Reversed meantone: Difference between revisions

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

== ~~REVERSE MEANTONE!~~ ==

== 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. ~~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.

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.

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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.

== Temperament data ==

<div style="line-height:1.6;">'''Reversed meantone (5&12, 2.3.41 subgroup)'''</div>

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[[Comma list]]: 82/81

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

[[Gencom]]: [2 4/3; 82/81]

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

[[POTE generator]]: ~4/3 = 494.5086

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[[TOP tuning|TOP generator]]s: ~2 = 1199.6961, ~4/3 = 494.3834

[[Optimal ~~GPV~~ sequence]]: {{~~Val list~~| 5, 12, 17 }}

</div></div>

<div style="line-height:1.6;">'''Reversed meantone (5&12, 2.3.7.41 subgroup)'''</div>

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

</div></div>

<div style="line-height:1.6;">'''Reversed meantone (5&12, 2.3.7.11.41 subgroup)'''</div>

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

</div></div>

[[Category:41-~~limit~~]]

=== Other tunings ===

[[Category:~~Temperaments~~]]

* [[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| ]]

[[Category:Subgroup temperaments]]

[[Category:Rank-2 temperaments]]

@@ Line 1: / Line 1: @@
 '''Reversed meantone''' is a temperament which tempers out the 41-limit comma [[82/81]].
-== REVERSE MEANTONE! ==
+== 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. 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.
+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.
@@ Line 8: / Line 8: @@
 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.
+== Temperament data ==
 <div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
 <div style="line-height:1.6;">'''Reversed meantone (5&amp;12, 2.3.41 subgroup)'''</div>
@@ Line 15: / Line 16: @@
 [[Comma list]]: 82/81
-[[Mapping]]: [{{val| 1 2 7 }}, {{val| 0 -1 -4 }}]
+[[Gencom]]: [2 4/3; 82/81]
+[[Mapping|Sval mapping]]: [{{val| 1 2 7 }}, {{val| 0 -1 -4 }}]
 [[POTE generator]]: ~4/3 = 494.5086
@@ Line 21: / Line 24: @@
 [[TOP tuning|TOP generator]]s: ~2 = 1199.6961, ~4/3 = 494.3834
-[[Optimal GPV sequence]]: {{Val list| 5, 12, 17 }}
+{{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>
-[[Category:41-limit]]
+=== Other tunings ===
-[[Category:Temperaments]]
+* [[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]]