Huygens vs meanpop
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"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 ||
Original HTML content:
<html><head><title>Meantone vs meanpop</title></head><body>"11-limit meantone" and "meanpop", both discussed at <a class="wiki_link" href="/Meantone%20family">Meantone family</a>, are two different temperaments in the 11 limit. This page compares and contrasts them in detail.<br />
<br />
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.<br />
<br />
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.<br />
<br />
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.<br />
<br />
Can meantone and meanpop be combined into a single temperament? Yes! It works wonderfully and that temperament is <a class="wiki_link" href="/31edo">31edo</a>. 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 <em>quadruply diminished third</em> 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.<br />
<br />
<table class="wiki_table">
<tr>
<th>JI interval<br />
</th>
<th>Meantone mapping<br />
</th>
<th>Meanpop mapping<br />
</th>
</tr>
<tr>
<td>12/11<br />
</td>
<td>Doubly diminished third (A-Cbb)<br />
</td>
<td>Doubly augmented prime (C-Cx)<br />
</td>
</tr>
<tr>
<td>11/10<br />
</td>
<td>Doubly augmented prime (C-Cx)<br />
</td>
<td>Doubly diminished third (A-Cbb)<br />
</td>
</tr>
<tr>
<td>11/9<br />
</td>
<td>Doubly augmented second (C-Dx)<br />
</td>
<td>Doubly diminished fourth (C-Fbb)<br />
</td>
</tr>
<tr>
<td>14/11<br />
</td>
<td>Diminished fourth (C-Fb), same as 9/7<br />
</td>
<td>Triply augmented second (C-Dx#)<br />
</td>
</tr>
<tr>
<td>11/8<br />
</td>
<td>Doubly augmented third (C-Ex)<br />
</td>
<td>Doubly diminished fifth (C-Gbb)<br />
</td>
</tr>
<tr>
<td>16/11<br />
</td>
<td>Doubly diminished sixth (A-Fbb)<br />
</td>
<td>Doubly augmented fourth (C-Fx)<br />
</td>
</tr>
<tr>
<td>11/7<br />
</td>
<td>Augmented fifth (C-G#), same as 14/9<br />
</td>
<td>Triply diminished seventh (A-Gbbb)<br />
</td>
</tr>
<tr>
<td>18/11<br />
</td>
<td>Doubly diminished seventh (A-Gbb)<br />
</td>
<td>Doubly augmented fifth (C-Gx)<br />
</td>
</tr>
<tr>
<td>20/11<br />
</td>
<td>Doubly diminished octave (C-Cbb)<br />
</td>
<td>Doubly augmented sixth (C-Ax)<br />
</td>
</tr>
<tr>
<td>11/6<br />
</td>
<td>Doubly augmented sixth (C-Ax)<br />
</td>
<td>Double diminished octave (C-Cbb)<br />
</td>
</tr>
</table>
<br />
<!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Tuning Spectra"></a><!-- ws:end:WikiTextHeadingRule:0 -->Tuning Spectra</h1>
<!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="Tuning Spectra-Spectrum of Undecimal Meantone Tunings by Eigenmonzos"></a><!-- ws:end:WikiTextHeadingRule:2 -->Spectrum of Undecimal Meantone Tunings by Eigenmonzos</h2>
<table class="wiki_table">
<tr>
<th>Eigenmonzo<br />
</th>
<th>Fifth<br />
</th>
</tr>
<tr>
<td>10/9<br />
</td>
<td>691.202<br />
</td>
</tr>
<tr>
<td>6/5<br />
</td>
<td>694.786<br />
</td>
</tr>
<tr>
<td>9/7<br />
</td>
<td>695.614<br />
</td>
</tr>
<tr>
<td>7/6<br />
</td>
<td>696.319<br />
</td>
</tr>
<tr>
<td>5/4<br />
</td>
<td>696.578<br />
</td>
</tr>
<tr>
<td>11/9<br />
</td>
<td>696.713 (minimax tuning)<br />
</td>
</tr>
<tr>
<td>8/7<br />
</td>
<td>696.883<br />
</td>
</tr>
<tr>
<td>12/11<br />
</td>
<td>697.021<br />
</td>
</tr>
<tr>
<td>7/5<br />
</td>
<td>697.085<br />
</td>
</tr>
<tr>
<td>11/8<br />
</td>
<td>697.295<br />
</td>
</tr>
<tr>
<td>11/10<br />
</td>
<td>697.500<br />
</td>
</tr>
<tr>
<td>14/11<br />
</td>
<td>697.812<br />
</td>
</tr>
<tr>
<td>4/3<br />
</td>
<td>701.955<br />
</td>
</tr>
</table>
<br />
<!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="Tuning Spectra-Spectrum of Meanpop Tunings by Eigenmonzos"></a><!-- ws:end:WikiTextHeadingRule:4 -->Spectrum of Meanpop Tunings by Eigenmonzos</h2>
<table class="wiki_table">
<tr>
<th>Eigenmonzo<br />
</th>
<th>Fifth<br />
</th>
</tr>
<tr>
<td>10/9<br />
</td>
<td>691.202<br />
</td>
</tr>
<tr>
<td>6/5<br />
</td>
<td>694.786<br />
</td>
</tr>
<tr>
<td>9/7<br />
</td>
<td>695.614<br />
</td>
</tr>
<tr>
<td>11/8<br />
</td>
<td>696.052<br />
</td>
</tr>
<tr>
<td>11/10<br />
</td>
<td>696.176<br />
</td>
</tr>
<tr>
<td>7/6<br />
</td>
<td>696.319<br />
</td>
</tr>
<tr>
<td>14/11<br />
</td>
<td>696.413<br />
</td>
</tr>
<tr>
<td>12/11<br />
</td>
<td>696.474<br />
</td>
</tr>
<tr>
<td>5/4<br />
</td>
<td>696.578 (minimax tuning)<br />
</td>
</tr>
<tr>
<td>11/9<br />
</td>
<td>696.839<br />
</td>
</tr>
<tr>
<td>8/7<br />
</td>
<td>696.883<br />
</td>
</tr>
<tr>
<td>7/5<br />
</td>
<td>697.085<br />
</td>
</tr>
<tr>
<td>4/3<br />
</td>
<td>701.955<br />
</td>
</tr>
</table>
</body></html>