HE on 18:1, 4 octaves

I am still working on the mathematical data you provided
in answer to my
<a href="/harmonic_entropy/topicId_707.html#707">
newbie question</a> as I need some time to "digest" them.
However, this
<a href=
"/harmonic_entropy/files/dyadic/heinz.gif">
new image
</a>
puzzles me quite a bit:

I was thinking of dyadic harmonic entropy as a pseudo-periodical function
where in logarithmic scales (in cents)
successive octaves of the fundamental would be the
strongest local minima
and the next local minima within
an octave would be for G/C, 3/2, 2:3, the fifth, 700 cents,
choose your pick..., everything being progressingly attenuated toward
higher entropies, which seems to be the case at first sight.

But, assuming the fundamental is C0 and looking at your image

1) O: C0 , HE(0) minimal , OK
2) 700: G0 , HE(0) < HE(700) OK
3) 1200: C1 , HE(0) < HE(1200) < HE(700) OK
4) 1900: G1, HE(1900) < HE(700) oops..., HE(G1) < HE(G0) ????
5) 2400: C2, HE(1900) < HE(2400) double oops, HE(G1) < HE(C2) ????

Above 2400 cents, all the minima are somewhat regularly "increasing",
for example HE(2800) (E2) < HE(3100) (G2)
which annoys me even more. Isn't there some kind of "aliasing",
I mean, a sampling problem which is understandable
when dealing with rational numbers? I understand it is a
difficult problem.
Hold on!

--- In harmonic_entropy@yahoogroups.com, Denix 13 <denix13@w...>
wrote:
> I am still working on the mathematical data you provided
> in answer to my
> <a
href="/harmonic_entropy/topicId_707.html#707">
> newbie question</a> as I need some time to "digest" them.
> However, this
> <a href=
> "/harmonic_entropy/files/dyadic/heinz.g
if">
> new image
> </a>
> puzzles me quite a bit:
>
> I was thinking of dyadic harmonic entropy as a pseudo-periodical
function
> where in logarithmic scales (in cents)
> successive octaves of the fundamental would be the
> strongest local minima
> and the next local minima within
> an octave would be for G/C, 3/2, 2:3, the fifth, 700 cents,
> choose your pick...,

This sounds like octave-equivalent harmonic entropy, which has been
explored here, but typically we don't assume octave-equivalence.

> everything being progressingly attenuated toward
> higher entropies, which seems to be the case at first sight.
>
> But, assuming the fundamental is C0 and looking at your image
>
> 1) O: C0 , HE(0) minimal , OK
> 2) 700: G0 , HE(0) < HE(700) OK
> 3) 1200: C1 , HE(0) < HE(1200) < HE(700) OK
> 4) 1900: G1, HE(1900) < HE(700) oops..., HE(G1) < HE(G0) ????

Yes, this is not surprising, considering that 3:1 is a simpler ratio
than 3:2.

> 5) 2400: C2, HE(1900) < HE(2400) double oops, HE(G1) < HE(C2) ????

3:1 is simpler than 4:1.

> Above 2400 cents, all the minima are somewhat
regularly "increasing",
> for example HE(2800) (E2) < HE(3100) (G2)
> which annoys me even more.

If you want to see larger intervals show up as more concordant
generally, you could 'seed' with a Farey series, which results in
pretty much the same graph but with an overall downward slope. There
are some examples in the files folder of this group and the tuning
group.

> Isn't there some kind of "aliasing",
> I mean, a sampling problem which is understandable
> when dealing with rational numbers?

I'm not sure what kind of sampling problem you're thinking of, but
perhaps it has something to do with the 'seeding' I refer to above --
which determines the set of ratios you'll be computing probabilities
for and summing over to get entropy.