Chordal harmonic entropy

I still don't have dyadic harmonic entropy calcs fully up to speed; I'm
lacking specifics for the error function (Ed Borasky, did you say you
could help?).

The Voronoi diagram that Paul E posted for triads is lovely, but it's
not clear to me how to go from that to a continuous function. Paul, is
it clear to you?

Tetrads would require a three-dimensional Voronoi diagram, a tough thing
to either program or display!

Also, Paul, I asked you in private email whether otonal and utonal
chords would be distinguished (from their reverse, corresponding
counterparts) by the Voronoi method. They wouldn't, would they? And if
not, that's a serious flaw in the process, is it not?

It makes sense to me to calculate chordal harmonic entropy as follows:

. start with a set of notes of known frequency.

. calculate the difference tone generated by each dyad.

. using both the actual tones and the difference tones, sum the
entropy of all dyads.

This process will distinguish otonal and utonal chords very nicely.

JdL

John deLaubenfels wrote,

>I still don't have dyadic harmonic entropy calcs fully up to speed; I'm
>lacking specifics for the error function (Ed Borasky, did you say you
>could help?).

Did you see Manuel's post for an approximation to the normal cdf, and my
post for a formula expressing the normal cdf in terms of the error function?

>The Voronoi diagram that Paul E posted for triads is lovely, but it's
>not clear to me how to go from that to a continuous function. Paul, is
>it clear to you?

Certainly. The axes would really be at a 60-degree, not 90-degree, angle.
Then you'd use the area under a bivariate normal distribution and chop in up
into columns above the Voronoi cells . . . and calculate the entropy.

>Tetrads would require a three-dimensional Voronoi diagram, a tough thing
>to either program or display!

You know it!

>Also, Paul, I asked you in private email whether otonal and utonal
>chords would be distinguished (from their reverse, corresponding
>counterparts) by the Voronoi method. They wouldn't, would they?

They are most clearly distinguished. Notice the difference between the red
dots and the blue dots in the diagram.

>It makes sense to me to calculate chordal harmonic entropy as follows:

> . start with a set of notes of known frequency.

> . calculate the difference tone generated by each dyad.

> . using both the actual tones and the difference tones, sum the
> entropy of all dyads.

>This process will distinguish otonal and utonal chords very nicely.

Sorry -- that's too cheap for me. I'll stick it out with the hard way.

[I wrote:]
>>It makes sense to me to calculate chordal harmonic entropy as follows:
>>
>> . start with a set of notes of known frequency.
>>
>> . calculate the difference tone generated by each dyad.
>>
>> . using both the actual tones and the difference tones, sum the
>> entropy of all dyads.
>>
>>This process will distinguish otonal and utonal chords very nicely.

[Paul E:]
>Sorry -- that's too cheap for me. I'll stick it out with the hard way.

Uh huh. Why is it that I suspect you WANT the problem to be hard?
Don't get me wrong: I love a good challenge I can sink my teeth into.
But, what are you going to do for 5-note chords? We're talking 4-D
space now: you gonna Voronoi THAT? I don't think so!

It strikes me that the method I propose, with suitable ratioing down
of the effective volume of the difference notes, would yield interesting
and possibly useful results. If I get time, I'll program it up and post
some values.

JdL

On Wed, 20 Sep 2000, John A. deLaubenfels wrote:

> I still don't have dyadic harmonic entropy calcs fully up to speed; I'm
> lacking specifics for the error function (Ed Borasky, did you say you
> could help?).

Yeah ... what language do you want? I can generate Basic, C, Pascal or
FORTRAN directly from Derive. They all look pretty much the same except
for how you code arrays -- parens or brackets -- and whether you code an
assignment as "=" or ":=".

--
znmeb@teleport.com (M. Edward Borasky) http://www.teleport.com/~znmeb

If they named a street after Picabo Street, would it be called Picabo
Street, Street Street or Picabo Street Street?

>The Voronoi diagram that Paul E posted for triads is lovely, but it's
>not clear to me how to go from that to a continuous function. Paul, is
>it clear to you?

Is there anything wrong with my earlier suggestion...

o Define the tuning resolution by the number of pixels on a planar section,
bounded on each axis by the largest intervals you want to consider.

o Every pixel represents one triad. Find all the triads belonging to a
series of a given limit -- Farey and Tenny limits work for triads too --
and call the pixels representing them "special".

o Center a gaussian umbrella over the pixel representing the "true" triad.

o For each special pixel, assign a probability by measuring the volume
under the umbrella and above the cell to which the pixel belongs.

o Express this probability in terms of the total volume of the umbrella.

o Sum the probabilities as usual.

...?

>Tetrads would require a three-dimensional Voronoi diagram, a tough thing
>to either program or display!

It's the 4-D gaussian that strikes me as the most difficult part. Though
it's all beyond my skills, none of it strikes me that it should be _hard_.

>>Also, Paul, I asked you in private email whether otonal and utonal
>>chords would be distinguished (from their reverse, corresponding
>>counterparts) by the Voronoi method. They wouldn't, would they? And if
>>not, that's a serious flaw in the process, is it not?
>
>It makes sense to me to calculate chordal harmonic entropy as follows:
>
> . start with a set of notes of known frequency.
>
> . calculate the difference tone generated by each dyad.
>
> . using both the actual tones and the difference tones, sum the
> entropy of all dyads.
>
>This process will distinguish otonal and utonal chords very nicely.

Aren't you forgetting sum tones? Anyway, the o/u distinction should
be made by the series that seeds the special chords. This will, it
strikes me, actually be unfair against the utonal chords. But when
sensory dissonance is added back into the mix, utonal chords will be
redeemed somewhat. Remember h.e. is only half of our picture of
concordance.

-Carl

>Uh huh. Why is it that I suspect you WANT the problem to be hard?
>Don't get me wrong: I love a good challenge I can sink my teeth into.
>But, what are you going to do for 5-note chords? We're talking 4-D
>space now: you gonna Voronoi THAT? I don't think so!

John, I want to do it right, because then I'm likely to find beautiful, deep
patterns like I am now in the diadic case, e.g., the linear relationship
with Tenney Harmonic Distance, etc.

>It strikes me that the method I propose, with suitable ratioing down
>of the effective volume of the difference notes, would yield interesting
>and possibly useful results. If I get time, I'll program it up and post
>some values.

Cool! You know, a certain second-order difference tone is the loudest
combination tone at low volumes . . .