|
|
| (5 intermediate revisions by 4 users not shown) |
| 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.
| | 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. |
|
| |
|
| 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. |
|
| |
|
| == 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;">
| |
| <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 | | See [[No-fives subgroup temperaments #Reversed meantone]] for technical data. |
|
| |
|
| [[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. |
|
| |
|
| [[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]]. |
|
| |
|
| [[POTE generator]]: ~4/3 = 494.5086
| | == Tunings == |
| | | === Other tunings === |
| [[TOP tuning|TOP generator]]s: ~2 = 1199.6961, ~4/3 = 494.3834 | | * [[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 }} | |
| </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
| | == Notes == |
| | | <references group="note"/> |
| [[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:41-limit]] | | [[Category:Reversed meantone| ]] <!-- main article --> |
| [[Category:Temperaments]]
| |
| [[Category:Subgroup temperaments]] | | [[Category:Subgroup temperaments]] |
| | [[Category:Rank-2 temperaments]] |