Does this make you horny? Randy, baby, yeah!

All number-fetishists, please take a look at

http://www.egroups.com/files/tuning/perlich/tenney/te01_13p2877.jpg.

If it's too big to fit on your screen, print it out.

It's the exponential of the harmonic entropy (shown for intervals less than
two octaves), which is the diadic discordance measure I settled on for a
ranking of the tetrads in Joseph's experiment.

I labeled all the ratios where there was a local minimum, as well as all the
ones where there was a visible bump in the curve. Given the hearing
resolution I assumed (1%), these happen to be exactly all the ratios within
this range for which numerator times denominator is less than 70.

Can you see any patterns in where the ratios fall? For example, pick a
numerator, and find all the ratios with that numerator. See any pattern? Now
do the same thing for a denominator. If you're really perverted, try it for
some value of numerator minus denominator. Feel free to respond either
off-list or on.

Paul,

Your curve reveals property I've noted on my logarithmic representation of
Stern-Brocot tree. Tones with same numerator or same denominator are almost
aligned. They are aligned by definition in my representation.

In my paper "Pourquoi les gammes natureles ne sont-elles pas symétriques ?"
I wrote :

<< Par ailleurs, on y trouve que deux familles de droites parallèles
qui relient beaucoup de tons. L'une correspond à l'harmonie majeure
(des suites d'harmoniques) et l'autre à une harmonie duale (des
suites de sous-harmoniques). >>

It's why I tend to think that symmetry harmonic/subharmonic could have
importance not only at macrotonal level but also at microtonal level where
your curve occurs. Since perception seems (it's not yet clear for me) in
contradiction with that, I question in view to eventually separate what
would be symmetric and what would be assymetric at sonance level.

Introduction of logarithm is worth shouting about.

Pierre Lamothe

Paul,

In reading my precedent post, I see that I talk of logarithm but it would
be an exponential on your y-axis. I'm astonished. Can you clarify that.

Pierre Lamothe

--- In tuning@egroups.com, Pierre Lamothe
<plamothe@a...> wrote:
>
> Paul,
>
> Your curve reveals property I've noted on my logarithmic
representation of
> Stern-Brocot tree. Tones with same numerator or same denominator
are almost
> aligned. They are aligned by definition in my representation.

Righto! On my curve, they are aligned up
until a certain complexity, and then they
start to flatten out.

>
> In my paper "Pourquoi les gammes natureles ne sont-elles pas
symétriques ?"
> I wrote :
>
> << Par ailleurs, on y trouve que deux familles de droites
parallèles
> qui relient beaucoup de tons. L'une correspond à l'harmonie
majeure
> (des suites d'harmoniques) et l'autre à une harmonie duale
(des
> suites de sous-harmoniques). >>
>
> It's why I tend to think that symmetry harmonic/subharmonic could
have
> importance not only at macrotonal level but also at microtonal
level where
> your curve occurs. Since perception seems (it's not yet clear for
me) in
> contradiction with that, I question in view to eventually separate
what
> would be symmetric and what would be assymetric at sonance level.

But Pierre, _any conceivable_ discordance
curve, no matter how ridiculous, would
imply harmonic/subharmonic symmetry _for
diads_. However, for triads (or larger
chords), a true measure of harmonic
entropy (as suggested by my Voronoi
diagrams) would _not_ imply harmonic/
subharmonic symmetry. Actually, I'm not
sure if I've responded to what you meant
above -- if it's not too painful, could you
clarify? Also, you might want to review what
I've written to you in the past about the
history of diatonic scale, Zarlino, and
tonality . . .
>
> Introduction of logarithm is worth shouting about.

Shout away! What have you to say?

--- In tuning@egroups.com, Pierre Lamothe
<plamothe@a...> wrote:
>
> Paul,
>
> In reading my precedent post, I see that I talk of logarithm but it
would
> be an exponential on your y-axis. I'm astonished. Can you clarify
that.

On that graph, I'm plotting the exponential of
the harmonic entropy on the y-axis. Entropy
is defined as

sum(p*log(p))

over a set of probabilities p. A few days ago I
found that for simple ratios, the harmonic
entropy was roughly proportional to the
Tenney Harmonic Distance,

log(n*d).

Then I saw you evaluating tetrads by
summing the six n*d complexities, rather
than the six log(n*d) complexities. The
analogy in the continuous world with
nonzero s (hearing standard error) would
be to use the exponential of harmonic
entropy rather than harmonic entropy. This
turned out to yield a more reasonable
ranking. Anyhow, the exponential of entropy
would be

exp(sum(p*log(p)))
=prod(exp(p*log(p)))
=prod(exp(log(p))^p)
=prod(p^p)

Did I do that right? That's a nice simple
formula, which is always a good sign that
you're dealing with a relevant quantity.

--- In tuning@egroups.com, "Paul Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/13229

> exp(sum(p*log(p)))
> =prod(exp(p*log(p)))
> =prod(exp(log(p))^p)
> =prod(p^p)
>
> Did I do that right? That's a nice simple
> formula, which is always a good sign that
> you're dealing with a relevant quantity.

This exponential business is the Tenney logarithmic method yes??
(no??)

__________ ___ __ __
Joseph Pehrson

I wrote:

> Entropy
> is defined as
>
> sum(p*log(p))

Whoops, that's -sum(p*log(p)).

> the exponential of entropy
> would be
>
> exp(sum(p*log(p)))
> =prod(exp(p*log(p)))
> =prod(exp(log(p))^p)
> =prod(p^p)

That should be:

exp(-sum(p*log(p)))
=prod(exp(-p*log(p)))
=prod(exp(log(p))^(-p))
=prod(exp(p^(-p)))

Joseph wrote,
>
> This exponential business is the Tenney logarithmic method yes??
> (no??)

Not exactly. Did you know that exp(log(x)) = x? Well, since I saw
that harmonic entropy of
simple ratios was proportional to the Tenney harmonic distance,
log(n*d), it follows that
the exponential of harmonic entropy is proportional to Lamonthe's
complexity, n*d.