Reversed meantone: Difference between revisions

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== 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.

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

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

</div></div>

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[[TOP tuning|TOP generator]]s: ~2 = 1197.2342, ~4/3 = 488.9029

[[Optimal ~~GPV~~ sequence~~]]: {{Val list~~| 5, 12, 17, 22, 49 }}

</div></div>

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[[TOP tuning|TOP generator]]s: ~2 = 1197.9683, ~4/3 = 491.3454

[[Optimal ~~GPV~~ sequence~~]]: {{Val list~~| 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| ]]

[[Category:Subgroup temperaments]]

[[Category:Rank-2 temperaments]]

@@ Line 2: / Line 2: @@
 == 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.
+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 24: / 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>
@@ Line 43: / Line 43: @@
 [[TOP tuning|TOP generator]]s: ~2 = 1197.2342, ~4/3 = 488.9029
-[[Optimal GPV sequence]]: {{Val list| 5, 12, 17, 22, 49 }}
+{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 49 }}
 </div></div>
@@ Line 62: / Line 62: @@
 [[TOP tuning|TOP generator]]s: ~2 = 1197.9683, ~4/3 = 491.3454
-[[Optimal GPV sequence]]: {{Val list| 5, 12, 17, 22, 39d }}
+{{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]]