄪♯)
| +24
|-
| 64/33
| Doubly diminished ninth (C–D𝄫♭)
| -19
| Augmented seventh (C–B♯)
| +12
|}


&lt;table class="wiki_table"&gt;
== Tuning spectra ==
    &lt;tr&gt;
=== Undecimal meantone ===
        &lt;th&gt;JI interval&lt;br /&gt;
{| class="wikitable center-all left-3"
&lt;/th&gt;
|-
        &lt;th&gt;Meantone mapping&lt;br /&gt;
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]
&lt;/th&gt;
! Generator (¢)
        &lt;th&gt;Meanpop mapping&lt;br /&gt;
! Comments
&lt;/th&gt;
|-
    &lt;/tr&gt;
| 10/9
    &lt;tr&gt;
| 691.202
        &lt;td&gt;12/11&lt;br /&gt;
|
&lt;/td&gt;
|-
        &lt;td&gt;Doubly diminished third (A-Cbb)&lt;br /&gt;
| 6/5
&lt;/td&gt;
| 694.786
        &lt;td&gt;Doubly augmented prime (C-Cx)&lt;br /&gt;
|
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 9/7
    &lt;tr&gt;
| 695.614
        &lt;td&gt;11/10&lt;br /&gt;
|
&lt;/td&gt;
|-
        &lt;td&gt;Doubly augmented prime (C-Cx)&lt;br /&gt;
| 15/14
&lt;/td&gt;
| 696.111
        &lt;td&gt;Doubly diminished third (A-Cbb)&lt;br /&gt;
|
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 7/6
    &lt;tr&gt;
| 696.319
        &lt;td&gt;11/9&lt;br /&gt;
|
&lt;/td&gt;
|-
        &lt;td&gt;Doubly augmented second (C-Dx)&lt;br /&gt;
| 5/4
&lt;/td&gt;
| 696.578
        &lt;td&gt;Doubly diminished fourth (C-Fbb)&lt;br /&gt;
| 5, 7, 9-odd-limit minimax
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 11/9
    &lt;tr&gt;
| 696.713
        &lt;td&gt;14/11&lt;br /&gt;
| 11-odd-limit minimax
&lt;/td&gt;
|-
        &lt;td&gt;Diminished fourth (C-Fb), same as 9/7&lt;br /&gt;
| 8/7
&lt;/td&gt;
| 696.883
        &lt;td&gt;Triply augmented second (C-Dx#)&lt;br /&gt;
|
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 12/11
    &lt;tr&gt;
| 697.021
        &lt;td&gt;11/8&lt;br /&gt;
|
&lt;/td&gt;
|-
        &lt;td&gt;Doubly augmented third (C-Ex)&lt;br /&gt;
| 7/5
&lt;/td&gt;
| 697.085
        &lt;td&gt;Doubly diminished fifth (C-Gbb)&lt;br /&gt;
|
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 15/11
    &lt;tr&gt;
| 697.158
        &lt;td&gt;16/11&lt;br /&gt;
|
&lt;/td&gt;
|-
        &lt;td&gt;Doubly diminished sixth (A-Fbb)&lt;br /&gt;
| 27/22
&lt;/td&gt;
| 697.159
        &lt;td&gt;Doubly augmented fourth (C-Fx)&lt;br /&gt;
|
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 22/21
    &lt;tr&gt;
| 697.22
        &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;
| 11/8
&lt;/td&gt;
| 697.295
        &lt;td&gt;Triply diminished seventh (A-Gbbb)&lt;br /&gt;
|
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 21/16
    &lt;tr&gt;
| 697.344
        &lt;td&gt;18/11&lt;br /&gt;
|
&lt;/td&gt;
|-
        &lt;td&gt;Doubly diminished seventh (A-Gbb)&lt;br /&gt;
| 11/10
&lt;/td&gt;
| 697.5
        &lt;td&gt;Doubly augmented fifth (C-Gx)&lt;br /&gt;
|
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 16/15
    &lt;tr&gt;
| 697.654
        &lt;td&gt;20/11&lt;br /&gt;
|
&lt;/td&gt;
|-
        &lt;td&gt;Doubly diminished octave (C-Cbb)&lt;br /&gt;
| 40/33
&lt;/td&gt;
| 697.797
        &lt;td&gt;Doubly augmented sixth (C-Ax)&lt;br /&gt;
|
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 14/11
    &lt;tr&gt;
| 697.812
        &lt;td&gt;11/6&lt;br /&gt;
|
&lt;/td&gt;
|-
        &lt;td&gt;Doubly augmented sixth (C-Ax)&lt;br /&gt;
| 33/28
&lt;/td&gt;
| 698.272
        &lt;td&gt;Double diminished octave (C-Cbb)&lt;br /&gt;
|
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 112/99
&lt;/table&gt;
| 698.640
|
|-
| 4/3
| 701.955
|
|}


