Some well-behaved brats for meantone

I checked integer brat ratios, looking for values where all three
brat ratios were integers or their inverses, and did the same for
inverse integer brats. The result was this list:

[-6, 3, -2], [-1, 1, -1], [1, 1/5, 5], [2, -1/5, -10], [4, -1, -4],
[9, -3, -3], [1/4, 1/2, 1/2]

Here the first number is the brat, the others the other beat ratios.
The b=1 fifth is very flat, but the others are all usable, and b=-6,
as we remarked before, is even poptimal. Robert's b=2 doesn't look to
be that good compared to some of the rest of these; the b=-1 is
particularly striking; it's pretty close to 2/7-comma, and even
closer to 5/17-comma. The exact value of the fifth is the positive
real root of f^4-2f-2.

By the way, Paul, have you any comments on the psychoacoustics of all
of this? What do you think of Robert's idea?

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

> I checked integer brat ratios, looking for values where all three
> brat ratios were integers or their inverses, and did the same for
> inverse integer brats. The result was this list:
>
> [-6, 3, -2], [-1, 1, -1], [1, 1/5, 5], [2, -1/5, -10], [4, -1, -4],
> [9, -3, -3], [1/4, 1/2, 1/2]

If we look for everything with either infinity or a Tenney height of
ten or less for all three ratios, we add to the above

[2/3, 1/3, 2], [-1/6, 2/3, -1/4], [0, 3/5, 0],
[infinity, infinity, -5/2], [3/2, 0, infinity].

The b=2/3 system is a "flattone" system, in the vicinity of the 26 et;
the two "infinity" systems are of course 1/4-comma meantone and JI.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith <gwsmith@s...>"
<gwsmith@s...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith <gwsmith@s...>"
> <gwsmith@s...> wrote:
>
> > I checked integer brat ratios, looking for values where all three
> > brat ratios were integers or their inverses, and did the same for
> > inverse integer brats. The result was this list:
> >
> > [-6, 3, -2], [-1, 1, -1], [1, 1/5, 5], [2, -1/5, -10], [4, -1, -4],
> > [9, -3, -3], [1/4, 1/2, 1/2]
>
> If we look for everything with either infinity or a Tenney height of
> ten or less for all three ratios, we add to the above
>
> [2/3, 1/3, 2], [-1/6, 2/3, -1/4], [0, 3/5, 0],
> [infinity, infinity, -5/2], [3/2, 0, infinity].
>
> The b=2/3 system is a "flattone" system, in the vicinity of the 26 et;
> the two "infinity" systems are of course 1/4-comma meantone and JI.

You mean Pythagorean. JI is not a linear temperament. I made this same
mistake myself recently. Of course you (and I, when I made the
mistake) meant that the fifths are just; so-called "3-limit JI"; but
this is an oxymoron to most people, and in any case we are talking
5-limit here.

Yes. I can see the same results in my spreadsheet. However, where you
find something to be very close to say 5/17-comma, I find it to be
exactly so (to the limit of accuracy of IEEE floating-point numbers).

This is because (and this answers a question of Paul Erlich's on the
tuning list in response to a 4 year old post of mine), like Bob
Wendell, I am using the approximation that frequency deviation is
proportional to comma fraction deviation (i.e. log frequency
deviation). This is an extremely good approximation in the range from
0 to about 20 cents. The few thousandths of a cent error that result
in the size of the generator are of no interest to anyone but a
mathematician, and it means that one can deal with linear equations
rather than polynomials.

And of course, just as you do with the polynomials, you could solve
these approximate linear equations algebraicly instead of numerically
to get the rational comma fraction solutions that are familiar to people.

--- In tuning-math@yahoogroups.com, "Dave Keenan <d.keenan@u...>"
<d.keenan@u...> wrote:
> You mean Pythagorean. JI is not a linear temperament.

I saw this after posting, but it didn't seem worth another posting to
correct it.

> Yes. I can see the same results in my spreadsheet. However, where
you
> find something to be very close to say 5/17-comma, I find it to be
> exactly so (to the limit of accuracy of IEEE floating-point
numbers).

It's the linearization business I just posted about on tuning.

> And of course, just as you do with the polynomials, you could solve
> these approximate linear equations algebraicly instead of
numerically
> to get the rational comma fraction solutions that are familiar to
people.

See above.