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'''Reversed meantone''' is a temperament which tempers out the 41-limit comma [[82/81]].
{{Mathematical interest}}


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


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


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.
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.


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.


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
See [[No-fives subgroup temperaments #Reversed meantone]] for technical data.  
<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
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.


[[Gencom]]: [2 4/3; 82/81]
A temperament in a simpler subgroup that has tunings around this range is [[supra]].


[[Mapping|Sval mapping]]: [{{val| 1 2 7 }}, {{val| 0 -1 -4 }}]
== 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}}


[[POTE generator]]: ~4/3 = 494.5086
== Notes ==
<references group="note"/>


[[TOP tuning|TOP generator]]s: ~2 = 1199.6961, ~4/3 = 494.3834
[[Category:Reversed meantone| ]] <!-- main article -->
 
[[Category:Subgroup temperaments]]
[[Optimal GPV sequence]]: {{Val list| 5, 12, 17 }}
[[Category:Rank-2 temperaments]]
</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 GPV sequence]]: {{Val list| 5, 12, 17, 22, 49 }}
</div></div>
 
[[Category:41-limit]]
[[Category:Temperaments]]
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