&lt;br /&gt;
==== Fokkertone ====
&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;
{| class="wikitable center-all left-3"
&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;
|-
! Eigenmonzo<br>(unchanged-interval)
! Generator (¢)
! Comments
|-
| 10/9
| 691.202
|
|-
| 14/13
| 694.340
|
|-
| 18/13
| 695.124
|
|-
| 15/13
| 695.226
|
|-
| 39/28
| 695.609
|
|-
| 13/12
| 695.612
|
|-
| 13/10
| 695.838
|
|-
| 16/13
| 696.035
|
|-
| 39/32
| 696.405
|
|-
| 5/4
| 696.578
| 5, 7, 9-odd-limit minimax
|-
| 11/9
| 696.713
| 11, 13, 15-odd-limit minimax
|-
| 4/3
| 701.955
|
|-
| 33/26
| 703.186
|
|-
| 13/11
| 703.597
|
|}


&lt;table class="wiki_table"&gt;
==== Grosstone ====
    &lt;tr&gt;
{| class="wikitable center-all left-3"
        &lt;th&gt;Eigenmonzo&lt;br /&gt;
|-
&lt;/th&gt;
! Eigenmonzo<br>(unchanged-interval)
        &lt;th&gt;Fifth&lt;br /&gt;
! Generator (¢)
&lt;/th&gt;
! Comments
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 10/9
        &lt;td&gt;10/9&lt;br /&gt;
| 691.202
&lt;/td&gt;
|
        &lt;td&gt;691.202&lt;br /&gt;
|-
&lt;/td&gt;
| 33/26
    &lt;/tr&gt;
| 693.178
    &lt;tr&gt;
|
        &lt;td&gt;6/5&lt;br /&gt;
|-
&lt;/td&gt;
| 5/4
        &lt;td&gt;694.786&lt;br /&gt;
| 696.578
&lt;/td&gt;
| 5, 7, 9-odd-limit minimax
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 11/9
        &lt;td&gt;9/7&lt;br /&gt;
| 696.713
&lt;/td&gt;
| 11-odd-limit minimax
        &lt;td&gt;695.614&lt;br /&gt;
|-
&lt;/td&gt;
| 39/32
    &lt;/tr&gt;
| 697.168
    &lt;tr&gt;
|
        &lt;td&gt;7/6&lt;br /&gt;
|-
&lt;/td&gt;
| 14/13
        &lt;td&gt;696.319&lt;br /&gt;
| 697.242
&lt;/td&gt;
| 13, 15-odd-limit minimax
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 13/10
        &lt;td&gt;5/4&lt;br /&gt;
| 697.289
&lt;/td&gt;
|
        &lt;td&gt;696.578&lt;br /&gt;
|-
&lt;/td&gt;
| 13/11
    &lt;/tr&gt;
| 697.376
    &lt;tr&gt;
|
        &lt;td&gt;11/9&lt;br /&gt;
|-
&lt;/td&gt;
| 16/13
        &lt;td&gt;696.713 (minimax tuning)&lt;br /&gt;
| 697.467
&lt;/td&gt;
|
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 15/13
        &lt;td&gt;8/7&lt;br /&gt;
| 697.511
&lt;/td&gt;
|
        &lt;td&gt;696.883&lt;br /&gt;
|-
&lt;/td&gt;
| 13/12
    &lt;/tr&gt;
| 697.731
    &lt;tr&gt;
|
        &lt;td&gt;12/11&lt;br /&gt;
|-
&lt;/td&gt;
| 18/13
        &lt;td&gt;697.021&lt;br /&gt;
| 697.966
&lt;/td&gt;
|
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 4/3
        &lt;td&gt;7/5&lt;br /&gt;
| 701.955
&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;
==== Meridetone ====
&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;
{| class="wikitable center-all left-3"
|-
! Eigenmonzo<br>(unchanged-interval)
! Generator (¢)
! Comments
|-
| 10/9
| 691.202
|
|-
| 5/4
| 696.578
| 5, 7, 9-odd-limit minimax
|-
| 11/9
| 696.713
| 11-odd-limit minimax
|-
| 18/13
| 697.465
| 13, 15-odd-limit minimax
|-
| 13/12
| 697.637
|
|-
| 16/13
| 697.797
|
|-
| 15/13
| 697.83
|
|-
| 39/32
| 697.946
|
|-
| 13/10
| 698.009
|
|-
| 14/13
| 698.335
|
|-
| 33/26
| 698.407
|
|-
| 13/11
| 698.801
|
|-
| 4/3
| 701.955
|
|}


