1/2-comma meantone

1/2-comma meantone is the tuning of meantone temperament in which each perfect fifth is tempered by a half of a syntonic comma from its just value of 3/2. This results in minor sevenths being exactly 9/5 (and major seconds being exactly 10/9).

In this system, the major thirds are exactly 100/81 or approximately 365 cents, thus bordering on neutral thirds. The fifths of this temperament fall between those of 26edo and 33edo, but closer to 33, which is the best small number candidate for a closed system approximating this meantone. If you want more precision, 125edo using the bb val flat 5th is audibly indistinguishable from a closed system of half-comma meantone, with an error measured in thousandths of a cent.

For a tuning with so flat a fifth, the 7-limit extension that preserves 7-odd-limit diamond monotone is flattertone. Otherwise the septimal meantone mapping has the sizes of 7/6 and 8/7 swapped, and the flattone mapping has the sizes of 7/5 and 10/7 swapped.

Tuning profile

Projection map:

[⟨	1	2	4	10	]
⟨	0	-1	-4	-17	]
⟨	0	1/2	2	17/2	]
⟨	0	0	0	0	]]

Tuning map: ⟨1200 1891.2019 2764.8074 3350.4316]

Error map: ⟨0 -10.7531 -21.5063 -18.3944]