Well tunings

By a well tuning I mean a regular tuning (a tuning of the primes in
some p-limit) which can be applied to p-limit JI scales in order to
produce a circulating temperament. It does not need to be a regular
temperament tuning, though the examples I give are.

For example, in the 5-limit for twelve note scales we have 12-et
commas 81/80, 128/125, and 2048/2025, and we want to deal with these
in a way which helps to produce a circulating temperament. One thing
we might do is to shrink the size of 81/80 and 2048/2025, so that
intervals off by these are closer to just. If we at the same time
decide to make them equal in size, we want to temper out
(81/80)/(2048/2025) = 32805/32768; hence, if we want this we need a
schismic temperament tuning. This gives us one constraint
on the tuning for 3 and 5, to get another, we might insist that 32/25,
which is 5/4 sharp by 128/125, come out to exactly 14/11. This gives
us a tuning of schismic (in fact of the 11-limit temperament tempering
out the schisma and 176/175) which works as a well tuning; the
values of which are

[1200, 1901.094, 2791.246]

which is pretty close to 89-equal.

More exotic plans are possible; for instance if we determine that
192/125 should come out as 20/13 and 162/125 as 13/10 we get a tuning
of the 13-limit temperament (or {2,3,5,13}-temperament) tempering out
325/324 and 625/624. Adding the schisma gives 53-et, but
the tuning described above with the exact exotic fifths and thirds is

[1200, 1902.809, 2785.674]

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> By a well tuning I mean a regular tuning (a tuning of the primes in
> some p-limit) which can be applied to p-limit JI scales in order to
> produce a circulating temperament. It does not need to be a regular
> temperament tuning, though the examples I give are.

Sorry, I meant this for tuning-math. You'll just have to live with it,
I guess. :)