&lt;table class="wiki_table"&gt;
=== Meanpop ===
    &lt;tr&gt;
{| class="wikitable center-all left-3"
        &lt;th&gt;Eigenmonzo&lt;br /&gt;
|-
&lt;/th&gt;
! Eigenmonzo<br>(unchanged-interval)
        &lt;th&gt;Fifth&lt;br /&gt;
! Generator (¢)
&lt;/th&gt;
! Comments
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 10/9
        &lt;td&gt;10/9&lt;br /&gt;
| 691.202
&lt;/td&gt;
|
        &lt;td&gt;691.202&lt;br /&gt;
|-
&lt;/td&gt;
| 6/5
    &lt;/tr&gt;
| 694.786
    &lt;tr&gt;
|
        &lt;td&gt;6/5&lt;br /&gt;
|-
&lt;/td&gt;
| 9/7
        &lt;td&gt;694.786&lt;br /&gt;
| 695.614
&lt;/td&gt;
|
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 40/33
        &lt;td&gt;9/7&lt;br /&gt;
| 695.815
&lt;/td&gt;
|
        &lt;td&gt;695.614&lt;br /&gt;
|-
&lt;/td&gt;
| 112/99
    &lt;/tr&gt;
| 695.886
    &lt;tr&gt;
|
        &lt;td&gt;11/8&lt;br /&gt;
|-
&lt;/td&gt;
| 11/8
        &lt;td&gt;696.052&lt;br /&gt;
| 696.052
&lt;/td&gt;
|
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 15/14
        &lt;td&gt;11/10&lt;br /&gt;
| 696.111
&lt;/td&gt;
|
        &lt;td&gt;696.176&lt;br /&gt;
|-
&lt;/td&gt;
| 11/10
    &lt;/tr&gt;
| 696.176
    &lt;tr&gt;
|
        &lt;td&gt;7/6&lt;br /&gt;
|-
&lt;/td&gt;
| 7/6
        &lt;td&gt;696.319&lt;br /&gt;
| 696.319
&lt;/td&gt;
|
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 27/22
        &lt;td&gt;14/11&lt;br /&gt;
| 696.3635
&lt;/td&gt;
|
        &lt;td&gt;696.413&lt;br /&gt;
|-
&lt;/td&gt;
| 14/11
    &lt;/tr&gt;
| 696.413
    &lt;tr&gt;
|
        &lt;td&gt;12/11&lt;br /&gt;
|-
&lt;/td&gt;
| 12/11
        &lt;td&gt;696.474&lt;br /&gt;
| 696.474
&lt;/td&gt;
|
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 15/11
        &lt;td&gt;5/4&lt;br /&gt;
| 696.497
&lt;/td&gt;
|
        &lt;td&gt;696.578 (minimax tuning)&lt;br /&gt;
|-
&lt;/td&gt;
| 5/4
    &lt;/tr&gt;
| 696.578
    &lt;tr&gt;
| 5, 7, 9, 11-odd-limit minimax
        &lt;td&gt;11/9&lt;br /&gt;
|-
&lt;/td&gt;
| 11/9
        &lt;td&gt;696.839&lt;br /&gt;
| 696.839
&lt;/td&gt;
|
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 8/7
        &lt;td&gt;8/7&lt;br /&gt;
| 696.883
&lt;/td&gt;
|
        &lt;td&gt;696.883&lt;br /&gt;
|-
&lt;/td&gt;
| 7/5
    &lt;/tr&gt;
| 697.085
    &lt;tr&gt;
|
        &lt;td&gt;7/5&lt;br /&gt;
|-
&lt;/td&gt;
| 16/15
        &lt;td&gt;697.085&lt;br /&gt;
| 697.654
&lt;/td&gt;
|
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 4/3
        &lt;td&gt;4/3&lt;br /&gt;
| 701.955
&lt;/td&gt;
|
        &lt;td&gt;701.955&lt;br /&gt;
|-
&lt;/td&gt;
| 22/21
    &lt;/tr&gt;
| 703.356
&lt;/table&gt;
|
|}


