!

>Hmmm. Have you ever seen Farey circles? If you take the rational numbers
>between 1 and 2, and for each p/q make a circle of radius 1/2q^2 and center
>[p/q, 1/2q^2], you get a fractal collection of circles which never
>intersect, but where the circles for adjacent Farey fractions always touch.
>Maybe that could inspire something.

!

-Carl

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> >
> > > Did either of you guys look at the loglog version of the moat-
of-
> 23 7-
> > > limit linear temperaments?
> >
> > I have a plot with unlabled axes and a curved red line on it.
> > Obviously, since I don't know what is being plotted, I draw no
> >conclusion.
>
> I've already clarified this for you!!

Again: The horizontal axis, as always, is *complexity*. The vertical
axis, as always, is *error*. We've already established we're on the
same page on those. It's easy to see, by the tick marks, if either or
both of the axes is scaled logarithmically. The red line is our
proposed moat. And again, the 7-limit 'linear' temperaments are
indexed as follows (I show the first three numbers in the val-wedgie,
since you feel they are the most important):

1) 1 4 10
2) 2 -4 -4
3) 5 1 12
4) 7 9 13
5) 1 4 -2
6) 3 0 -6
7) 4 -3 2
8) 2 8 1
9) 6 5 3
10) 1 9 -2
11) 2 8 8
12) 6 -7 -2
13) 6 10 10
14) 7 -3 8
15) 4 4 4
16) 1 -8 -14
17) 3 0 6
18) 0 0 12
19) 1 4 -9
20) 0 5 0
21) 3 12 -1
22) 10 9 7
23) 3 5 -6
24) 9 5 -3
25) 8 6 6
26) 6 -2 -2
27) 6 5 22
28) 3 12 11
29) 2 -9 -4
30) 11 13 17
31) 6 10 3
32) 4 2 2

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> Again: The horizontal axis, as always, is *complexity*. The vertical
> axis, as always, is *error*.

I don't want complexity and error, I want log(complexity) and
log(error), and labeling the axes if possible is always a good plan.

We've already established we're on the
> same page on those. It's easy to see, by the tick marks, if either or
> both of the axes is scaled logarithmically.

I prefer knowing to guessing.

The red line is our
> proposed moat. And again, the 7-limit 'linear' temperaments are
> indexed as follows (I show the first three numbers in the val-wedgie,
> since you feel they are the most important):

Putting in the names would have made it a hell of a lot clearer than
numbers I had no clue about. I'm looking at a Windows file in Linux,
so the name comes out as "7lin23~1.gif"; I hope the actual name means
something I can understand. The convex hull of your selected
temperaments looks by eyeballing to be 20, 17, 23, 21, 22, 12 which
corresponds to blackwood, augmented, porcupine, supermajor seconds,
nonkleismic, and miracle. This ought to correspond to what I found by
doing the same thing using Maple, but I'm not sure it does. I'm not
impressed with the idea of arbitarily cooking the books to keep
miralce on but leave ennealimmal off, incidentally, if this is part of
the plan. The slope from blackwood to augmented is not much different
than the slope from augmented to porcupine, and it seems you could
start off simply with the slope of -1, and then drop it off gradually.
That would already began to make more sense out of this; I don't think
imagining you see a moat curve when there are in fact lots of ways to
draw the curve and they look quite different is a very good start.

Here is TOP complexity times TOP error for the above:

Blackwood: 46.9
Augmented: 48.8
Porcupine: 45.8
Supermajor seconds: 31.3
Nonkleismic: 23.8
Miracle: 13.3

I checked my big list of 32201 temperaments, and found exactly four
where the error times the complexity was less than 50, and the
complexity was less than 10: blackwood, augmented, diminished, and
dominant seventh. You could simply start out like this, and roll down
gradually, and you'd already be making a hell of a lot more sense.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > Again: The horizontal axis, as always, is *complexity*. The
vertical
> > axis, as always, is *error*.
>
> I don't want complexity and error, I want log(complexity) and
> log(error), and labeling the axes if possible is always a good plan.
>
> We've already established we're on the
> > same page on those. It's easy to see, by the tick marks, if
either or
> > both of the axes is scaled logarithmically.
>
> I prefer knowing to guessing.
>
> The red line is our
> > proposed moat. And again, the 7-limit 'linear' temperaments are
> > indexed as follows (I show the first three numbers in the val-
wedgie,
> > since you feel they are the most important):
>
> Putting in the names would have made it a hell of a lot clearer than
> numbers I had no clue about. I'm looking at a Windows file in Linux,
> so the name comes out as "7lin23~1.gif"; I hope the actual name
means
> something I can understand.

It does.

> The convex hull of your selected
> temperaments looks by eyeballing to be 20, 17, 23, 21, 22, 12 which
> corresponds to blackwood, augmented, porcupine, supermajor seconds,
> nonkleismic, and miracle. This ought to correspond to what I found
by
> doing the same thing using Maple, but I'm not sure it does. I'm not
> impressed with the idea of arbitarily cooking the books to keep
> miralce on but leave ennealimmal off,

If you followed the discussion, you'd see that the only reason this
happened was that your list of 126 was too thin in the high-
complexity region to find a moat; otherwise ennealimmal might well
have made it in. Rather than assuming ill intentions on the part of
everyone else, try slowing down and taking a few extra moments to
survey the writings of those who are painstakingly working on this
stuff, rather than firing off a knee-jerk reaction upon a sorely
incomplete impression.

> incidentally, if this is part of
> the plan. The slope from blackwood to augmented is not much
different
> than the slope from augmented to porcupine, and it seems you could
> start off simply with the slope of -1, and then drop it off
gradually.
> That would already began to make more sense out of this;

Why is that?

> I don't think
> imagining you see a moat curve when there are in fact lots of ways
to
> draw the curve and they look quite different is a very good start.

But if it's done quantitatively? Dave said he was unable to find a
better moat according to his *quantitative* measure; perhaps it's
possible to verify this computationally?

> Here is TOP complexity times TOP error for the above:
>
> Blackwood: 46.9
> Augmented: 48.8
> Porcupine: 45.8
> Supermajor seconds: 31.3
> Nonkleismic: 23.8
> Miracle: 13.3
>
> I checked my big list of 32201 temperaments, and found exactly four
> where the error times the complexity was less than 50, and the
> complexity was less than 10: blackwood, augmented, diminished, and
> dominant seventh. You could simply start out like this, and roll
down
> gradually, and you'd already be making a hell of a lot more sense.

In terms of having your inclusion list be insensitive to a fair range
of variation in the inclusion criterion? If so, I'm all ears.