21/109-comma meantone

A more regular way to obtain 14/11 thirds is as follows: if a circle of
fifths is [a, a, a, sqrt(176/7)/a^3, a, a, a, sqrt(176/7)/a^3,
a, a, a, 56/(11*a^3)] then the major thirds are as we want them, namely
[sqrt(77)/7, sqrt(77)/7, sqrt(77)/7, sqrt(77)/7, sqrt(77)/7,
sqrt(77)/7, sqrt(77)/7, sqrt(77)/7, 14/11, 14/11, 14/11, 14/11].
Instead of making some of the fifths pure, we can take all but the wolf
to be of the same size. This gives us fifths of size (176/7)^(1/8),
or 697.81 cents, which is more or less 21/109-comma meantone. The wolf
is not too bad at 2^(3/2) (7/11)^(11/8), or 724 cents.

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

>
> Instead of making some of the fifths pure, we can take all but the wolf
> to be of the same size. This gives us fifths of size (176/7)^(1/8),
> or 697.81 cents, which is more or less 21/109-comma meantone.

By way of tying it all together, note that the same analysis carried
out for a fifth of (224/9)^(1/8) makes the four large thirds equal to
9/7. This fifth is close to the brat = -1 or 5/17-comma fifth of Erv
Wilson--the Wilson fifth f satisfying f^4-2f-2=0 is 695.63 cents, only
0.016 cents sharper than (224/7)^(1/8). Put another way, the Wilson
fifth, used as a meantone fifth, gives four thirds of size 434.96
cents, only 0.1276 cents shy of a pure 9/7, *and* it has synchronized
beating with its "real" major thirds! Quite an interesting temperament,
particularly given what we've seen before.

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

Put another way, the Wilson
> fifth, used as a meantone fifth, gives four thirds of size 434.96
> cents, only 0.1276 cents shy of a pure 9/7, *and* it has
synchronized
> beating with its "real" major thirds! Quite an interesting
temperament,
> particularly given what we've seen before.

that's remarkable! did wilson ever notice this?

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

> Put another way, the Wilson
> > fifth, used as a meantone fifth, gives four thirds of size 434.96
> > cents, only 0.1276 cents shy of a pure 9/7, *and* it has
> synchronized
> > beating with its "real" major thirds! Quite an interesting
> temperament,
> > particularly given what we've seen before.
>
> that's remarkable! did wilson ever notice this?

It would be nice to know.

The idea of squeezing extra intervals, such as 9/7, out of a
temperament, like the idea of synchronized beating, can be extended
beyond meantone. I've found one example--if we temper 15 notes in
kleismic temperament, using a minor third generator of (56/9)^(1/10),
we get five 9/7 thirds after the ten major thirds. Alas, the sharp
thirds go with sharp fifths, which could pass for 17/11's but not
very well for 3/2's.

>

I am not sure which aspect of this is being questioned as being noticed. Sychronized beating he
was definately going for. i missed the post from where this is from too.
he tends to be overlook his material over and over quite a bit. he also does concider this quite
an "artistic" accomplishment as it was for him quite "creative"

>
> From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
>
>
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> Put another way, the Wilson
> > fifth, used as a meantone fifth, gives four thirds of size 434.96
> > cents, only 0.1276 cents shy of a pure 9/7, *and* it has
> synchronized
> > beating with its "real" major thirds! Quite an interesting
> temperament,
> > particularly given what we've seen before.
>
> that's remarkable! did wilson ever notice this?

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> Put another way, the Wilson
> > fifth, used as a meantone fifth, gives four thirds of size 434.96
> > cents, only 0.1276 cents shy of a pure 9/7, *and* it has
> synchronized
> > beating with its "real" major thirds! Quite an interesting
> temperament,
> > particularly given what we've seen before.
>
> that's remarkable! did wilson ever notice this?

I happened to notice that this is extremely close to the fifth that I
described for a portion of my 19+3 temperament:

/tuning/topicId_38076.html#38287

A portion of that message has the following:

<< The tuning is a 19-tone well temperament (19-WT) with 3 auxiliary
tones to supply the 11 factors for the three 15-limit otonalities,
which are built on F, C, and G. The circle of 19 fifths has twelve
fifths from F to E-sharp of ~695.61 cents, or ~6.34 cents narrow
(such that 8 consective fifths give an exact 9:7). The remaining
seven fifths are of equal size and start on F-flat (same pitch as E-
sharp, but renamed) and end at F; these are ~693.23 cents, or ~8.72
cents narrow. The three auxiliary tones (B-semiflat, F-semisharp,
and C-semisharp) are tuned as just 11:9's above G, D, and A,
respectively. >>

This would seem to make that portion of the cirle of fifths starting
on F and ending on E-sharp the virtual equivalent of Erv Wilson's
temperment, i.e., for all practical purposes equal-beating. This is
something that I didn't notice until your messages pointed this out,
although at the time I devised the 19+3 temperament (1978) I *did*
notice that the major triads have a (harmonic) consonance much more
akin to 1/4-comma meantone temperament than to 19-ET, but melodically
the intervals sound more like 19-ET than 1/4-comma meantone. Now I
understand why.

Would Paul (or any others) consider using this instead of 19-ET,
particularly in view of the 15-limit otonalities that the 3
additional tones provide? It's true that a guitar would require some
zig-zag fretting, but it should still be easier to play than a 31-ET
instrument.

--George