&lt;/body&gt;&lt;/html&gt;</pre></div>
==== Tridecimal meanpop ====
{| class="wikitable center-all left-3"
|-
! Eigenmonzo<br>(unchanged-interval)
! Generator (¢)
! Comments
|-
| 10/9
| 691.202
|
|-
| 14/13
| 694.340
|
|-
| 18/13
| 695.124
|
|-
| 15/13
| 695.226
|
|-
| 39/28
| 695.609
|
|-
| 13/12
| 695.612
|
|-
| 33/26
| 695.824
|
|-
| 13/10
| 695.838
|
|-
| 16/13
| 696.035
|
|-
| 13/11
| 696.043
| 13 and 15-odd-limit minimax
|-
| 39/32
| 696.405
|
|-
| 5/4
| 696.578
| 5, 7, 9 and 11-odd-limit minimax
|-
| 4/3
| 701.955
|
|}
 
<!-- formerly known as meanplop
==== Meanpop variant ====
{| class="wikitable center-all left-3"
|-
! Eigenmonzo<br>(unchanged-interval)
! Generator (¢)
! Comments
|-
| 16/13
| 689.868
|
|-
| 10/9
| 691.202
|
|-
| 13/12
| 692.285
|
|-
| 13/10
| 693.223
|
|-
| 18/13
| 693.897
|
|-
| 15/13
| 694.193
|
|-
| 14/13
| 694.878
|
|-
| 11/8
| 696.052
| 13 and 15-odd-limit minimax
|-
| 5/4
| 696.578
| 5, 7, 9 and 11-odd-limit minimax
|-
| 33/26
| 698.407
|
|-
| 13/11
| 698.801
|
|-
| 4/3
| 701.955
|
|}
-->
 
[[Category:Meantone]]
[[Category:Temperament extensions]]

Latest revision as of 13:15, 24 December 2025

< Meantone
English Wikipedia has an article on:

Undecimal meantone (also known as huygens) 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 (for standard meantone tunings) which does not split the fifth that is not too complex and adds hardly any additional error (so we are not talking about dominant 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: huygens (12 & 31) and meanpop (19 & 31).

In 11-limit huygens, 11/8 is represented by the doubly augmented third, for example C–E𝄪. This is 18 fifths along the chain of fifths; E𝄪 is 18 fifths up from C. Huygens is tuned best sharp of 31edo, around 697 cents.

