triads behave as i thought

even with the primitive voronoi algorithm,

/harmonic_entropy/files/trivoro.gif

the area associated with each triad a:b:c is, as i suspected, inversely proportional to the geometric mean of the terms, i.e., proportional to 1/cuberoot(a*b*c) -- as long as a*b*c is not too close to the limit used for all the triads in the computation:

/harmonic_entropy/files/triadic.gif

this is in perfect agreement with the guess i posted in this list some time ago . . .

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--- In harmonic_entropy@y..., wally paulrus <wallyesterpaulrus@y...>
wrote:

> /harmonic_entropy/files/triadic.gif

oops, that's supposed to say slope = -1, not slope = 1 . . .

>> /harmonic_entropy/files/triadic.gif

Ehhhxcellent.

Where does this leave the validation exercise?

-Carl

(Before, many thanks for the Cariani reference)

Did first attempts with voronoi cells seem to point towards the awaited u/o-tonal symmetry breaking?

Pierre

--- In harmonic_entropy@y..., "Carl Lumma" <clumma@y...> wrote:
> >> /harmonic_entropy/files/triadic.gif
>
> Ehhhxcellent.
>
> Where does this leave the validation exercise?
>
> -Carl

i'm not sure if i referred to this or something else as the
validation exercise. can you provide me with a reference (probably in
these archives)? i'm getting fuzzy in my old age (except on top).

--- In harmonic_entropy@y..., "Pierre Lamothe" <plamothe@a...> wrote:
> (Before, many thanks for the Cariani reference)
>
> Did first attempts with voronoi cells seem to point towards the
>awaited u/o-tonal symmetry breaking?
>
> Pierre

yes, they sure did. i provided some clear illustrations of this fact,
some of which you may find here in the files section.

>>Where does this leave the validation exercise?
>
>i'm not sure if i referred to this or something else as the
>validation exercise. can you provide me with a reference
>(probably in these archives)? i'm getting fuzzy in my old age
>(except on top).

It was something else. IIUC, when using 1/...rt(a*b...) to
get 'widths' for the harmonic entropy calculation, is the
sum of these 'widths' under the distribution relatively
independent of where the distribution is centered in n-ad
space?

See message # 215.

Incidentally, what did you ever decide as far as reduced vs.
unreduced fractions in h.e. calculations? Msg. 271 & 362.

-Carl

--- In harmonic_entropy@y..., "Carl Lumma" <clumma@y...> wrote:
> >>Where does this leave the validation exercise?
> >
> >i'm not sure if i referred to this or something else as the
> >validation exercise. can you provide me with a reference
> >(probably in these archives)? i'm getting fuzzy in my old age
> >(except on top).
>
> It was something else. IIUC, when using 1/...rt(a*b...) to
> get 'widths' for the harmonic entropy calculation, is the
> sum of these 'widths' under the distribution relatively
> independent of where the distribution is centered in n-ad
> space?
>
> See message # 215.

ok . . . so now, armed with the 1/cuberoot(a*b*c) rule that i just
verified, i'll still be well advised to do the validation exercise in
most cases. i wouldn't have to do it only in cases where i'm using
the actual area of each polygonal cell, but i like using the formula
for approximate widths instead, especially as an alternative to the
voronoi cells which are way too large for triads right up against the
limit (i.e., a*b*c real close to a million). so yes, validation
exercises are coming . . .

> Incidentally, what did you ever decide as far as reduced vs.
> unreduced fractions in h.e. calculations? Msg. 271 & 362.

it's a tough call. only reduced fractions can be assigned non-
overlapping, exhaustive portions of interval space, while unreduced
fractions would seem to better model what goes on with virtual pitch.