acceptace regions

We might try in analyzing or plotting 7-limit linear temperaments a
transformation like this:

u = 4 - ln(complexity) - ln(error)
v = 12 - 4 ln(complexity) - ln(error)

We can obtain a fine list simply by taking everything in the first
quadrant and leaving the rest. Morover, while the cornet here is not
sharp, if we want to smooth it we can easily accomodate such a desire
by taking everything above a hyperpola uv = constant in the first
quadrant--in other words, use uv as a goodness function, and insist
on a goodness higher than zero.

Think the resulting list is too small? Try moving the origin
elsewhere, by setting

u' = A - ln(complexity) - ln(error)
v' = B - 4 ln(complexity) - ln(error)

Still unhappy? I think the slopes of -1 and -4 I use work well, but
you could try changing slopes *and* origins in order to better get
what you think is a moat, or are willing to claim is one.

I think a uv plot of 7-limit linears would be interesting. I'd also
like some kind of feedback, so I don't get the feeling I am talking
to myself here.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> We might try in analyzing or plotting 7-limit linear temperaments a
> transformation like this:
>
> u = 4 - ln(complexity) - ln(error)
> v = 12 - 4 ln(complexity) - ln(error)
>
> We can obtain a fine list simply by taking everything in the first
> quadrant and leaving the rest. Morover, while the cornet here is
not
> sharp, if we want to smooth it we can easily accomodate such a
desire
> by taking everything above a hyperpola uv = constant in the first
> quadrant--in other words, use uv as a goodness function, and insist
> on a goodness higher than zero.
>
> Think the resulting list is too small? Try moving the origin
> elsewhere, by setting
>
> u' = A - ln(complexity) - ln(error)
> v' = B - 4 ln(complexity) - ln(error)
>
> Still unhappy? I think the slopes of -1 and -4 I use work well, but
> you could try changing slopes *and* origins in order to better get
> what you think is a moat, or are willing to claim is one.
>
> I think a uv plot of 7-limit linears would be interesting. I'd also
> like some kind of feedback, so I don't get the feeling I am talking
> to myself here.

It sounds interesting, but what is the basic idea, and where are
these numbers and parameters coming from?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > We might try in analyzing or plotting 7-limit linear temperaments a
> > transformation like this:
> >
> > u = 4 - ln(complexity) - ln(error)
> > v = 12 - 4 ln(complexity) - ln(error)

In the terms Paul, Carl and I have been using, this is a cutoff relation

max[ln(complexity)/4 + ln(error)/4,
4 * ln(complexity)/3 + ln(error)/12] < 1

I wish you'd told us what units you're assuming for error here. I
can't possibly consider this because I don't know whether it's cents
or octaves? ... Only teasing, ... to make a point ;-)

So this is a pair of lines that together take a triangular bite out of
the lower left edge of the sheet of temperaments. I'm guessing they
are designed to depart at equal but opposite angles from a log-flat
line tangent to their corner.

It seems we may be moving towards some kind of agreement. :-)

> > We can obtain a fine list simply by taking everything in the first
> > quadrant and leaving the rest. Morover, while the cornet here is
> not
> > sharp, if we want to smooth it

Yes. Definitely.

> we can easily accomodate such a desire
> > by taking everything above a hyperpola uv = constant in the first
> > quadrant--in other words, use uv as a goodness function, and insist
> > on a goodness higher than zero.
> >
> > Think the resulting list is too small? Try moving the origin
> > elsewhere, by setting
> >
> > u' = A - ln(complexity) - ln(error)
> > v' = B - 4 ln(complexity) - ln(error)
> >
> > Still unhappy? I think the slopes of -1 and -4 I use work well, but
> > you could try changing slopes *and* origins in order to better get
> > what you think is a moat, or are willing to claim is one.
> >
> > I think a uv plot of 7-limit linears would be interesting. I'd also
> > like some kind of feedback, so I don't get the feeling I am talking
> > to myself here.

OK. But I don't think it will help to do a u v plot. I'd prefer to see
it on the existing log log plot, and I'd really like to see if you can
come up with one of these hyperbolic-log beasties that gives the same
list as Pauls red curve. This is exciting. :-)

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> > wrote:
> > > We might try in analyzing or plotting 7-limit linear temperaments a
> > > transformation like this:
> > >
> > > u = 4 - ln(complexity) - ln(error)
> > > v = 12 - 4 ln(complexity) - ln(error)
>
> In the terms Paul, Carl and I have been using, this is a cutoff relation
>
> max[ln(complexity)/4 + ln(error)/4,
> 4 * ln(complexity)/3 + ln(error)/12] < 1

Only if you choose to use it for one. It's a coordinate
transformation, primarily.

