1/6 syntonic comma meantone

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1/6 comma meantone is the tuning of meantone temperament which tunes the fifth as the sixth root of 45/4, or in other words 698.371 cents. This means the fifth is flattened by 1/6 of the syntonic comma (81/80 ratio) of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. In 1/6-comma meantone, the diatonic tritone 45/32 is tuned justly, and it can be characterized fully as the regular tuning tempering out 81/80 and tuning 2 and 45/32 justly. It is closely related to 55edo in terms of its tuning.

Fractional projection matrix

The fractional projection map defining 7-limit 1/6 comma meantone is

[1 0 0 0>
[2/3 1/3 1/6 0>
[-4/3 4/3 2/3 0>
[-19/3 10/3 5/3 0>

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