Yahoo Tuning Groups Ultimate Backup tuning Well temperaments done right this time

I goofed last round, and when I went back and recalculated this, I got
reasonably non-squirrelly candidates, which however were really only
variations on four well-temperaments. One is Wendell Well; the other
three are probably ones Robert tossed back in the water.

WW is the only one which allows us to put the slightly off ratio with
the best third, but it seems to me it makes more sense to choose the
odd beat ratio most nearly equal to 3/2 or 2. For WW this would be a
3016/2019 in place of 3/2, but there are two places to put this,
leading to the scale I list below, and another one where the eighth
degree, 13155/8306 in the version I give here, is raised by
23680/23679 to 59200/37377.

Wendell Well

[1, 4385/4153, 13970/12459, 14800/12459, 15640/12459, 4/3, 5865/4153,
18662/
12459, 13155/8306, 20902/12459, 7400/4153, 7820/4153],

Smith 1

[1, 9500/8961, 6709/5974, 17776/14935, 11260/8961, 3991/2987,
8445/5974, 13418/8961, 4750/2987,
5020/2987, 26664/14935, 5630/2987]

Smith 2

[1, 9500/8961, 6709/5974, 7125/5974,11260/8961, 3991/2987, 8445/5974,
3/2, 4750/2987, 5020/2987,
26664/14935, 5630/2987]

Smith 3

[1, 9500/8961, 10040/8961, 17776/14935, 11260/8961, 3991/2987,
38000/26883, 13418/8961, 4750/2987, 5020/2987, 15964/8961, 5630/2987]

/tuning-math/files/well%20temperament%20plots/wendellwell.jpg

/tuning-math/files/well%20temperament%20plots/smith1.jpg

/tuning-math/files/well%20temperament%20plots/smith2.jpg

/tuning-math/files/well%20temperament%20plots/smith3.jpg

🔗Robert Wendell <rwendell@cangelic.org>

HI, Gene! Pardon me, but I'm having trouble figuring out just what
the charts these links point to represent.

Cheers,

Bob

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> I goofed last round, and when I went back and recalculated this, I
got
> reasonably non-squirrelly candidates, which however were really only
> variations on four well-temperaments. One is Wendell Well; the other
> three are probably ones Robert tossed back in the water.
>
> WW is the only one which allows us to put the slightly off ratio
with
> the best third, but it seems to me it makes more sense to choose the
> odd beat ratio most nearly equal to 3/2 or 2. For WW this would be a
> 3016/2019 in place of 3/2, but there are two places to put this,
> leading to the scale I list below, and another one where the eighth
> degree, 13155/8306 in the version I give here, is raised by
> 23680/23679 to 59200/37377.
>
>
>
> Wendell Well
>
> [1, 4385/4153, 13970/12459, 14800/12459, 15640/12459, 4/3,
5865/4153,
> 18662/
> 12459, 13155/8306, 20902/12459, 7400/4153, 7820/4153],
>
> Smith 1
>
> [1, 9500/8961, 6709/5974, 17776/14935, 11260/8961, 3991/2987,
> 8445/5974, 13418/8961, 4750/2987,
> 5020/2987, 26664/14935, 5630/2987]
>
> Smith 2
>
> [1, 9500/8961, 6709/5974, 7125/5974,11260/8961, 3991/2987,
8445/5974,
> 3/2, 4750/2987, 5020/2987,
> 26664/14935, 5630/2987]
>
> Smith 3
>
> [1, 9500/8961, 10040/8961, 17776/14935, 11260/8961, 3991/2987,
> 38000/26883, 13418/8961, 4750/2987, 5020/2987, 15964/8961,
5630/2987]
>
>
>
> /tuning-math/files/well%20temperament%
20plots/wendellwell.jpg
>
> /tuning-math/files/well%20temperament%
20plots/smith1.jpg
>
> /tuning-math/files/well%20temperament%
20plots/smith2.jpg
>
> /tuning-math/files/well%20temperament%
20plots/smith3.jpg

🔗Robert Wendell <rwendell@cangelic.org>

From looking at your graphic for my temperament, I conclude that the
vertical scale on the left side is the width of major thirds in
cents. Since the horizontal scale at the bottom goes from zero to
twelve and the low point is at the value of the third on C, I assume
this axis represents the cycle of fifths, but I'm puzzled by the
significance of the continuum. This is not a set of discrete values,
but a continuum. ??? Please explain.

I'm also wondering where the beat ratios come into the picture in
these charts, if at all. It seems they may be somehow implicit but
not represented explicitly. Are these graphic representations of a
continuum of possibilities that are a function of maintaining the
previously stated precise beat ratios? ??? This doesn't add up for
me, so the only thing I can imagine that makes sense to me right now
is that the continuum is an artifice introduced by some graphics
package and there is no significance outside of the intersections
with discrete points along the cycle of fifths axis. ? :)

Cheers,

Bob

🔗Robert Wendell <rwendell@cangelic.org>

Why does it make more sense to put the small remainder (0.0058 cents)
from the Pythagorean comma on a 3/2 beat ratio? The whole philosophy
behind the beat synchrony employed in this design approach is to
provide the thirds most severly compromised in terms of intonation
the most precise synchronization in terms of the beat ratios they
form with the other third in the major triads of which they are a
part.

Since C is by far the least compromised third in the whole
temperament, it seems the best place to put this small remainder,
although it is so small it probably doesn't matter very much, unless
I'm missing something. Your graphics seem to vary quite significantly
from each other, so I suspect I'm missing something fundamental here.

Also, I would have thought that if we're going to place the remainder
on something other than C, why not on another beat ratio of 2.0,
since this ratio is, to my ear, the most robust in terms of how much
tolerance the ear forgives over a range of deviations from a perfect
2.0? I suspect this is because it is a much simpler ratio than 3/2.
I'm curious to follow your reaction to all this and understand the
thinking behind your statements.

Cheers,

Bob

P.S. I'm also wondering whether you have a convenient way to convert
your format here to provid offsets from ET instead of ratios. My
spreadsheets are set up so that the offsets drive all the
calculations. I have the GUI version of Scala downloaded now, so
maybe I could use that, but although I find the program extremely
rich, I also find the interface quite unfriendly. I spent forever
trying to figure out how to achieve some very simple results, and I'm
not a software neophyte. I used to do tech support professionally for
DOS products.

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "Robert Wendell" <rwendell@c...> wrote:

> Why does it make more sense to put the small remainder (0.0058 cents)
> from the Pythagorean comma on a 3/2 beat ratio?

Because that is where it is smallest.

> P.S. I'm also wondering whether you have a convenient way to convert
> your format here to provid offsets from ET instead of ratios.

That would be very easy to do.

What do you think of the "Wendellized" versions of Werckmeister III and
Vallotti, by the way? Have you looked at that?

Well temperaments done right this time

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Robert Wendell <rwendell@cangelic.org>

🔗Robert Wendell <rwendell@cangelic.org>

🔗Robert Wendell <rwendell@cangelic.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗manuel.op.de.coul@eon-benelux.com