Reversed meantone: Difference between revisions

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~~<h2>IMPORTED REVISION FROM WIKISPACES</h2>~~

~~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 19:00:01 UTC</tt>.<br>~~

~~: The original revision id was <tt>336215312</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. 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.

'''Reversed meantone''' is a [[regular temperament|temperament]] which tempers out the [[41-limit]] [[comma]] [[82/81]].

~~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>~~

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.

~~<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"><html><head><title>Reverse Meantone</title></head><body><h1 id="toc0"><a name="REVERSE MEANTONE!"></a>REVERSE MEANTONE!</h1>

Related to this idea, [[162/161]] is a 23-limit comma (specifically 161 = 7 × 23), and [[163/162]] with the numerator being prime would indeed be ridiculous.

~~<br />~~

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. ~~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.~~<br />~~

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.

~~<br />~~

Related to this idea, 162/161 is a 23-limit comma (specifically 161=7*23), and 163/162 being prime would indeed be ridiculous.~~<~~/~~body><~~/~~html>~~</~~pre~~></~~div~~>

See [[No-fives subgroup temperaments #Reversed meantone]] for technical data.

Reversed meantone may be extended to the 2.3.23.25.41 subgroup by mapping 32/25 and 23/18 to the major third, resulting in the '''shrub''' temperament.

A temperament in a simpler subgroup that has tunings around this range is [[supra]].

== Tunings ==

=== Other tunings ===

* [[DKW theory|DKW]] (2.3.41): ~2 = 1200.0000{{c}}, ~3/2 = 706.8411{{c}}

* DKW (2.3.6561/160<ref group="note">Mathematically identical to [[meantone]], but optimized for the "retroptolemaic" thirds, [[2560/2187]] and [[6561/5120]], rather than 6/5 and 5/4</ref>): ~2 = 1200.0000{{c}}, ~3/2 = 706.8984{{c}}

== Notes ==

[[Category:Reversed meantone| ]]

[[Category:Subgroup temperaments]]

[[Category:Rank-2 temperaments]]

@@ Line 1: / Line 1: @@
-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+{{Mathematical interest}}
-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 19:00:01 UTC</tt>.<br>
-: The original revision id was <tt>336215312</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. 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.
+'''Reversed meantone''' is a [[regular temperament|temperament]] which tempers out the [[41-limit]] [[comma]] [[82/81]].
-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>
+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.
-<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;
+Related to this idea, [[162/161]] is a 23-limit comma (specifically 161 = 7 × 23), and [[163/162]] with the numerator being prime would indeed be ridiculous.
- &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. 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;
+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;
-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>
+See [[No-fives subgroup temperaments #Reversed meantone]] for technical data.
+Reversed meantone may be extended to the 2.3.23.25.41 subgroup by mapping 32/25 and 23/18 to the major third, resulting in the '''shrub''' temperament.
+A temperament in a simpler subgroup that has tunings around this range is [[supra]].
+== Tunings ==
+=== Other tunings ===
+* [[DKW theory|DKW]] (2.3.41): ~2 = 1200.0000{{c}}, ~3/2 = 706.8411{{c}}
+* DKW (2.3.6561/160<ref group="note">Mathematically identical to [[meantone]], but optimized for the "retroptolemaic" thirds, [[2560/2187]] and [[6561/5120]], rather than 6/5 and 5/4</ref>): ~2 = 1200.0000{{c}}, ~3/2 = 706.8984{{c}}
+== Notes ==
+<references group="note"/>
+[[Category:Reversed meantone| ]] <!-- main article -->
+[[Category:Subgroup temperaments]]
+[[Category:Rank-2 temperaments]]