Well temperaments for divisions other than 12

Well temperaments work for 12-equal because it has a good fifth; we
have a Pythagorean comma to distibute among twelve fifths, which on
average is a schisma of tempering per fifth. Hence a well-temperament
for twelve notes should involve tempering a fifth, and so really can
only be for
meantone or schismic temperament.

Well temperings for other divisions of the octave will not normally
involve fifths; instead, they will be for temperaments which have
generators for which the division in question has particularly good
values. This means a well-tempering of 19 notes should be a tempering
of hanson, and involve distributing the 19-comma among 19 notes, which
comes out as on average just 1/7 of a cent per note. This doesn't give
us much to play with but certainly we could use it to well-temper hanson.

22-equal has a good 9/7, which suggests well-tempering hedgehog, the
temperament with period 600 cents and generator a slightly sharp 9/7.
We get a 14.1 cent comma to distibute among 11 supermajor third
generators, or 1.28 cents per generator, which should allow us to do
something. 31-equal has a nice major third, and this suggests
well-tempering wuerschmidt; another possibility would be
mothra/supermajor seconds.

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

31-equal has a nice major third, and this suggests
> well-tempering wuerschmidt; another possibility would be
> mothra/supermajor seconds.

We might also well-temper hemiwuerschmidt or hemimothra, the latter
better known undet the name of miracle. It seems to me this could be
interesting.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> Well temperaments work for 12-equal because it has a good fifth; we
> have a Pythagorean comma to distibute among twelve fifths, which on
> average is a schisma of tempering per fifth. Hence a well-
temperament
> for twelve notes should involve tempering a fifth, and so really can
> only be for meantone or schismic temperament.
>
> Well temperings for other divisions of the octave will not normally
> involve fifths; instead, they will be for temperaments which have
> generators for which the division in question has particularly good
> values. This means a well-tempering of 19 notes should be a
tempering
> of hanson, and involve distributing the 19-comma among 19 notes,
which
> comes out as on average just 1/7 of a cent per note. This doesn't
give
> us much to play with but certainly we could use it to well-temper
hanson.

Yes, not much room at all to maneuver.

My approach to a 19-tone well-temperament (with 3 auxiliary tones):
/tuning/topicId_38076.html#38287
treats the division as a meantone system and simply accepts that the
19 minor 3rds will have a total error greater than with 19-ET when
the majority of fifths are altered to ~5/17 comma. Since that
posting I've concluded that I might as well make them exactly 5/17-
comma to make the 9 best major triads equal-beating, which also
serves to improve the other primes (7 and 13.)

> 22-equal has a good 9/7, which suggests well-tempering hedgehog, the
> temperament with period 600 cents and generator a slightly sharp
9/7.
> We get a 14.1 cent comma to distibute among 11 supermajor third
> generators, or 1.28 cents per generator, which should allow us to do
> something. 31-equal has a nice major third, and this suggests
> well-tempering wuerschmidt; another possibility would be
> mothra/supermajor seconds.

Had you considered that a circle of 17 fifths could be well-tempered
as a 13-limit non-5 tuning? (You'll have to wait for Xenharmonikon
18 to see how I did it back in 1978.)

--George

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

> Had you considered that a circle of 17 fifths could be well-tempered
> as a 13-limit non-5 tuning? (You'll have to wait for Xenharmonikon
> 18 to see how I did it back in 1978.)

It might be interesting to look at non-5 Fokker blocks for 17.

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

> Had you considered that a circle of 17 fifths could be well-tempered
> as a 13-limit non-5 tuning? (You'll have to wait for Xenharmonikon
> 18 to see how I did it back in 1978.)

I suppose you could start from 17&29, 17&46, 17&80 which as full
13-limit temperaments are all different but which as no-fives are the
same. Another interesting starting point is 41&58, or hemififths,
which has a nice no-fives MOS on 17 notes; that has a circle of 17
half-fifths.