Reversed meantone

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

Reversed meantone (5&12, 2.3.41 subgroup)

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

Reversed meantone (5&12, 2.3.7.41 subgroup)

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

Reversed meantone (5&12, 2.3.7.11.41 subgroup)

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

Other tunings

• DKW (2.3.41): ~2 = 1\1, ~3/2 = 706.8411 (~4/3 = 493.1589)
• DKW (2.3.6561/160[1]): ~2 = 1\1, ~3/2 = 706.8984 (~4/3 = 493.1016)
1. Mathematically identical to meantone, but optimized for the "retroptolemaic" thirds, 2560/2187 and 6561/5120, rather than 6/5 and 5/4