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| '''Reversed meantone''' is a temperament which tempers out the 41-limit comma [[82/81]].
| | {{Mathematical interest}} |
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| == 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.
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| 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. |
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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.
| | 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. |
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| == Temperament data ==
| | 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;">
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| <div style="line-height:1.6;">'''Reversed meantone (5&12, 2.3.41 subgroup)'''</div>
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| <div class="mw-collapsible-content">
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| Subgroup: 2.3.41
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| [[Comma list]]: 82/81 | | See [[No-fives subgroup temperaments #Reversed meantone]] for technical data. |
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| [[Gencom]]: [2 4/3; 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. |
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| [[Mapping|Sval mapping]]: [{{val| 1 2 7 }}, {{val| 0 -1 -4 }}] | | A temperament in a simpler subgroup that has tunings around this range is [[supra]]. |
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| [[POTE generator]]: ~4/3 = 494.5086 | | == 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}} |
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| [[TOP tuning|TOP generator]]s: ~2 = 1199.6961, ~4/3 = 494.3834
| | == Notes == |
| | <references group="note"/> |
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| [[Optimal GPV sequence]]: {{Val list| 5, 12, 17 }} | | [[Category:Reversed meantone| ]] <!-- main article --> |
| </div></div>
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| <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>
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| <div class="mw-collapsible-content">
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| Subgroup: 2.3.7.41
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| [[Comma list]]: 64/63, 82/81
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| [[Gencom]]: [2 4/3; 64/63 82/81]
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| [[Mapping|Sval mapping]]: [{{val| 1 2 2 7 }}, {{val| 0 -1 2 -4 }}]
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| [[POTE generator]]: ~4/3 = 490.0323
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| [[TOP tuning|TOP generator]]s: ~2 = 1197.2342, ~4/3 = 488.9029
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| [[Optimal GPV sequence]]: {{Val list| 5, 12, 17, 22, 49 }}
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| </div></div>
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| <div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
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| <div style="line-height:1.6;">'''Reversed meantone (5&12, 2.3.7.11.41 subgroup)'''</div>
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| <div class="mw-collapsible-content">
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| Subgroup: 2.3.7.11.41
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| [[Comma list]]: 64/63, 82/81, 99/98
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| [[Gencom]]: [2 4/3; 64/63 82/81 99/98]
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| [[Mapping|Sval mapping]]: [{{val| 1 2 2 1 7 }}, {{val| 0 -1 2 6 -4 }}]
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| [[POTE generator]]: ~4/3 = 492.1787
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| [[TOP tuning|TOP generator]]s: ~2 = 1197.9683, ~4/3 = 491.3454
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| [[Optimal GPV sequence]]: {{Val list| 5, 12, 17, 22, 39d }}
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| </div></div>
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| ==Non-octave==
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| As [[meantone]] is based within the octave and on the syntonic comma, 81/80, tempering the fifth flat, and the thirds to be simple rational intervals, basing it within the third or fifth harmonic or the perfect fifth or the major tenth, sixth or third instead still may temper 81/80, even if it might as well be called reverse meantone. Of these, the major third of 5/4 is the least recognizable as an interval, though still very recognizable. Additionally, the major third of 5/4 is a very narrow interval to base a temperament within, so if its use is intended as more than a joke it might as well be the [[Augmented family|augmented]] temperament 4ned5/2.
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| ==Temperament data==
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| <div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
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| <div style="line-height:1.6;">'''Reversed meantone (b11&b19, 3.2.5 subgroup)'''
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| Subgroup: 3.2.5
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| [[Comma list]]: 81/80
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| [[Gencom]]: [3 3/2; 81/80]
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| [[Mapping|Sval mapping]]: [{{val| 1 1 0 }}, {{val| 0 -1 4 }}]
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| [[POTE generator]]: ~3/2 = 698.3375
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| [[TOP tuning|TOP generator]]s: ~3 = 1898.4460, ~3/2 = 697.0491
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| [[Optimal GPV sequence]]: {{Val list| b11, b19, b30 }}
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| </div></div>
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| <div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;"><div style="line-height:1.6;">'''Reversed meantone (b11&b19, 3.2.5.7 subgroup)'''
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| Subgroup: 3.2.5.7
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| [[Comma list]]: 64/63, 81/80
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| [[Gencom]]: [3 3/2; 64/63 81/80]
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| [[Mapping|Sval mapping]]: [{{val| 1 1 0 3}}, {{val| 0 -1 4 -13}}]
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| [[POTE generator]]: ~3/2 = 698.5001
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| [[TOP tuning|TOP generator]]s: ~3 = 1898.4580, ~3/2 = 697.2159
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| [[Optimal GPV sequence]]: {{Val list| b11, b19, b30, b41, b71 }}