In meanpop, 11/8 is represented by the doubly diminished fifth, for example C–G𝄫. This is in the opposite direction along the circle of fifths – 13 fifths down. Meanpop is tuned best flat of 31edo, around 696 cents.

In the 13-limit, meanpop extends by 105/104, whereas meantone forks into fokkertone, grosstone, and meridetone.

Can huygens 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 E𝄪 and G𝄫 are the same note. (In other words, the interval of the quadruply diminished third is tuned to 0 cents, setting a minor third equal to four chromatic semitones. Expressed in tempered fifths and octave-reduced, this interval is the 31-comma [-49 31⟩, which is the 3-limit comma tempered out in 31edo.) This makes everything much simpler and results in 121/120 and 243/242 being tempered out, so that 12/11~11/10 is a true neutral second (exactly half of a minor third), and 11/9 is a true neutral third (exactly half of a perfect fifth). Keep in mind that neither of these things are true in either huygens or meanpop.

Interval chain

# Cents* Approximate ratios
7-limit 11-limit extensions
Meantone Meanpop
0 0.0 1/1
1 696.7 3/2
2 193.3 9/8, 10/9, 28/25
3 890.0 5/3
4 386.6 5/4
5 1083.3 15/8, 28/15
6 579.9 7/5, 25/18
7 76.6 21/20, 25/24, 28/27 22/21
8 773.2 14/9, 25/16 11/7
9 269.9 7/6
10 966.6 7/4
11 463.2 21/16
12 1159.9 35/18, 49/25, 63/32 55/28, 88/45 64/33
13 656.5 35/24 22/15 16/11
14 153.2 35/32 11/10 12/11
15 849.8 49/30 33/20, 44/27 18/11
16 346.5 49/40 11/9 27/22, 40/33
17 1043.2 49/27 11/6 20/11
18 539.8 49/36 11/8 15/11
19 36.5 49/48 33/32 45/44, 56/55

* In 7-limit CWE tuning, octave reduced

Selected intervals

JI interval Huygens mapping Meanpop mapping
Nominals Fifth steps Nominals Fifth steps
33/32 Doubly augmented seventh minus an octave (C–B𝄪) +19 Diminished second (C–D𝄫) -12
22/21 Augmented unison (C–C♯), same as 25/24 +7 Triply diminished third (C–E𝄫𝄫) -24
12/11 Doubly diminished third (C–E𝄫♭) -17 Doubly augmented unison (C–C𝄪) +14
11/10 Doubly augmented unison (C–C𝄪) +14 Doubly diminished third (C–E𝄫♭) -17
112/99 Diminished third (C–E𝄫), same as 8/7 -10 Triply augmented unison (C–C𝄪♯) +21
33/28 Augmented second (C–D♯), same as 7/6 +9 Triply diminished fourth (C–F𝄫♭) -22
27/22, 40/33 Doubly diminished fourth (C–F𝄫) -15 Doubly augmented second (C–D𝄪) +16
11/9 Doubly augmented second (C–D𝄪) +16 Doubly diminished fourth (C–F𝄫) -15
14/11 Diminished fourth (C–F♭), same as 9/7 -8 Triply augmented second (C–D𝄪♯) +23
15/11 Doubly diminished fifth (C–G𝄫) -13 Doubly augmented third (C–E𝄪) +18
11/8 Doubly augmented third (C–E𝄪) +18 Doubly diminished fifth (C–G𝄫) -13
16/11 Doubly diminished sixth (C–A𝄫♭) -18 Doubly augmented fourth (C–F𝄪) +13
22/15 Doubly augmented fourth (C–F𝄪) +13 Doubly diminished sixth (C–A𝄫♭) -18
11/7 Augmented fifth (C–G♯), same as 14/9 +8 Triply diminished seventh (C–B𝄫𝄫) -23
18/11 Doubly diminished seventh (C–B𝄫♭) -16 Doubly augmented fifth (C–G𝄪) +15
33/20, 44/27 Doubly augmented fifth (C–G𝄪) +15 Doubly diminished seventh (C–B𝄫♭) -16
56/33 Diminished seventh (C–B𝄫), same as 12/7 -9 Triply augmented fifth (C–G𝄪♯) +22
99/56 Augmented sixth (C–A♯), same as 7/4 +10 Triply diminished octave (C–C𝄫♭) -21
20/11 Doubly diminished octave (C–C𝄫) -14 Doubly augmented sixth (C–A𝄪) +17
11/6 Doubly augmented sixth (C–A𝄪) +17 Doubly diminished octave (C–C𝄫) -14
21/11 Diminished octave (C–C♭), same as 48/25 -7 Triply augmented sixth (C–A𝄪♯) +24
64/33 Doubly diminished ninth (C–D𝄫♭) -19 Augmented seventh (C–B♯) +12

