Quarts of meantone

If you take 36/35 and raise it by a schisma to 59049/57344 you get an
interval I will call, for the purposes of this article, a "quart".
It's a quarter-tone of 50.7 cents.

The point of quarts is that just as 1/4 comma meatone gives pure major
thirds and 1/3 comma meantone pure minor thirds, 1/10 quart meantone
gives pure 7/4s and 1/9 quart meantone gives pure 7/6s. Unfortunately,
the meantone fifth which gives pure 7/5s does not come out evenly in
quarts; instead we need (81/80)*(225/224) = 3645/3584. Taking a sixth
of this off of a fifth gives a meantone with pure 7/5s; that's 1/6
comma + 1/6 kleisma meantone, close to 1/10 quart but not quite the
same, the Barton tuning. It's (3136/3125)^(1/30), or a fifth of a
cent, sharper than 1/10 quart meantone, and (3136/3125)^(1/12) cents
sharper than 1/4 comma meantone.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@...>
wrote:
>
> If you take 36/35 and raise it by a schisma to 59049/57344 you get an
> interval I will call, for the purposes of this article, a "quart".
> It's a quarter-tone of 50.7 cents.
>
> The point of quarts is that just as 1/4 comma meatone gives pure major
> thirds and 1/3 comma meantone pure minor thirds, 1/10 quart meantone
> gives pure 7/4s and 1/9 quart meantone gives pure 7/6s.

And 1/8 quart gives pure 9/7s.

So:

1/10-quart pure 7/4s
1/9-quart pure 7/6s
1/8-quart pure 9/7s

Plus stuff like 2/19 quart gives pure 49/48s, 1/7 quart gives pure
28/27s and so forth.

one quart = 2.3857 commas