Reversed meantone: Difference between revisions

(21 intermediate revisions by 11 users not shown)

Line 1:

~~<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 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.

'''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 ~~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.~~<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 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.
+'''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 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;
+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]]