Tuning spectra

Undecimal meantone

Eigenmonzo
(unchanged-interval) 		Generator (¢) Comments
10/9 691.202
6/5 694.786
9/7 695.614
15/14 696.111
7/6 696.319
5/4 696.578 5, 7, 9-odd-limit minimax
11/9 696.713 11-odd-limit minimax
8/7 696.883
12/11 697.021
7/5 697.085
15/11 697.158
27/22 697.159
22/21 697.22
11/8 697.295
21/16 697.344
11/10 697.5
16/15 697.654
40/33 697.797
14/11 697.812
33/28 698.272
112/99 698.640
4/3 701.955

Fokkertone

Eigenmonzo
(unchanged-interval) 		Generator (¢) Comments
10/9 691.202
14/13 694.340
18/13 695.124
15/13 695.226
39/28 695.609
13/12 695.612
13/10 695.838
16/13 696.035
39/32 696.405
5/4 696.578 5, 7, 9-odd-limit minimax
11/9 696.713 11, 13, 15-odd-limit minimax
4/3 701.955
33/26 703.186
13/11 703.597

Grosstone

Eigenmonzo
(unchanged-interval) 		Generator (¢) Comments
10/9 691.202
33/26 693.178
5/4 696.578 5, 7, 9-odd-limit minimax
11/9 696.713 11-odd-limit minimax
39/32 697.168
14/13 697.242 13, 15-odd-limit minimax
13/10 697.289
13/11 697.376
16/13 697.467
15/13 697.511
13/12 697.731
18/13 697.966
4/3 701.955

Meridetone

Eigenmonzo
(unchanged-interval) 		Generator (¢) Comments
10/9 691.202
5/4 696.578 5, 7, 9-odd-limit minimax
11/9 696.713 11-odd-limit minimax
18/13 697.465 13, 15-odd-limit minimax
13/12 697.637
16/13 697.797
15/13 697.83
39/32 697.946
13/10 698.009
14/13 698.335
33/26 698.407
13/11 698.801
4/3 701.955

Meanpop

Eigenmonzo
(unchanged-interval) 		Generator (¢) Comments
10/9 691.202
6/5 694.786
9/7 695.614
40/33 695.815
112/99 695.886
11/8 696.052
15/14 696.111
11/10 696.176
7/6 696.319
27/22 696.3635
14/11 696.413
12/11 696.474
15/11 696.497
5/4 696.578 5, 7, 9, 11-odd-limit minimax
11/9 696.839
8/7 696.883
7/5 697.085
16/15 697.654
4/3 701.955
22/21 703.356

Tridecimal meanpop

Eigenmonzo
(unchanged-interval) 		Generator (¢) Comments
10/9 691.202
14/13 694.340
18/13 695.124
15/13 695.226
39/28 695.609
13/12 695.612
33/26 695.824
13/10 695.838
16/13 696.035
13/11 696.043 13 and 15-odd-limit minimax
39/32 696.405
5/4 696.578 5, 7, 9 and 11-odd-limit minimax
4/3 701.955
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