Acceptance regions

For a linear temperament, let e be the absolute value of the error and
c the complexity. Then the log-flat badness is e c^(n/(n-2)), where n
is the number of primes (with 2 being the number of generators; in
general it becomes e c^(n/(n-g))). If we set x = log(e), y = log(c)
then bounds on error, complexity and badness become

x <= a, y <= b, y+(n/(n-2))x <= c

This defines a triangular region in the xy plane. We could define a
region with smooth boundry instead, in particular an ellipse. If we
took a set of temperaments we wanted on our list, and analyzed them
statistically, we might have an idea of what region we are looking
for. One way might be to do principle component analysis, and convert
the data set into something we can draw a nice circle around. All of
which leads to the quesiton, which temperaments do we start out with?

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

> This defines a triangular region in the xy plane. We could define a
> region with smooth boundry instead, in particular an ellipse. If we
> took a set of temperaments we wanted on our list, and analyzed them
> statistically, we might have an idea of what region we are looking
> for. One way might be to do principle component analysis, and convert
> the data set into something we can draw a nice circle around. All of
> which leads to the quesiton, which temperaments do we start out with?

Here is Paul's true L1 top list of 32 temperaments. I hope I made this
so Paul can process it easily. On each line is the temperament, the
coordinates after performing a principle component analysis, and the
radius--distance from the midpoint. It is sorted in order of
increasing radius, but this is not a badness figure per se; the
question is, however, if a good list can be obtained by reducing the
radius, so you want to check the bottom part of the list and see what
is essential. I started with stuff like father and ennealimmal, so
they are still in there. I don't know what temperaments not on the
list fall inside the circle.

The results strike me as a little weird, in the sense that meantone
comes in at number 29. However the last three temperaments on the list
are outliers in terms of radius, and we could get a list of 29 by
deep-sixing them. Since the temperaments in question are beep, father
and the {21/20, 28/27} temperament, that would definately please some
people.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> > This defines a triangular region in the xy plane. We could define
a
> > region with smooth boundry instead, in particular an ellipse. If
we
> > took a set of temperaments we wanted on our list, and analyzed
them
> > statistically, we might have an idea of what region we are looking
> > for. One way might be to do principle component analysis, and
convert
> > the data set into something we can draw a nice circle around. All
of
> > which leads to the quesiton, which temperaments do we start out
with?
>
> Here is Paul's true L1 top list of 32 temperaments.

Where?

> I hope I made this

What?

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

> Here is Paul's true L1 top list of 32 temperaments.

Somehow it didn't actually get added; here it is. Blackwood rules, but
if we toss beep and father it won't anymore, I imagine.

[0, 5, 0, 8, 0, -14] [.2064288959e-1, -.1043499280]
[4, 2, 2, -6, -8, -1] [.1314173477, -.5776059996e-1]
[3, 0, 6, -7, 1, 14] [-.9759478278e-1, -.1184137769]
[6, 5, 22, -6, 18, 37] [-.1480758164, .5924299839e-1]
[4, 4, 4, -3, -5, -2] [-.1139119995, -.1276836173]
[9, 5, -3, -13, -30, -21] [-.1902784347, .1548210912e-1]
[16, 2, 5, -34, -37, 6] [.7687685623e-1, .1955158230]
[10, 9, 7, -9, -17, -9] [-.2605524521, -.2903797753e-1]
[2, 8, 8, 8, 7, -4] [-.2717885664, -.1286298091]
[1, -8, -14, -15, -25, -10] [-.2989669753, -.4080104036e-1]
[6, -7, -2, -25, -20, 15] [-.3049734797, -.3330484564e-1]
[7, -3, 8, -21, -7, 27] [-.3074384258, -.4688241454e-1]
[1, 4, -2, 4, -6, -16] [-.2696069971, -.1733403969]
[2, 8, 1, 8, -4, -20] [-.2896647403, -.1420876517]
[6, 5, 3, -6, -12, -7] [-.3075281408, -.1335897048]
[4, -3, 2, -14, -8, 13] [-.3195568516, -.1404043854]
[1, 9, -2, 12, -6, -30] [-.3339264533, -.1182544459]
[2, 1, 6, -3, 4, 11] [.3610308982, .5104682762e-2]
[0, 0, 7, 0, 11, 16] [.3740392389, .1111744330e-1]
[3, 0, -6, -7, -18, -14] [-.3548963487, -.1508618972]
[2, 25, 13, 35, 15, -40] [.2647446362, .3022722620]
[1, -3, -4, -7, -9, -1] [.4175560613, .2406964027e-1]
[2, -4, -4, -11, -12, 2] [-.4040260830, -.1851377308]
[5, 1, 12, -10, 5, 25] [-.4482934949, -.1394673397]
[7, 9, 13, -2, 1, 5] [-.4461970788, -.1511829310]
[5, 13, -17, 9, -41, -76] [.3651340235, .3600565223]
[18, 27, 18, 1, -22, -34] [.4028591494, .3906980840]
[13, 14, 35, -8, 19, 42] [.4248301718, .3944244021]
[1, 4, 10, 4, 13, 12] [-.5476384245, -.2120574236]
[2, 3, 1, 0, -4, -6] [.9385116395, .1474599147]
[1, -1, 3, -4, 2, 10] [.9519941420, .1568469925]
[1, 4, 3, 4, 2, -4] [.9852784843, .1709570383]

