5/16 kleisma marvel

If we take a marvel (225/224 planar) tempering with pure 7s, the
fifths and major thirds will be of the form (3/2)(224/225)^k,
(5/4)(224/225)^(1/2-k), the square of the product of these being exactly
7/2. These tunings I call k-kleismic, in analogy to k-comma for
meantone; here the fifth is flattened by the kth part of a septimal
kleisma of 225/224 instead of the kth part of 81/80.

We might ask which values of k now lead to interesting beat ratios
(brats.) The most interesting brats are 3/2, with pure fifths, 0, with
pure minor thirds, infinity, with pure major thirds, and -1, where the
intervals of a major triad in close position all beat together.

These correspond to 0-kleismic for pure fifths, with major thirds flat
by half a kleisma, 1/4-kleismic for 0, with pure minor thirds,
1/2-kleismic for infinity, with pure major thirds, and very nearly
5/16-kleismic for a brat of -1.

This tuning is analogous to the Wilson meantone. That is very close to
being 5/17 comma meantone, but the exact value is an algebraic integer
of degree four, the positive root of an irreducible monic polynomial
with two real roots, one negative and one positive: x^4-2x-2. Brat -1
for pure-7s marvel is also the positive root of a monic polynomial of
degree four with two real roots, one negative and one positive, the
polynomial in this case being x^4+2x^3+x^2-14. One difference is that
here the positive root is not the largest in absolute value, so you
don't get the same sort of recurrence relationships. The major third
also has a polynomial, 8x^4-8x^3+2x^2-7; this is not an algebraic
integer but the tuning for 5/2 is, with polynomial x^4-2x^3+x^2-14.
Comparing these two, we see the roots of the first polynomial are the
tunings for 3/2 and -5/2. The conjugate root of the Wilson fifth is a
-4/5 for a meantone tuning, but the tuning isn't Wilson--it's
approximately 4/21-comma meantone, and the exact value of the fifth is
the positive root of x^16-4x^12-128. Why do we get two real roots in
all these cases, with the conjugate root having some sort of tuning
interpretation? I dunno; it might be worth thinking about.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
One difference is that
> here the positive root is not the largest in absolute value, so you
> don't get the same sort of recurrence relationships.

We do get a recurrence relationship for the tuning of 5/2, however:

a[n+4] = 2a[n+3] - a[n+2] + 14a[n]

The ratios a[n+1]/a[n] converge to the tuning of 5/2.