Insight

Say you're standing in the middle of a forest, but it's like, infinite (or
just really big so you can't see an end to the trees). But they're not just
any old trees, they're perfect trees, all perfect cylinders exactly one unit
in radius. And they're lined up in a grid, like on a tree farm. So you're
standing in this perfect, imaginary forest, exactly on the spot a tree would
be, if it were not missing, replaced by you. Now if you look around you
notice some strange things. If you look parallel to the lines of trees,
"with the grain" so to speak, you can see between the lines of trees forever
(it's infinite, remember?). If you look at a 45 degree angle to the line,
the trees form diagonal lines and you can also see infinitely that way. In
fact, if you look at any tree close to you there will be a straight line of
trees behind it and you can see through them. But if you look in a random
direction, the trees won't form any lines and will obscure your view. Seem
familiar? Okay, how about if some madman went around and tacked a sign on
each tree with two numbers, how many trees up from you and how many over,
with a slash between them, like a fraction. The tree in front of you has
"1/0", the one two your left has "0/1" the one up and to the right of you
has "1/1", the one to the right of that one has "1/2", you get the idea. Now
you notice that whenever a fraction is not in simplest form, it is hidden
behind another tree that has the same fraction in simplest form (like, 4/6
is behind 2/3). "That's pretty neat", you think to yourself. Also, the
closest tree in between two others has the mediant of the their two
fractions. But that's still not the coolest part. Get this, the distance a
tree is from you exactly corresponds to its musical consonance. How well you
can see through the trees is harmonic entropy.

You might have to read that again to understand it,
Keenan P.

Keenan Pepper wrote,

>Okay, how about if some madman went around and tacked a sign on each tree
>with two numbers, how many trees up from you and how many over, with a
>slash between them, like a fraction. The tree in front of you has "1/0",
>the one two your left has "0/1" the one up and to the right of you has
>"1/1", the one to the right of that one has "1/2", you get the idea.

How do you plan on dealing with negatives here? S'thinks you only need
the upper-right (+,+) quadrant. In which case, you've got a strict
"lambdoma" (like Partch's tonality diamond, but starting with 0/0 and
proceeding by integer-limit instead of odd-limit). And since this
structure has a plane of mirror symmetry intersecting its line of 1/1's,
you really only need the lower 45-degrees of the quadrant.

>Also, the closest tree in between two others has the mediant of the their
>two fractions.

Closest tree? Isn't it the middle tree, only working when there are an
odd number of trees involved?

>Get this, the distance a tree is from you exactly corresponds to its
>musical consonance.

Inversely, I'm sure you meant.

For a fraction, a/b, that's sqrt(a^2 + b^2) -- an integer-limit version of
the prime-factor measure you've posted here before. This new version also
requires we ignore ratios that aren't in lowest form.

One interesting thing about this metric is that, like a numerator limit
for proper fractions, it assigns very low consonance to some very wide
intervals (like 7/1). I guess it's a matter of definition what happens
to consonance as intervals get very wide. We're used to thinking of
consonance and dissonance as mutually-exclusive inverses, but perhaps
wide intervals are neither consonant or dissonant. . .

>How well you can see through the trees is harmonic entropy.

Hurm! This leaves a lot to the imagination -- perhaps you suggest using
your distance metric of a given radius, instead of a Farey series of a
given order, to obtain the initial template of ratios. Since how well
you see through depends on how far the target tree is from you (as you
pointed out), this is not unreasonable. In fact, it's been proposed on
this list before.

IIRC, the Farey series was liked because the entropy curves it produces
are well-behaved as its order is changed, and because a Farey series of
a given order is a list of all the intervals in a harmonic series of a
certain limit, and it is tempting to think that the ear/mind uses a
harmonic series as _its_ template. The a*b series also had some things
going for it... none of which I can remember. And I can't recall if
sqrt(a^2 + b^2) was ever tried. Paul?

-Carl

--- In tuning@egroups.com, Carl Lumma <CLUMMA@N...> wrote:

> IIRC, the Farey series was liked because the entropy curves it
produces
> are well-behaved as its order is changed, and because a Farey
series of
> a given order is a list of all the intervals in a harmonic series
of a
> certain limit, and it is tempting to think that the ear/mind uses a
> harmonic series as _its_ template. The a*b series also had some
things
> going for it... none of which I can remember. And I can't recall if
> sqrt(a^2 + b^2) was ever tried. Paul?

Carl, You must have missed my response to Keenan's post in
http://www.egroups.com/message/tuning/11859. The
"nice spacing properties" I mention there, which the Farey series
and "Tenney" series had, means that there is a simple rule for
predicting with good accuracy the amount of space that a ratio takes
up in the series, based on the numbers in the ratio, and thus one
doesn't see abrupt changes in the harmonic entropy curve as one
increments the order of the series used.

>>IIRC, the Farey series was liked because the entropy curves it produces
>>are well-behaved as its order is changed, and because a Farey series of
>>a given order is a list of all the intervals in a harmonic series of a
>>certain limit, and it is tempting to think that the ear/mind uses a
>>harmonic series as _its_ template. The a*b series also had some things
>>going for it... none of which I can remember. And I can't recall if
>>sqrt(a^2 + b^2) was ever tried. Paul?
>
>Carl, You must have missed my response to Keenan's post in
>http://www.egroups.com/message/tuning/11859.

Nope- I just get the list in digest form.

>The "nice spacing properties" I mention there, which the Farey series
>and "Tenney" series had, means that there is a simple rule for
>predicting with good accuracy the amount of space that a ratio takes
>up in the series, based on the numbers in the ratio, and thus one
>doesn't see abrupt changes in the harmonic entropy curve as one
>increments the order of the series used.

Ah- that's right. Thanks. It's d for the Farey series, and 1/sqrt(n*d)
for the Tenney, right? I don't believe I ever saw the proof for the
Tenney series, or why you thought it would be easier to apply to triads
than the Farey series (the topic of stellation was also hot at the time).

-Carl

>>or why you thought it would be easier to apply to triads than the Farey
>>series
>
>Did I ever say that?

Perhaps not as strongly as I remembered...

>I would really like to know! More interesting, though, and perhaps easier
>to extend to triads (?), is my very recent (Friday?) posting to the effect
>that the domain of a dyad in the set of dyads within a certain numerator
>times-denominator limit is inversely proportional to the square root of the
>numerator times the denominator.

-C.