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| </div></div>
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| <div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
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| <div style="line-height:1.6;">'''Reversed meantone (b11de&b19e, 3.2.5.7.11 subgroup)'''
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| Subgroup: 3.2.5.7.11
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| [[Comma list]]: 64/63, 81/80, 99/98
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| [[Gencom]]: [2 3/2; 64/63 81/80 99/98]
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| [[Mapping|Sval mapping]]: [{{val| 1 1 0 4 7}}, {{val| 0 -1 4 -6 -13}}]
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| [[POTE generator]]: ~3/2 = 702.7738
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| [[TOP tuning|TOP generator]]s: ~3 = 1893.9031, ~3/2 = 699.7986
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| [[Optimal GPV sequence]]: {{Val list| b8, b11de, b19e, b27, b38dee, b46cde }}
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| </div></div>
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| <div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
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| <div style="line-height:1.6;">'''Reversed meantone (c16&c28, 5.2.3 subgroup)'''
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| Subgroup: 5.2.3 (3/2.5)
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| [[Comma list]]: 81/80
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| [[Gencom]]: [3/2 10/9; 81/80]
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| [[Mapping|Sval mapping]]: [{{val| 4 2 3 }}, {{val| 0 -1 —1 }}]
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| [[POTE generator]]: ~3/2 = 696.5784, ~10/9 = 192.5712
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| [[TOP tuning|TOP generator]]s: ~3/2 = 697.0491, ~10/9 = 192.7014
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| [[Optimal GPV sequence]]: {{Val list| c16, c28, c44 }}
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| </div></div>
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| <div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;"><div style="line-height:1.6;">'''Reversed meantone (c12d&c16dee, 5.2.3.7 subgroup)'''Subgroup: 5.2.3.7
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| [[Comma list]]: 64/63, 81/80
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| [[Gencom]]: [3/2 10/9; 64/63 81/80]
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| [[Mapping|Sval mapping]]: [{{val| 4 2 3 6}}, {{val| 0 -1 —1 -4}}]
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| [[POTE generator]]: ~3/2 = 696.5784, ~10/9 = 201.7001
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| [[TOP tuning|TOP generator]]s: ~3/2 = 697.2159, ~10/9 = 202.3697
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| [[Optimal GPV sequence]]: {{Val list| c12d, c16dee, c28, c40, c52d, c64 }}
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| </div></div>
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| <div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
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| <div style="line-height:1.6;">'''Reversed meantone (c12de&c16dee, 5.2.3.7.11 subgroup)'''
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| Subgroup: 5.2.3.7.11
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| [[Comma list]]: 64/63, 81/80, 99/98
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| [[Gencom]]: [3/2 10/9; 64/63 81/80 99/98]
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| [[Mapping|Sval mapping]]: [{{val| 4 2 3 6 8}}, {{val| 0 -1 —1 -4 -7}}]
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| [[POTE generator]]: ~3/2 = 696.5784, ~10/9 = 204.5471
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| [[TOP tuning|TOP generator]]s: ~3/2 = 699.7986, ~10/9 = 205.4927
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| [[Optimal GPV sequence]]: {{Val list| c12de, c16dee, c28, c40e, c52dee, c64 }})
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| </div></div>
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| <div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
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| <div style="line-height:1.6;">'''Reversed meantone (q4&q7, 3/2.2.3.5 subgroup)'''
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| Subgroup: 3/2.2.3.5
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| [[Comma list]]: 81/80
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| [[Gencom]]: [3/2 9/8; 81/80]
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| [[Mapping|Sval mapping]]: [{{val| 1 2 3 4 }}, {{val| 0 -1 -1 0 }}]
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| [[POTE generator]]: ~9/8 = 198.7378
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| [[TOP tuning|TOP generator]]s: ~3/2 = 699.3710, ~9/8 = 198.0062
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| [[Optimal GPV sequence]]: {{Val list| q4, q7, q11}}
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| </div></div>
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| <div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;"><div style="line-height:1.6;">'''Reversed meantone (q3&q7, 3/2.2.3.5.7 subgroup)'''
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| Subgroup: 3/2.2.3.5.7
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| [[Comma list]]: 64/63, 81/80
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| [[Gencom]]: [3/2 9/8; 64/63 81/80]
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| [[Mapping|Sval mapping]]: [{{val| 1 2 3 4 6}}, {{val| 0 -1 -1 0 -4}}]
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| [[POTE generator]]: ~9/8 = 205.3121
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| [[TOP tuning|TOP generator]]s: ~3/2 = 700.2305, ~9/8 = 204.8077
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| [[Optimal GPV sequence]]: {{Val list| q3, q7, q10, q17, q24, q31 }}
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| </div></div>
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| <div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
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| <div style="line-height:1.6;">'''Reversed meantone (q7e&q11, 3/2.2.3.5.7.11 subgroup)'''
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| Subgroup: 3/2.2.3.5.7.11
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| [[Comma list]]: 64/63, 81/80, 99/98
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| [[Gencom]]: [3/2 9/8; 64/63 81/80 99/98]
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| [[Mapping|Sval mapping]]: [{{val| 1 2 3 4 4 4}}, {{val| 0 -1 -1 0 3 7}}]
