Grackle and well-tuning

The well tuning I used on Schumann

/makemicromusic/topicId_7252.html#7253

is a schismic temperament which equates 32/25 with 14/11, which means
it is a planar temperament tempering out the schisma and
(32/25)/(14/11) = 176/175. The Ellis duodene is a modmos as defined
recently here:

/tuning-math/message/10805

If we look at what it becomes in schismic, we have

-9,-8,-7,-6; -1,0,1,2; 7,8,9,10

Minor thirds appear as an interval of +9 fifths, and since we have a
couple of 18s in this list, it makes sense to equate them with 10/7s,
which they are close to in any tuning range near 2^(267/457) fifths.
This would mean adding (6/5)^2/(10/7) = 126/125 to our commas. The
11-limit linear temperament tempering out the schisma, 126/125, and
176/175 is 11-limit grackle, so this well-tuning can be regarded as
tuning in grackle. Grackle is done well by 89-et; the version of
101-equal we would want to use for it is h89+h12, where these are
standard vals; the result differes only in the 11-limit. The version
of 457-et we would want would be 5*h89+h12; obviously we could use
anything in between.

<89 141 207 250 308|
<101 160 235 284 250|
<457 724 1063 1284 1582|

In between we have the semiconvergents 190, 279, 368 any of which
could also be used. A poptimal grackle fifth, such as from 255-et,
would give something closer to the duodene and less circulating.

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

> Minor thirds appear as an interval of +9 fifths, and since we have a
> couple of 18s in this list, it makes sense to equate them with 10/7s,
> which they are close to in any tuning range near 2^(267/457) fifths.
> This would mean adding (6/5)^2/(10/7) = 126/125 to our commas. The
> 11-limit linear temperament tempering out the schisma, 126/125, and
> 176/175 is 11-limit grackle, so this well-tuning can be regarded as
> tuning in grackle.

If you take 7-limit meantone, and replace the 81/80 with a
32805/32768, you get grackle if you pair it with 126/125, and
garibaldi from 225/224, so these are logical places to start looking
for well-tunings. Grackle is closer to meantone and shrinks 81/80
more, so it seems like the more convinicing candidate. If we look at
commas going up to the 11-limit, 99/98 does not give as good a version
of grackle as 176/175 does, so if we choose this we've got something
up to the 11-limit which might work.

This isn't how I came up with grackle for well-tunings, but it could
have been. We can do the same sort of thing in other connections; for
instance, if we replace 3125/3072 in 7-limit magic with the
wuerschmidt comma 393216/390625 we get 7-limit wuerschmidt which we
might try to adapt to a well-tuning for magic blocks. This probably
won't work since it would lead to sharp major thirds rather than flat
ones; so a search for all decent 5/4 generator temperaments like the
one I did for 3/2 generator temperaments might be in order.

Alternatively, we could simply proceed as before and look for
temperaments which shrink 3125/3072 without converting it to a unison;
for example the standard 138-equal, or 91 or 94. If I look at 138, it
suggests superkleismic tempering which has 6/5 generators, so some
connection with 5/4 generators would need to be established. There's a
lot of room left for exploration, I think.