width and entropy

>Zero divided by zero is one. It's also a terribly convenient way for
>setting unisons apart from all other intervals.

Gee, I thought things divided by 0 were usually infinite.

>The conceptual advantage of the Farey series (which leads to a denominator
>rule) is that is seems reasonable to believe that the brain has a
"template" >of sorts of the harmonic series up to a certain limit. Then
all intervals >within that template are possible interpretations, and
others aren't. If >this is the way it works, it doesn't matter if you are
holding the lower >note, upper note, arithmetic mean, or geometric mean
constant.

So the denominator is the thing for the Farey series, eh? No matter which
frequency you hold constant? What of Keenan's idea of measuring the period
of the VF?

>What I just found for the last case is that in fact the width is, to an
even >better approximation that the one for the first (Farey) case,
inversely >proportional to the _square root_ of the product of the
numerator and >denominator.

Drat! Just when I was getting comfy with the Farey entropy formula.

The nifty thing about the mediant is that it divides the space between two
fractions by finding the simplest one in between. That makes sense.

I don't understand this new geometric mean entropy... The geometric mean
may split the space between two ratios into portions of equal logarithmic
size, but why should the brain/ear care about that?

Carl

<>
<From: Carl Lumma <clumma@nni.com>
<
<>Zero divided by zero is one. It's also a terribly convenient way for
<>setting unisons apart from all other intervals.
<
<Gee, I thought things divided by 0 were usually infinite.<

Dr. Akkoc is right. 0/0 is undefined. I should have said: "For this table,
0/0 is defined as 1."

The analogies with silence and white noise are worth responding to. My
table was one of all the possible rational dyads in lowest terms , ordered
in terms of d, and then n (thus making it related to but not identical to
Farey series). Noise is not a dyad. Silence could be a dyad with amplitude
zero, but that presents an ontological question well beyond my meagre
talents.

>>Zero divided by zero is one. It's also a terribly convenient way for
>>setting unisons apart from all other intervals.

>Gee, I thought things divided by 0 were usually infinite.

But 0 divided by things is usually zero. Zero divided by zero is one of
the seven undefined terms of arithmetic. This convention is very helpful
in many areas of mathematics, including calculus. If I recall correctly,
the seven undefined terms are:

0/0
inf/inf
0*inf
inf-inf
1^inf
inf^0
0^0

(where inf means infinity). Each of these can take on a different value
based on the "usually" argument, but the first six of them can in fact
be thought of as taking on any possible value, call it x, due to the
following "checks":

x*0 = 0 -> 0/0 = x
x*inf = inf -> inf/inf = x
x/0 = inf (or x/inf = 0) -> 0*inf = x
x+inf = inf -> inf-inf = x
infth root of x = 1 -> 1^inf = x
0th root of x = inf -> inf^0 = x

The last one is undefined because it can be either zero or one: things
to the 0 are usually 1, but 0 to the things is usually 0.

(Someone correct me if this isn't the right seven.)

>The nifty thing about the mediant is that it divides the space between
two
>fractions by finding the simplest one in between. That makes sense.

>I don't understand this new geometric mean entropy... The geometric
mean
>may split the space between two ratios into portions of equal
logarithmic
>size, but why should the brain/ear care about that?

I'm not sure if you're confused about something: the fact is that the
mediant is used in all these formulations, and the parameter
(denominator, numerator, arithmetic mean, geometric mean) only defines
the set of fractions for which mediants are computed.

Assuming you weren't confused about that . . .

Well, if you are comparing intevals by starting with a unison and then
spreading the two notes apart by equal amounts, then the set of all
ratios whose VF is above a fixed lower bound will be the set for which
the geometric mean of the numerator and denominator is below a fixed
upper bound.