> It seems we may be moving towards some kind of agreement. :-)

I've been *trying* to help you and Paul here, with all of this stuff
about convex hulls and what not which I have been told is useless. I
hope I am finally communicating *something*. This idea, by the way, is
an old one but not much heed was paid to it by anyone, including me,
and I proposed it. The new aspects are to use it as a coordinate
transformation in a loglog context, and to draw hyperbolas if you want
to smooth corners.

> OK. But I don't think it will help to do a u v plot.

Why not?

I'd prefer to see
> it on the existing log log plot,

Why not both?

and I'd really like to see if you can
> come up with one of these hyperbolic-log beasties that gives the same
> list as Pauls red curve. This is exciting. :-)

Probably that can be done, but what is special about Paul's red curve?
I didn't like it, and in any case Paul tells us that the reason it
drops off so fast at the end is not because he was trying to nuke
ennealimmal, but because he ran out of things to plot. The massive
subjectivity of it all is what I'm hoping to avoid.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> > > wrote:
> > > > We might try in analyzing or plotting 7-limit linear
temperaments a
> > > > transformation like this:
> > > >
> > > > u = 4 - ln(complexity) - ln(error)
> > > > v = 12 - 4 ln(complexity) - ln(error)
> >
> > In the terms Paul, Carl and I have been using, this is a cutoff
relation
> >
> > max[ln(complexity)/4 + ln(error)/4,
> > 4 * ln(complexity)/3 + ln(error)/12] < 1
>
> Only if you choose to use it for one. It's a coordinate
> transformation, primarily.
>
>
> > It seems we may be moving towards some kind of agreement. :-)
>
> I've been *trying* to help you and Paul here, with all of this stuff
> about convex hulls and what not which I have been told is useless. I
> hope I am finally communicating *something*. This idea, by the way, is
> an old one but not much heed was paid to it by anyone, including me,
> and I proposed it. The new aspects are to use it as a coordinate
> transformation in a loglog context, and to draw hyperbolas if you want
> to smooth corners.
>
> > OK. But I don't think it will help to do a u v plot.
>
> Why not?

Well log(error) and log(complexity) are already so far away from any
reasonable kind of pain(error) and pain(complexity) that I fear this
will just make it worse. You convinced us to humour you with the
log-log thing, but this seems to be going too far.

What is the point of putting all the math-complexity of the cutoff
relation into the coordinate transformation just so the cutoff
relation looks simple. The math-complexity is still there.
>
> I'd prefer to see
> > it on the existing log log plot,
>
> Why not both?

Sure. I'd like to see it on linear-linear too. But since I'm not doing
it, I figure I can't ask for too much.

> and I'd really like to see if you can
> > come up with one of these hyperbolic-log beasties that gives the same
> > list as Pauls red curve. This is exciting. :-)
>
> Probably that can be done, but what is special about Paul's red curve?
> I didn't like it, and in any case Paul tells us that the reason it
> drops off so fast at the end is not because he was trying to nuke
> ennealimmal, but because he ran out of things to plot.

OK. So long as its something close to Paul's list, otherwise we're
just wasting our time.

> The massive
> subjectivity of it all is what I'm hoping to avoid.

I'm not sure why I'm not getting through with this, but you can't
avoid subjectivity if you're trying to make a list of things that are
likely to be of use to human musicians, not mathematician, not aliens
from outer space, not humpback whales, but humans.

We're not trying to _avoid_ subjectivity, we're trying to _model_ it.
And we're trying to do so in such a way that
(a) the model has only a small number of parameters, preferably no
more than 3.
(b) the list chosen is as insensitive as possible to the values of the
parameters
(c) we have agreement from as many people, as possible.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> Well log(error) and log(complexity) are already so far away from any
> reasonable kind of pain(error) and pain(complexity) that I fear this
> will just make it worse. You convinced us to humour you with the
> log-log thing, but this seems to be going too far.

One posting you use the word "exciting" and in the next you aren't
even willing to look at a plot, because doing that might be to go too
far in humoring an obviously confused person. Which is it?

> What is the point of putting all the math-complexity of the cutoff
> relation into the coordinate transformation just so the cutoff
> relation looks simple. The math-complexity is still there.

It's to make clear what and how I am doing. An objection to clarity
and simplicity, which is what this amounts to, seems to me to be very
misplaced. And Paul and you both tbink *I* am making spurious excuses!

> Sure. I'd like to see it on linear-linear too. But since I'm not doing
> it, I figure I can't ask for too much.

People ask me to do things on this list all the time. Sometimes I even
do them.

> OK. So long as its something close to Paul's list, otherwise we're
> just wasting our time.

Does this mean you didn't even *look* at my lists??