I don't understand. Those clearly aren't error and complexity numbers
you're giving after the wedgie. What am I to do with them?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> > Here is Paul's true L1 top list of 32 temperaments.
>
> Somehow it didn't actually get added; here it is. Blackwood rules,
but
> if we toss beep and father it won't anymore, I imagine.
>
> [0, 5, 0, 8, 0, -14] [.2064288959e-1, -.1043499280]
> [4, 2, 2, -6, -8, -1] [.1314173477, -.5776059996e-1]
> [3, 0, 6, -7, 1, 14] [-.9759478278e-1, -.1184137769]
> [6, 5, 22, -6, 18, 37] [-.1480758164, .5924299839e-1]
> [4, 4, 4, -3, -5, -2] [-.1139119995, -.1276836173]
> [9, 5, -3, -13, -30, -21] [-.1902784347, .1548210912e-1]
> [16, 2, 5, -34, -37, 6] [.7687685623e-1, .1955158230]
> [10, 9, 7, -9, -17, -9] [-.2605524521, -.2903797753e-1]
> [2, 8, 8, 8, 7, -4] [-.2717885664, -.1286298091]
> [1, -8, -14, -15, -25, -10] [-.2989669753, -.4080104036e-1]
> [6, -7, -2, -25, -20, 15] [-.3049734797, -.3330484564e-1]
> [7, -3, 8, -21, -7, 27] [-.3074384258, -.4688241454e-1]
> [1, 4, -2, 4, -6, -16] [-.2696069971, -.1733403969]
> [2, 8, 1, 8, -4, -20] [-.2896647403, -.1420876517]
> [6, 5, 3, -6, -12, -7] [-.3075281408, -.1335897048]
> [4, -3, 2, -14, -8, 13] [-.3195568516, -.1404043854]
> [1, 9, -2, 12, -6, -30] [-.3339264533, -.1182544459]
> [2, 1, 6, -3, 4, 11] [.3610308982, .5104682762e-2]
> [0, 0, 7, 0, 11, 16] [.3740392389, .1111744330e-1]
> [3, 0, -6, -7, -18, -14] [-.3548963487, -.1508618972]
> [2, 25, 13, 35, 15, -40] [.2647446362, .3022722620]
> [1, -3, -4, -7, -9, -1] [.4175560613, .2406964027e-1]
> [2, -4, -4, -11, -12, 2] [-.4040260830, -.1851377308]
> [5, 1, 12, -10, 5, 25] [-.4482934949, -.1394673397]
> [7, 9, 13, -2, 1, 5] [-.4461970788, -.1511829310]
> [5, 13, -17, 9, -41, -76] [.3651340235, .3600565223]
> [18, 27, 18, 1, -22, -34] [.4028591494, .3906980840]
> [13, 14, 35, -8, 19, 42] [.4248301718, .3944244021]
> [1, 4, 10, 4, 13, 12] [-.5476384245, -.2120574236]
> [2, 3, 1, 0, -4, -6] [.9385116395, .1474599147]
> [1, -1, 3, -4, 2, 10] [.9519941420, .1568469925]
> [1, 4, 3, 4, 2, -4] [.9852784843, .1709570383]

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> I don't understand. Those clearly aren't error and complexity numbers
> you're giving after the wedgie. What am I to do with them?

