--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > #1
> >
> > 2^-90 3^-15 5^49
> >
> > This is not only the the one with lowest badness on the list, it
is
> the smallest comma, which suggests we are not tapering off, and is
> evidence for flatness.
> >
> > Map:
> >
> > [ 0 1]
> > [49 -6]
> > [15 0]
> >
> > Generators: a = 275.99975/1783 = 113.00046/730; b = 1
> >
> > I suggest the "Woolhouse" as a name for this temperament, because
> of the 730. Other ets consistent with this are 84, 323, 407, 1053
and
> 1460.
> >
> > badness: 34
> > rms: .000763
> > g: 35.5
> > errors: [-.000234, -.001029, -.000796]
> >
> > #2 32805/32768 Schismic badness=55
> >
> > #3 25/24 Neutral thirds badness=82
> >
> > #4 15625/15552 Kleismic badness=97
> >
> > #5 81/80 Meantone badness=108
> >
> > It looks pretty flat so far as this method can show, I think.
>
> How well do these results back up my now-famous (I hope) heuristic,
> which involves only the size of the numbers in, and the difference
> between numerator and denominator of, the unison vector? How might
we
> weight the gens and/or cents measures so that the heuristic will
work
> perfectly?

Let's start with #5 and work our way up to #1:

U V W X Y Z
unisonvector rms(oct) |n-d|/(d*log(d)) V/U g log(d) X/Y
------------ -------- --------------- ---- ---- ------ ---
80/81 0.003517 0.002809 .799 2.944 4.394 .67
15552/15625 0.0008583 0.0004838 .564 4.546 9.657 .471
24/25 0.024083 0.012427 .516 1.414 3.219 .439
32768/32805 0.00013475 0.00010847 .805 6.976 10.398 .671
[90 15 -49] 6.358e-007 3.44e-007 .541 35.5 78.862 .450

Our current "g" measure is clearly too large when the generator is a
fifth, as I've been trying to complain for quite a while now and only
Dave Keenan has changes his ways accordingly, and the comparison with
the heuristic, though agreeable, suggests that the heuristic is in
fact better than our current measure. What if we weighted the
intervals unequally in both the g and in the "rms" calculations?
Could we get the heuristic to work perfectly? I think that would be
very interesting for our paper.

Moving on to some relatively "bad" examples . . . Gene,
your "Enneadecal" comma should have a power of 2 equal to 14, not 15
as you said, right?

U V W X Y Z
unisonvector rms(oct) |n-d|/(d*log(d)) V/U g log(d) X/Y
------------ -------- --------------- ---- ---- ------ ---
[52 17 -34] 2.864e-005 1.5102e-005 .527 24.042 54.721 .439
128/135 0.01508 0.010571 .701 2.94 4.905 .599
[14 19 -19] 8.733e-005 5.314e-005 .608 15.513 30.579 .507
648/625 0.009217 0.005716 .620 3.27 6.438 .508
[8 14 -13] 0.0002305 0.0001463 .635 11.045 20.923 .528

Here the heuristic seems to work even better (and the V/U and X/Y are
well within the range of their values for the top 5).