> We're not trying to _avoid_ subjectivity, we're trying to _model_ it.

The best model for total subjectivity is simply to pick any
temperametns you like for any reason you like. We could try that.

> And we're trying to do so in such a way that
> (a) the model has only a small number of parameters, preferably no
> more than 3.

Moving the origin of my uv stuff is two. Adding a hyperbola gives you
three.

> (c) we have agreement from as many people, as possible.

Has this last been attempted? It seems to me it's got to be whatever
you and Paul want, lately.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> > Well log(error) and log(complexity) are already so far away from any
> > reasonable kind of pain(error) and pain(complexity) that I fear this
> > will just make it worse. You convinced us to humour you with the
> > log-log thing, but this seems to be going too far.
>
> One posting you use the word "exciting" and in the next you aren't
> even willing to look at a plot, because doing that might be to go too
> far in humoring an obviously confused person. Which is it?

Wow! You're good at putting offensive words in people's mouths aren't
you? All I said was I don't thing it's worth the trouble of doing a u
v plot of it. I'd rather see it on log log and even better, linear
linear. But if you've done the plot, I'm happy to look at it.

> > OK. So long as its something close to Paul's list, otherwise we're
> > just wasting our time.
>
> Does this mean you didn't even *look* at my lists??

Of course I looked at it, but I didn't line it up against Paul's and
check off the differences. It looks pretty close from memory, but it
will mean a lot more if I see the two cutoff lines plotted on the same
graph, preferably linear linear, but I'm happy to look at the others too.

> > We're not trying to _avoid_ subjectivity, we're trying to _model_ it.
>
> The best model for total subjectivity is simply to pick any
> temperametns you like for any reason you like. We could try that.

I mean we are trying to model some kind of statistical average of
human subjectivity on the question of the musical usefulness of a
temperament.

It should be obvious that humans will agree with each other as to what
is a good temperament, infinitely better than any of them will agree
with a random temperament selector.

>
> > And we're trying to do so in such a way that
> > (a) the model has only a small number of parameters, preferably no
> > more than 3.
>
> Moving the origin of my uv stuff is two. Adding a hyperbola gives you
> three.

Good. I'm just waiting to see it plotted.

> > (c) we have agreement from as many people, as possible.
>
> Has this last been attempted? It seems to me it's got to be whatever
> you and Paul want, lately.

It's what this whole thing is about. We're arguing. Trying to convince
each other. I've moved a considerable way towards accepting more
extreme errors and complexities than I would have thought acceptable
two years ago, thanks to the arguments of people on this list.
Sometimes the process is painful but we do seem like we might be
converging.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> > > wrote:
> > > > We might try in analyzing or plotting 7-limit linear
temperaments a
> > > > transformation like this:
> > > >
> > > > u = 4 - ln(complexity) - ln(error)
> > > > v = 12 - 4 ln(complexity) - ln(error)
> >
> > In the terms Paul, Carl and I have been using, this is a cutoff
relation
> >
> > max[ln(complexity)/4 + ln(error)/4,
> > 4 * ln(complexity)/3 + ln(error)/12] < 1
>
> Only if you choose to use it for one. It's a coordinate
> transformation, primarily.
>
>
> > It seems we may be moving towards some kind of agreement. :-)
>
> I've been *trying* to help you and Paul here, with all of this stuff
> about convex hulls and what not which I have been told is useless.

Gene, I said "this looks interesting, please help me understand what
it means and what you are doing", not "this is useless". If you can't
distinguish the two, how is anyone ever going to have any hope of
communicating with you?

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

> Gene, I said "this looks interesting, please help me understand what
> it means and what you are doing", not "this is useless". If you can't
> distinguish the two, how is anyone ever going to have any hope of
> communicating with you?

What you said was "this isn't helping", which seemed to me to mean the
same thing. It sounded like I was being shut out of the conversation,
and asked to concentrate on calculating lists instead.

On an unrelated point, changing the font size in Mozilla changes the
font size for this reply message, and changes the font size for
everything outside of the posting, but doesn't change the small font
of the posting itself. I'm tired of squinting and would like a Big
Print font if possible. Carl?

>On an unrelated point, changing the font size in Mozilla changes the
>font size for this reply message, and changes the font size for
>everything outside of the posting, but doesn't change the small font
>of the posting itself. I'm tired of squinting and would like a Big
>Print font if possible. Carl?

My advice is to have the list delivered by e-mail, where you can
control this and many other things to your heart's content. You
do miss out a bit on the threading, is the only drawback.

If you insist on using the web interface, I'm not sure anything can
be done if yahoo is using absolute font sizes in their html. You
might try your user options to see if there is a 'display formatted
message / display plain text message' option, and toggle it to see
if that helps.

-Carl