I thought you might like to plot them. However what I really need is a
list of temperaments you think we ought to include. Don't worry about
excluding.

The numbers have a mean value of zero, and are adjusted so that the 32
temperaments are in a more-or-less circular blob. Apparently blackwood
was near the average values for complexity and error, and hence ended
up closest to the center (in a suitably adjusted sense of "closest"
the principle component analysis gave.) Center of the accpetance
region is not a goodness measure; all it means is that you are stuck
with blackwood if you use this sort of region. No getting rid of it
other than by starting with a new set of temperaments, which moves the
midpoint somewhere else and changes the ellipses you draw.

Have you listed your must-have 7-limit linear temperaments already?

It occurs to me we aren't getting a triangular region, we are getting
a quadrant and then sawing off a corner using the badness cutoff
line, so we have an unbounded region. It acts like it's bounded
because we can only get so far when trying simultaneously for small
error and small compexity, but no error and no complexity is down at
-infinity, -infinity. This makes the idea of using an ellipse pretty
dubious; a parabolic region might make more sense.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > I don't understand. Those clearly aren't error and complexity
numbers
> > you're giving after the wedgie. What am I to do with them?
>
> I thought you might like to plot them. However what I really need
is a
> list of temperaments you think we ought to include.

I'm not omniscient.

> Don't worry about
> excluding.

This isn't helping.

> The numbers have a mean value of zero, and are adjusted so that the
32
> temperaments are in a more-or-less circular blob. Apparently
blackwood
> was near the average values for complexity and error, and hence
ended
> up closest to the center (in a suitably adjusted sense of "closest"
> the principle component analysis gave.) Center of the accpetance
> region is not a goodness measure; all it means is that you are stuck
> with blackwood if you use this sort of region. No getting rid of it
> other than by starting with a new set of temperaments, which moves
the
> midpoint somewhere else and changes the ellipses you draw.
>
> Have you listed your must-have 7-limit linear temperaments already?

No; the idea was to do a complete search within an extra-large region
and then look for the widest moats. Dave and I have done this for
equal temperaments, 5-limit linear temperaments, 7-limit planar
temperaments. Now we're asking for your help.

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

> No; the idea was to do a complete search within an extra-large
region
> and then look for the widest moats. Dave and I have done this for
> equal temperaments, 5-limit linear temperaments, 7-limit planar
> temperaments. Now we're asking for your help.

And the reason why we care about moats is?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > No; the idea was to do a complete search within an extra-large
> region
> > and then look for the widest moats. Dave and I have done this for
> > equal temperaments, 5-limit linear temperaments, 7-limit planar
> > temperaments. Now we're asking for your help.
>
> And the reason why we care about moats is?

To come up with a list of temperaments which would not change even if
our cutoff criterion were to be altered by a fair amount.

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

> > And the reason why we care about moats is?
>
> To come up with a list of temperaments which would not change even if
> our cutoff criterion were to be altered by a fair amount.

I thought these moats were gerrymandered, so how is that going to
work? Anyway, isn't it more important to have a list with the good
stuff on it, moat or no moat?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > > And the reason why we care about moats is?
> >
> > To come up with a list of temperaments which would not change
even if
> > our cutoff criterion were to be altered by a fair amount.
>
> I thought these moats were gerrymandered, so how is that going to
> work?

Unclear on your question . . .

> Anyway, isn't it more important to have a list with the good
> stuff on it,

That's obviously the starting point.

> moat or no moat?

Without a moat, there would be questionable cases, of "if those are
in, why isn't this in" and "if those are out, why isn't this out".

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

> Without a moat, there would be questionable cases, of "if those are
> in, why isn't this in" and "if those are out, why isn't this out".

With a moat, there might be a question of why you are using a
seemingly unmotivated, ad hoc criterion. Maybe we could formalize it
to a similarity circle or something that could be justified?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > Without a moat, there would be questionable cases, of "if those
are
> > in, why isn't this in" and "if those are out, why isn't this out".
>
> With a moat, there might be a question of why you are using a
> seemingly unmotivated, ad hoc criterion. Maybe we could formalize it
> to a similarity circle or something that could be justified?

If the two agree, all the better.