Reversed meantone - Revision history

Overthink: word differently

2026-05-24T02:31:56Z

word differently

← Older revision		Revision as of 02:31, 24 May 2026
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	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]] would indeed be ridiculous ~~with 163 being prime~~.		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.		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.

Overthink: supra

2026-05-23T22:45:05Z

supra

← Older revision		Revision as of 22:45, 23 May 2026
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	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.		Related to this idea, [[162/161]] is a 23-limit comma (specifically 161 = 7 × 23), and [[163/162]] would indeed be ridiculous with 163 being prime.

	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.		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.
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	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.		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 ==		== Tunings ==

VectorGraphics at 05:35, 23 May 2026

2026-05-23T05:35:57Z

← Older revision		Revision as of 05:35, 23 May 2026
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	See [[No-fives subgroup temperaments #Reversed meantone]] for technical data.		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.

	== Tunings ==		== Tunings ==

FloraC: - data. Mark as mathematical interest. Misc. cleanup

2026-05-11T17:34:52Z

- data. Mark as mathematical interest. Misc. cleanup

@@ Line 1: / Line 1: @@
-'''Reversed meantone''' is a temperament which tempers out the 41-limit comma [[82/81]].
+{{Mathematical interest}}
-== Properties ==
+'''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. 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.
 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 ==
+See [[No-fives subgroup temperaments #Reversed meantone]] for technical data.
-[[Comma list]]: 82/81
+== Tunings ==
+=== Other tunings ===
-[[Gencom]]: [2 4/3; 82/81]
+* [[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}}
-{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 39d }}
+== Notes ==
-</div></div>
+<references group="note"/>
 [[Category:Reversed meantone| ]] <!-- main article -->
 [[Category:Subgroup temperaments]]
 [[Category:Rank-2 temperaments]]

FloraC: - parent category

2025-04-28T14:37:47Z

- parent category

← Older revision		Revision as of 14:37, 28 April 2025
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	* 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)		* 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:~~41-limit]]~~		[[Category:Reversed meantone\| ]] <!-- main article -->
	~~[[Category:Temperaments~~]]
	[[Category:Subgroup temperaments]]		[[Category:Subgroup temperaments]]
	[[Category:Rank-2 temperaments]]		[[Category:Rank-2 temperaments]]

Lériendil at 23:40, 19 April 2025

2025-04-19T23:40:54Z

← Older revision		Revision as of 23:40, 19 April 2025
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	[[Category:Temperaments]]		[[Category:Temperaments]]
	[[Category:Subgroup temperaments]]		[[Category:Subgroup temperaments]]
			[[Category:Rank-2 temperaments]]

Lériendil at 08:08, 14 August 2024

2024-08-14T08:08:54Z

← Older revision		Revision as of 08:08, 14 August 2024
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	{{Optimal ET sequence\|legend=1\| 5, 12, 17, 22, 39d }}		{{Optimal ET sequence\|legend=1\| 5, 12, 17, 22, 39d }}
	</div></div>		</div></div>

			=== Other tunings ===
			* [[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:41-limit]]		[[Category:41-limit]]
	[[Category:Temperaments]]		[[Category:Temperaments]]
	[[Category:Subgroup temperaments]]		[[Category:Subgroup temperaments]]

Cmloegcmluin: "optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence

2023-05-07T16:26:29Z

"optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence

← Older revision		Revision as of 16:26, 7 May 2023
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	[[TOP tuning\|TOP generator]]s: ~2 = 1199.6961, ~4/3 = 494.3834		[[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></div>

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

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

FloraC: Undo. The data added are simply meantone and the "non-octave" section doesn't make much sense.

2022-06-10T13:57:19Z

Undo. The data added are simply meantone and the "non-octave" section doesn't make much sense.

Show changes

Moremajorthanmajor: /* Temperament data */

2022-06-10T06:04:12Z

Temperament data

Show changes