Reversed meantone: Difference between revisions
Wikispaces>Kosmorsky **Imported revision 336214684 - Original comment: ** |
word differently |
||
| (21 intermediate revisions by 11 users not shown) | |||
| Line 1: | Line 1: | ||
{{Mathematical interest}} | |||
'''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]] with the numerator 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 | 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 | 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. | |||
A temperament in a simpler subgroup that has tunings around this range is [[supra]]. | |||
== 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}} | |||
== Notes == | |||
<references group="note"/> | |||
[[Category:Reversed meantone| ]] <!-- main article --> | |||
[[Category:Subgroup temperaments]] | |||
[[Category:Rank-2 temperaments]] | |||
Latest revision as of 02:31, 24 May 2026
|
This page presents a topic of primarily mathematical interest.
While it is derived from sound mathematical principles, its applications in terms of utility for actual music may be limited, highly contrived, or as yet unknown. |
Reversed meantone is a 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 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.
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.
A temperament in a simpler subgroup that has tunings around this range is supra.
Tunings
Other tunings
- DKW (2.3.41): ~2 = 1200.0000 ¢, ~3/2 = 706.8411 ¢
- DKW (2.3.6561/160[note 1]): ~2 = 1200.0000 ¢, ~3/2 = 706.8984 ¢