Another co-tuning example

The meantone-superpythagorean co-tuning is the pair of tunings for
meantone and superpythagorean such that the 7-limit is a linear
combination of the meantone and superpythagorean values for any
7-limit interval. The co-tuning has good, practical tuning values for
both meantone and superpythagorean, with the meantone fifth of 696.039
cents a smidgen above the 696 cents of 50-equal, and the
superpythagorean fifth of 709.913 being between that of 22 and 27, and
in the 9-limit poptimal range.

To see if this is a usual situation, I checked another such pair, the
myna-superkleismic pair. Myna is 27&31, with wedgie <<10 9 7 -9 -17
-9|| and superkleismic is 15&41 with wedgie <<9 10 -3 -5 -30 -35||;
they both have 6/5 as a generator. The myna co-tuning generator is
310.183 cents, right in the 7-limit poptimal range near the side where
it borders the 9-limit poptimal range, an excellent choice. The
superkleismic co-tuning generator is 321.989 cents, which is between
the 7-limit poptimal range and the 9-limit poptimal range, another
excellent choice. Again, some amount of optimality seems to be
involved in the values of a co-tuning pair. It might be interesting to
see what happens if one of the temperaments is much more precise than
the other.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@c...> wrote:
> It might be interesting to
> see what happens if one of the temperaments is much more precise than
> the other.

What that tends to do is screw things up, especially for the less
accurate temperament. I looked at 53 co-fifths for meantone, and most
were good to excellent. However, ennealimmal-meantone gave 677.310,
considerably flatter even than 7-equal and not acceptable. Hystrix,
which is not a high-accuracy temperament, managed to get a fifth even
flatter than 19-et, at 694.764. However, hystrix is a marginal system
with a very, very sharp fifth (it is the {36/35, 160/147} temperament
with wedgie <<3 5 1 1 -7 -12|| and a rms fifth of 727.687 cents, and
so for enthusiasts only.) Gurion, a microtemperament with a fifth only
very slightly flat, leads to an absurdly sharp 704.909 meantone
co-fifth, and
kwai, another microtemperament with a fifth only a little sharp leads
to a flat 693.914 fifth. Unidec, another micro, leads to the more
reasonable 694.568, and amity, also a micro, to 694.956, which is
actually in Blackwood's acceptable range, being sharper than the 19-et
fifth. However generally results were much better than this.