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Reversed meantone: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
'''Reversed meantone''' is a temperament which tempers out the 41-limit comma [[82/81]].
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:Kosmorsky|Kosmorsky]] and made on <tt>2012-05-16 18:58:35 UTC</tt>.<br>
: The original revision id was <tt>336214684</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">=REVERSE MEANTONE!=


As meantone is based on the syntonic comma, 81/80, tempering the fifth flat, tempering 82/81 instead results in a sharper fifth, and a major third equivalent to the 41st harmonic instead of the 5th, so it might as well be called reverse meantone unless you got a better name. The 41st is very delicate however and mistuning by several cents destroys it, so if its use is intended as more than a joke exact quarter comma tempering is best, although 39edo does a fair job.
== Properties ==
As [[meantone]] is based on the syntonic comma, 81/80, tempering the fifth flat, tempering [[82/81]] instead results in a sharper fifth, and a major third equivalent to the 41st harmonic instead of the 5th, so it might as well be called reverse meantone. As a very high limit interval, however, that [[41/32]] is far less recognizable as an interval than meantone’s 5/4, and would more likely be heard as a flat 9/7. Additionally, the 41st is very delicate, and mistuning by several cents destroys it, so if its use is intended as more than a joke exact quarter comma tempering is best, although [[39edo]] does a fair job.


Related to this idea, 162/161 is a 23-limit comma (specifically 161=7*23), and 163/162 being prime would indeed be ridiculous.</pre></div>
Related to this idea, [[162/161]] is a 23-limit comma (specifically 161 = 7 × 23), and [[163/162]] being prime would indeed be ridiculous.
<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;Reverse Meantone&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="REVERSE MEANTONE!"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;REVERSE MEANTONE!&lt;/h1&gt;
The more well known [[64/63]] comma equates 9/8 with 8/7 instead of 10/9, which also results in a sharper fifth, and the major third is equivalent to 9/7.
&lt;br /&gt;
 
As meantone is based on the syntonic comma, 81/80, tempering the fifth flat, tempering 82/81 instead results in a sharper fifth, and a major third equivalent to the 41st harmonic instead of the 5th, so it might as well be called reverse meantone unless you got a better name. The 41st is very delicate however and mistuning by several cents destroys it, so if its use is intended as more than a joke exact quarter comma tempering is best, although 39edo does a fair job.&lt;br /&gt;
== Temperament data ==
&lt;br /&gt;
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Related to this idea, 162/161 is a 23-limit comma (specifically 161=7*23), and 163/162 being prime would indeed be ridiculous.&lt;/body&gt;&lt;/html&gt;</pre></div>
<div style="line-height:1.6;">'''Reversed meantone (5&amp;12, 2.3.41 subgroup)'''</div>
<div class="mw-collapsible-content">
Subgroup: 2.3.41
 
[[Comma list]]: 82/81
 
[[Gencom]]: [2 4/3; 82/81]
 
[[Mapping|Sval mapping]]: [{{val| 1 2 7 }}, {{val| 0 -1 -4 }}]
 
[[POTE generator]]: ~4/3 = 494.5086
 
[[TOP tuning|TOP generator]]s: ~2 = 1199.6961, ~4/3 = 494.3834
 
{{Optimal ET sequence|legend=1| 5, 12, 17 }}
</div></div>
 
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">'''Reversed meantone (5&amp;12, 2.3.7.41 subgroup)'''</div>
<div class="mw-collapsible-content">
Subgroup: 2.3.7.41
 
[[Comma list]]: 64/63, 82/81
 
[[Gencom]]: [2 4/3; 64/63 82/81]
 
[[Mapping|Sval mapping]]: [{{val| 1 2 2 7 }}, {{val| 0 -1 2 -4 }}]
 
[[POTE generator]]: ~4/3 = 490.0323
 
[[TOP tuning|TOP generator]]s: ~2 = 1197.2342, ~4/3 = 488.9029
 
{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 49 }}
</div></div>
 
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">'''Reversed meantone (5&amp;12, 2.3.7.11.41 subgroup)'''</div>
<div class="mw-collapsible-content">
Subgroup: 2.3.7.11.41
 
[[Comma list]]: 64/63, 82/81, 99/98
 
[[Gencom]]: [2 4/3; 64/63 82/81 99/98]
 
[[Mapping|Sval mapping]]: [{{val| 1 2 2 1 7 }}, {{val| 0 -1 2 6 -4 }}]
 
[[POTE generator]]: ~4/3 = 492.1787
 
[[TOP tuning|TOP generator]]s: ~2 = 1197.9683, ~4/3 = 491.3454
 
{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 39d }}
</div></div>
 
=== Other tunings ===
* [[DKW theory|DKW]] (2.3.41): ~2 = 1\1, ~3/2 = 706.8411 (~4/3 = 493.1589)
* DKW (2.3.6561/160<ref>Mathematically identical to [[meantone]], but optimized for the "retroptolemaic" thirds, [[2560/2187]] and [[6561/5120]], rather than 6/5 and 5/4</ref>): ~2 = 1\1, ~3/2 = 706.8984 (~4/3 = 493.1016)
 
[[Category:Reversed meantone| ]] <!-- main article -->
[[Category:Subgroup temperaments]]
[[Category:Rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/Reversed_meantone"