Fun with Matlab

/harmonic_entropy/files/Erlich/fun.gif

My guess that a triad l:m:n occupies an area proportional to (l*m*n)^
(1/3) seems to be borne out pretty well, nay?

Paul Erlich schrieb:
>
> /harmonic_entropy/files/Erlich/fun.gif
>
> My guess that a triad l:m:n occupies an area proportional to
> (l*m*n)^
> (1/3) seems to be borne out pretty well, nay?

What a great chart!

Obviously you selected a string of ratios that keeps the fifth
just for the diagonal. What happens along the other spokes? and
what are the large triangular areas, where the new spokes start?
In fact, what are the large areas directly surrounding the major
chord? Did you put 6:5:4 in the hub or did it end up there by
itself; in other words, is there just one hub in the middle of
it all, and that's 6:5:4?

Who, by the way, are the Voronoi? Some Star Trek creatures (I
don't have TV, I'm washing the rabio)? And is it possible to
make such a graph one-dimensional for dyads?

(the reason i'm getting lively suddenly instead of reading along
in silent awe or incomprehension is that i often wondered
whether it is possible to express and/or quantize the fact that
the lower-number ratios (odd, no fancy recombinations allowed)
are surrounded by obvious empty spaces and remain that way as
you up the limit.) (one sentence)

in awe

klaus

>What a great chart!

>Obviously you selected a string of ratios that keeps the fifth
>just for the diagonal.

Not really.

>What happens along the other spokes?

Some linear function of l, m, and n is a constant. Essentially the same
graph, without the voronoi, can be seen here:

/harmonic_entropy/files/Erlich/closeup.jpg

The spokes are not precisely straight, as you can see on the more zoomed-out
diagrams, like

/harmonic_entropy/files/perlich/trimap.jpg

>and
>what are the large triangular areas, where the new spokes start?

The last triad in a particular linear series before you hit the l*m*n <= 1
million limit.

>In fact, what are the large areas directly surrounding the major
>chord?

The voronoi cell simply contains all points in the plane that are closer to
a particular point in a set than to any other point in the set. In this
case, the set is l,m,n such that l:m:n is in lowest terms and l*m*n <= 1
million.

>Did you put 6:5:4 in the hub or did it end up there by
>itself;

By itself.

>in other words, is there just one hub in the middle of
>it all, and that's 6:5:4?

>Who, by the way, are the Voronoi? Some Star Trek creatures (I
>don't have TV, I'm washing the rabio)?

I hope it's not infectious!

>And is it possible to
>make such a graph one-dimensional for dyads?

Easy -- just cut the interval between each pair of neighboring points, in
half.

>(the reason i'm getting lively suddenly instead of reading along
>in silent awe or incomprehension is that i often wondered
>whether it is possible to express and/or quantize the fact that
>the lower-number ratios (odd, no fancy recombinations allowed)
>are surrounded by obvious empty spaces and remain that way as
>you up the limit.) (one sentence)

Klaus, that's THE WHOLE IDEA of harmonic entropy, from the very beginning.
Of course one-dimensional graphs don't excite a lot of people!

"Paul H. Erlich" schrieb:

> >and
> >what are the large triangular areas, where the new spokes
> start?
>
> The last triad in a particular linear series before you hit
> the l*m*n <= 1
> million limit.
so close to the 3rd root of a million? and amounting to
augmented triads?

>
> >In fact, what are the large areas directly surrounding the
> major
> >chord?
>
> The voronoi cell simply contains all points in the plane that
> are closer to
> a particular point in a set than to any other point in the
> set. In this
> case, the set is l,m,n such that l:m:n is in lowest terms and
> l*m*n <= 1
> million.
Well, I hope that most of my questions amount to: What changes
from one point to the other? do you have the cubic roots of a
million in the center and approximations, getting worse in
various ways, extending from it?

> >Who, by the way, are the Voronoi? Some Star Trek creatures (I
> >don't have TV, I'm washing the rabio)?
>
> I hope it's not infectious!
I caught it from John Lennon...

> >(the reason i'm getting lively suddenly instead of reading
> along
> >in silent awe or incomprehension is that i often wondered
> >whether it is possible to express and/or quantize the fact
> that
> >the lower-number ratios (odd, no fancy recombinations
> allowed)
> >are surrounded by obvious empty spaces and remain that way as
> >you up the limit.) (one sentence)
>
> Klaus, that's THE WHOLE IDEA of harmonic entropy, from the
> very beginning.
> Of course one-dimensional graphs don't excite a lot of people!

If this means that the curves in the diagrams I have seen so far
represent the space around an interval, then I'm not thinking
totally different from the rest of the world after all. This
would also mean that things like the Tenney harmonic distance
are not related to this at all. (I used to think - still do -
of the "space around a ratio" as relating to "harmonic
plausibility", and a variant of Tenney's distance as "melodic
plausibility". Of course, I just couldn't see a way to quantify
this.)

klaus

>so close to the 3rd root of a million? and amounting to
>augmented triads?

Only in one particular region in the plot. The plot has points corresponding
to all conceivable triads.

>Well, I hope that most of my questions amount to: What changes
>from one point to the other? do you have the cubic roots of a
>million in the center and approximations, getting worse in
>various ways, extending from it?

I think it's all much simpler than you think -- but I can't quite make out
what you're thinking. There's a dot for every l:m:n such that l*m*n is less
than a million. If l*m*n is less than 40,000 then l:m:n is typed out
explicitly. Is this making sense?

>If this means that the curves in the diagrams I have seen so far
>represent the space around an interval,

Yes.

>This
>would also mean that things like the Tenney harmonic distance
>are not related to this at all.

You can obtain Tenney harmonic _from_ harmonic entropy -- go to
/harmonic_entropy/files/dyadic/ and download
het006_16.zip. Notice how, for the simple ratios, exp(entropy) goes as
numerator times denominator. Therefore, for the simple ratios, entropy goes
as log(numerator times denominator) = Tenney harmonic distance.

Thanks a lot. I think my idea of the triadic diagram is clear
enough now. And yes, the diagram for the dyads also looks quite
plausible. Then my "plausibility" measure, which weights the
prime factors, is really something different (and probably
yields a chaotic function).

klaus