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| [[POTE generator]]: ~9/8 = 195.0236
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| [[TOP tuning|TOP generator]]s: ~3/2 = 698.2079, ~9/8 = 193.9826
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| [[Optimal GPV sequence]]: {{Val list| q7e, q11, q18, q25 }}
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| </div></div>
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| <div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
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| <div style="line-height:1.6;">'''Reversed meantone (q16&q25, 5/2.2.3.5 subgroup)'''
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| Subgroup: 5/2.2.3.5
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| [[Comma list]]: 81/80
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| [[Gencom]]: [5/2 3/2; 81/80]
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| [[Mapping|Sval mapping]]: [{{val| 1 -1 -1 0 }}, {{val| 0 4 5 4 }}]
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| [[POTE generator]]: ~3/2 = 696.9241
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| [[TOP tuning|TOP generator]]s: ~5/2 = 1586.4184, ~3/2 = 696.9702
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| [[Optimal GPV sequence]]: {{Val list| q16, q25, q41 }}
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| </div></div>
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| <div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;"><div style="line-height:1.6;">'''Reversed meantone (q16&q25, 5/2.2.3.5.7 subgroup)'''
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| Subgroup: 5/2.2.3.5.7
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| [[Comma list]]: 64/63, 81/80
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| [[Gencom]]: [5/2 3/2; 64/63 81/80]
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| [[Mapping|Sval mapping]]: [{{val| 1 -1 -1 0 3}}, {{val| 0 4 5 4 -2}}]
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| [[POTE generator]]: ~3/2 = 696.7420
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| [[TOP tuning|TOP generator]]s: ~5/2 = 1587.1426, ~3/2 = 697.1060
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| [[Optimal GPV sequence]]: {{Val list| q16, q25, q41 }}
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| </div></div>
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| <div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
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| <div style="line-height:1.6;">'''Reversed meantone (q16&q41, 5/2.2.3.5.7.11 subgroup)'''
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| Subgroup: 5/2.2.3.5.7.11
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| [[Comma list]]: 81/80, 126:125, 176:175
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| [[Gencom]]: [5/2 3/2; 81/80 126:125 176:175]
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| [[Mapping|Sval mapping]]: [{{val| 1 -1 -1 0 3 7}}, {{val| 0 4 5 4 -2 -10}}]
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| [[POTE generator]]: ~3/2 = 696.3353
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| [[TOP tuning|TOP generator]]s: ~5/2 = 1587.1426, ~3/2 = 697.2531
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| [[Optimal GPV sequence]]: {{Val list|q16, q25, q41, q57}}
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| </div></div>
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| <div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
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| <div style="line-height:1.6;">'''Reversed meantone (q9&q14, 5/3.2.3.5 subgroup)'''
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| Subgroup: 5/3.2.3.5
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| [[Comma list]]: 81/80
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| [[Gencom]]: [5/3 10/9; 81/80]
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| [[Mapping|Val mapping]]: [{{val| 1 2 3 4 }}, {{val| 0 -3 -4 -4 }}]
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| [[POTE generator]]: ~10/9 = 189.0028
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| [[TOP tuning|TOP generator]]s: ~5/3 = 884.9987, ~10/9 = 189.1396
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| [[Optimal GPV sequence]]: {{Val list| q9, q14, q23 }}
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| </div></div>
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| <div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;"><div style="line-height:1.6;">'''Reversed meantone (q9&q14d, 5/3.2.3.5.7 subgroup)'''
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| Subgroup: 5/3.2.3.5.7
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| [[Comma list]]: 126/125, 81/80
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| [[Gencom]]: [5/3 10/9; 126/125 81/80]
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| [[Mapping|Sval mapping]]: [{{val| 1 2 3 4 6}}, {{val| 0 -3 -4 -4 -10}}]
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| [[POTE generator]]: ~10/9 = 192.2011
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| [[TOP tuning|TOP generator]]s: ~5/3 = 887.4720, ~10/9 = 192.8777
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| [[Optimal GPV sequence]]: {{Val list| q9, q14d, q23 }}
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| </div></div>
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| <div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
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| <div style="line-height:1.6;">'''Reversed meantone (q9&q14, 5/3.2.3.5.7.11 subgroup)'''
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| Subgroup: 5/3.2.3.5.7.11
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| [[Comma list]]: 81/80, 126/125, 100/99
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| [[Gencom]]: [5/3 10/9; 81/80 126/125 100/99]
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| [[Mapping|Sval mapping]]: [{{val| 1 2 3 4 4 6}}, {{val| 0 -3 -4 -4 -1 -6}}]
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| [[POTE generator]]: ~10/9 = 190.2709
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| [[TOP tuning|TOP generator]]s: ~5/3 = 886.3893, ~10/9 = 190.7078
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| [[Optimal GPV sequence]]: {{Val list| q9, q14, q23, q37}}
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| </div></div>
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| [[Category:41-limit]]
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| [[Category:Temperaments]]
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| [[Category:Subgroup temperaments]] | | [[Category:Subgroup temperaments]] |
| | [[Category:Rank-2 temperaments]] |
