What are we talking about?

The creation of this list has at least served its purpose. Beardsley's convinced
that the main list's quietened down. Now I'll continue my quest to find out what
it all means.

The basic idea is that a chord is rated according to the ambiguity in assigning a
ratio to it. So, could we divide pitch space into bins, calculate a load of
ratios, and plot a histogram of the results. The more ratios there were in a bin,
the more complex chords that fall into the bin would be.

Is this what you're doing, then? With a Gaussian to allow more than one bin to
contribute to the complexity measure of a point in pitch space?

As for these Voronoi-cell type graphs. I think I could draw one with each cell
sized according to a complexity measure for the interval inside. This would mean
having a complexity field around each point, and drawing the lines where the field
strength of the to strongest ratios are equal. But I expect it would be easier to
generate loads and loads of ratios, and go for the histogram.

It would be good to use hexagonal cells, although I suppose we could still draw 60
degree lattices on square cells. I'll have to work out hexagons anyway before I
can write a generalized keyboard applet. One problem would be that you need a lot
of square pixels to draw one hexagon.

Well, what do people think?

Graham

>The basic idea is that a chord is rated according to the ambiguity in
assigning a
>ratio to it. So, could we divide pitch space into bins, calculate a load
of
>ratios, and plot a histogram of the results. The more ratios there were in
a bin,
>the more complex chords that fall into the bin would be.

>Is this what you're doing, then? With a Gaussian to allow more than one
bin to
>contribute to the complexity measure of a point in pitch space?

Not exactly. It's not just how many ratios in a bin or under a Gaussian,
it's how they're distributed. If one is much wider than the others, the
listener has a much easier time guessing how to interpret that interval. If
they're all equally wide, then the listener is very confused. Entropy
measures the degree of compressibility in a signal. If the signal has a much
higher probability of taking on one value than any of the other values, it
is very compressible, and the entropy is small. If all values are equally
probable, it is incompressible, and the entropy is maximal. You might want
to find an information theory book to learn more.

>As for these Voronoi-cell type graphs. I think I could draw one with each
cell
>sized according to a complexity measure for the interval inside. This
would mean
>having a complexity field around each point, and drawing the lines where
the field
>strength of the to strongest ratios are equal.

Don't understand, but go for it -- I'd like to see it.

>But I expect it would be easier to
>generate loads and loads of ratios, and go for the histogram.

Huh?

>It would be good to use hexagonal cells, although I suppose we could still
draw 60
>degree lattices on square cells.

The Voronoi cells include hexagons, pentagons, heptagons, quadrilaterals . .
. I think forcing only one shape will result in a highly unnatural solution.

In-Reply-To: <CE80F17667E4D211AE530090274662729C545D@...>
Paul Erlich wrote:

> You might
> want
> to find an information theory book to learn more.

Any recommendations? I might be going past the central library tomorrow,
so I'll see what they've got.

> >As for these Voronoi-cell type graphs. I think I could draw one with
> each
> cell
> >sized according to a complexity measure for the interval inside. This
> would mean
> >having a complexity field around each point, and drawing the lines
> where
> the field
> >strength of the to strongest ratios are equal.
>
> Don't understand, but go for it -- I'd like to see it.

It could be interesting, but we'll have to see if I find the time.

> >But I expect it would be easier to
> >generate loads and loads of ratios, and go for the histogram.
>
> Huh?

Well, to do even a standard Voronoi cell plot, you need to decide which
ratios constitute the near neighbours of others, and draw the lines at
the correct angles. That all means tedious programming, although I don't
forsee any show-stoppers. I could modify an existing Voronoi cell program
more easily, but haven't been able to find any.

For a histogram, all I need is a function for producing a set of ratios,
which I already have, and another for converting ratios to cents, which I
also have. Then I need a 2-D array for storing the totals, which is
trivial, and a way of displaying the results, which I've half worked out.
So it's something I could get done over the weekend ... although I don't
know which.

About the ratios that get generated -- can they include unisons? I didn't
before, thinking that tetrads should be tetrads, but I think I may as well
add them so there's an origin.

> >It would be good to use hexagonal cells, although I suppose we could
> still
> draw 60
> >degree lattices on square cells.
>
> The Voronoi cells include hexagons, pentagons, heptagons,
> quadrilaterals . .
> . I think forcing only one shape will result in a highly unnatural
> solution.

No, I don't mean for the voronoi-type cells, but the bins of whatever
kind. With a hexagon, all adjacent hexagons will be equivalent, but with
squares some share an edge, but some only a corner. Hexagons can also
tesselate for 60 degree axes.

For the Voronoi-type diagram, it may still be useful to store ratios in
bins, so that a ratio's near neighbours will always be in the same or
adjacent bins. That should speed up the computatation, as fewer points
will need to be considered to decide which pair are closest. Hexagons may
be an advantage again, and wouldn't affect the result. Or they may be too
much trouble.

Graham

Graham Breed wrote,

>Any recommendations? I might be going past the central library tomorrow,
>so I'll see what they've got.

The only one I know of is _Elements of Information Theory_ by Thomas Cover
and Joy Thomas.

>Well, to do even a standard Voronoi cell plot, you need to decide which
>ratios constitute the near neighbours of others, and draw the lines at
>the correct angles. That all means tedious programming, although I don't
>forsee any show-stoppers. I could modify an existing Voronoi cell program
>more easily, but haven't been able to find any.

The one I use seems to make use of some very special geometrical properties
of Voronoi cells that I see no way of generalizing to "weighted"
modifications.

>About the ratios that get generated -- can they include unisons? I didn't
>before, thinking that tetrads should be tetrads, but I think I may as well
>add them so there's an origin.

Oh yes -- and don't forget to extend the diagram to regions where one or
both of the intervals are negative.