Reversed meantone: Difference between revisions
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''' | '''Reversed meantone''' is a temperament which tempers out the 41-limit comma [[82/81]]. | ||
== | == 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 [[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]] 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. | 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. | ||
[[Category: | == Temperament data == | ||
[[Category: | <div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;"> | ||
<div style="line-height:1.6;">'''Reversed meantone (5&12, 2.3.41 subgroup)'''</div> | |||
<div class="mw-collapsible-content"> | |||
Subgroup: 2.3.41 | |||
[[Comma list]]: 82/81 | |||
[[Gencom]]: [2 4/3; 82/81] | |||
[[Mapping|Sval mapping]]: [{{val| 1 2 7 }}, {{val| 0 -1 -4 }}] | |||
[[POTE generator]]: ~4/3 = 494.5086 | |||
[[TOP tuning|TOP generator]]s: ~2 = 1199.6961, ~4/3 = 494.3834 | |||
{{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&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&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> | |||
=== 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:Reversed meantone| ]] <!-- main article --> | |||
[[Category:Subgroup temperaments]] | |||
[[Category:Rank-2 temperaments]] | |||
Latest revision as of 14:37, 28 April 2025
Reversed meantone is a temperament which tempers out the 41-limit comma 82/81.
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.
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
Subgroup: 2.3.41
Comma list: 82/81
Gencom: [2 4/3; 82/81]
Sval mapping: [⟨1 2 7], ⟨0 -1 -4]]
POTE generator: ~4/3 = 494.5086
TOP generators: ~2 = 1199.6961, ~4/3 = 494.3834
Optimal ET sequence: 5, 12, 17
Subgroup: 2.3.7.41
Comma list: 64/63, 82/81
Gencom: [2 4/3; 64/63 82/81]
Sval mapping: [⟨1 2 2 7], ⟨0 -1 2 -4]]
POTE generator: ~4/3 = 490.0323
TOP generators: ~2 = 1197.2342, ~4/3 = 488.9029
Subgroup: 2.3.7.11.41
Comma list: 64/63, 82/81, 99/98
Gencom: [2 4/3; 64/63 82/81 99/98]
Sval mapping: [⟨1 2 2 1 7], ⟨0 -1 2 6 -4]]
POTE generator: ~4/3 = 492.1787
TOP generators: ~2 = 1197.9683, ~4/3 = 491.3454