Harmonic Entropy for the Sophisticated Ear

The "N" part of the harmonic entropy algorithm (how many ratios are
included) has been changed many times, with little or no effect, as has been
observed before. But in every graph I've seen "s" (how baldy an interval can
be mistuned for it to be acceptable) is always 1%. Wouldn't this have a much
greater affect on the local extrema than N? Seems to me that a large value
of s would smooth out the graph, losing extrema, and a small one would make
it more wrinkly, adding new ones. Once N is high enough, it doesn't matter
very much what it is exactly, because all those complex ratios are always
snuggled in between a bunch of others and covered up. It would be
interesting to see a "Harmonic Entropy Graph for the Tone Deaf" with s
around 10%, and a "Harmonic Entropy Graph for the Sophisticated Ear" with s
around 0.1%. These would also be usefull for different registers, because
pitches in a higher register are more easily descriminated.

Keenan P.

P.S. Could someone give me the actual formula for harmonic entropy, so I
won't have to bother the List with my inane requests? A have a hunch it uses
a Gaussian distrbution around each ratio, but I'm not sure how.

--- In tuning@egroups.com, "Keenan Pepper" <mtpepper@p...> wrote:
> The "N" part of the harmonic entropy algorithm (how many ratios are
> included) has been changed many times, with little or no effect, as
has been
> observed before. But in every graph I've seen "s" (how baldy an
interval can
> be mistuned for it to be acceptable) is always 1%.

I've done some for 0.6% -- I recently posted a link to an old post
with the results for N=100, s=0.6%.

Wouldn't this have a much
> greater affect on the local extrema than N? Seems to me that a
large value
> of s would smooth out the graph, losing extrema, and a small one
would make
> it more wrinkly, adding new ones. Once N is high enough, it doesn't
matter
> very much what it is exactly, because all those complex ratios are
always
> snuggled in between a bunch of others and covered up.

All correct.

It would be
> interesting to see a "Harmonic Entropy Graph for the Tone Deaf"
with s
> around 10%, and a "Harmonic Entropy Graph for the Sophisticated
Ear" with s
> around 0.1%. These would also be usefull for different registers,
because
> pitches in a higher register are more easily descriminated.

All true. I would say, based on Goldstein's experiments, that the
most
sophisticated ear listening in the best register (around 3000 Hz
IIRC)
would probably not do much better than 0.6%. I will work on making
some graphs for different s values when I get back to the office (I
also "owe" the list some sound files of tetrads and searches of
scales
with 6 or more notes . . .)

>
> Keenan P.
>
> P.S. Could someone give me the actual formula for harmonic entropy,
so I
> won't have to bother the List with my inane requests? A have a
hunch it uses
> a Gaussian distrbution around each ratio, but I'm not sure how.

Each ratio in the Farey (or whatever) series is given a "range" equal
to the distance between the two mediants adjacent to that ratio. This
range is then multiplied by the height of the Gaussian (with standard
deviation s and centered around the true interval) at the ratio. This
gives a probability value (p) to each ratio, and the sum of these
probabilities should be 1. Then the harmonic entropy of the true
interval is: -sum(p*log(p)), where the sum runs over all the ratios
in
the Farey (or whatever) series.

> [Keenan Pepper]
> http://www.egroups.com/message/tuning/12238
>
> ... It would be interesting to see a "Harmonic Entropy Graph
> for the Tone Deaf" with s around 10%, and a "Harmonic Entropy
> Graph for the Sophisticated Ear" with s around 0.1%. These would
> also be usefull for different registers, because pitches in a
higher register are more easily descriminated.

Hello, Keenan. First I wanted to tell you how impressed I am
that at your young age, you can grasp so much of some really
complex tuning theory.

When I visited Paul Erlich last year, he explained his 'harmonic
entropy' postings to me in greater detail (which is what led
to my creation of the webpages), and he showed me a graph he'd
made where s = 0.6%. It was interesting to see the difference
from s = 1%, and you have indeed understood precisely how s works
in the calculation: it is a measure of how finely a listener can
discriminate discreet pitches.

> P.S. Could someone give me the actual formula for harmonic
> entropy, so I won't have to bother the List with my inane
> requests? A have a hunch it uses a Gaussian distrbution around
> each ratio, but I'm not sure how.

Paul is your man, Keenan - this whole concept was formulated
by him. You're right about the Gaussian distribution, but I
could never explain it to you. Paul?

-monz
http://www.ixpres.com/interval/monzo/homepage.html

Thanks, Paul E, for the latest chart, at:

http://www.egroups.com/files/tuning/perlich/ent_006.jpg

The curve is detailed and interesting. But I'm having a bit of trouble
correlating some of the entropy dips with my perception of interval
concordance.

Compare, for example, 5/4 and 4/3 from the first octave, to their
counterparts, 5/2 and 8/3, in the second octave. The curve shows 4/3
as having less entropy then 5/4, but 8/3 as having more entropy than
5/2. What does this mean?

When I think of relating the harmonic entropy curve to actual tuning,
my first thought would be to match higher entropy with higher tuning
"pain". Your own recent scale-finding posts attempt to minimize total
entropy, in alignment with this correlation. But this leads to the
following (correct me if I'm wrong): places with deeper entropy dips,
such as 5/2 vs. 8/3, would need to be tuned closer to JI than ones with
less pronounced dips. The slope of the entropy curve is equivalent to
the spring constant (I should say "constant" because it changes) in my
energy model, d(pain)/d(tuning).

Does this really make tuning sense? That is, is it more important to
tune 4/3 close to true than 5/4, yet LESS important to tune 8/3 close
to true than 5/2?

JdL

[I wrote:]
>The slope of the entropy curve is equivalent to
>the spring constant (I should say "constant" because it changes) in my
>energy model, d(pain)/d(tuning).

which is a muddle of my own model! The slope of the entropy curve
is equivalent to the force generated by one of my springs; it IS
d(pain)/d(tuning), but the spring "constant" is d(force)/d(tuning),
another derivative higher.

JdL

Looking some more at Paul E's harmonic entropy graph at:

http://www.egroups.com/files/tuning/perlich/ent_006.jpg

Here's another concern I have: 9/7 vs. 11/9. The graph shows 9/7 as
the more concordant of the two intervals, with a pronounced dip of its
own; 11/9 is perched at the top of an almost-flat mesa between 6/5
and 5/4.

But 11/9 is the "neutral third", which several people have described
as relatively pleasant, whereas 9/7 is the "car horn third", quite
irritating-sounding on its own, though part of a well-tuned 7-limit
dominant 9th chord.

The graph "makes sense" just looking at the larger numbers in 11/9 vs.
9/7, yet for some reason the impression of the intervals does not
match, to my ears.

JdL