Re: [harmonic_entropy] Digest Number 66

> /harmonic_entropy/files/dyadic/partch.bmp

Very impressive, Paul! It's so detailed! I never noticed before (until you stuck it in my face) how the ratios line up
almost in layers. Do you have to draw me a picture? Yes, you do. :-)

You know, for a number cruncher who doesn't understand the mystery behind the numbers, you're all right. ;-)

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
--

--- In harmonic_entropy@y..., "David J. Finnamore" <daeron@b...>
wrote:
> >
/harmonic_entropy/files/dyadic/partch.bmp
>
> Very impressive, Paul! It's so detailed! I never noticed before
(until you stuck it in my face) how the ratios line up
> almost in layers. Do you have to draw me a picture? Yes, you
do. :-)

Octave-equivalent harmonic entropy agrees with how Partch lined up
the ratios in his consonance graph, the One-Footed Bride.

For the non-octave-equivalent case, go to
/harmonic_entropy/files/dyadic/ and
download het006_16.zip. A lot of things are lining up here -- along
diagonal lines. Do you see them?

Paul wrote:

> For the non-octave-equivalent case, go to
> /harmonic_entropy/files/dyadic/ and
> download het006_16.zip. A lot of things are lining up here -- along
> diagonal lines. Do you see them?

Sure enough - at steeper angles for lower numbers and shallower angles for
higher numbers.

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
--

But can you see what exactly lines up? A set of fractions falls on a
straight line if . . .

-----Original Message-----
From: David J. Finnamore [mailto:daeron@...]
Sent: Wednesday, March 14, 2001 10:25 PM
To: harmonic_entropy@yahoogroups.com
Subject: [harmonic_entropy] Re: Digest Number 66

Paul wrote:

> For the non-octave-equivalent case, go to
> /harmonic_entropy/files/dyadic/ and
> download het006_16.zip. A lot of things are lining up here -- along
> diagonal lines. Do you see them?

Sure enough - at steeper angles for lower numbers and shallower angles for
higher numbers.

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
--

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--- In harmonic_entropy@y..., "Paul H. Erlich" <PERLICH@A...> wrote:
> But can you see what exactly lines up? A set of fractions falls on a
> straight line if . . .

The same amount gets added to (or subtracted from the other
direction) both the numerator and denominator?

Well, that's one set of lines, but there's plenty more . . . unless what you
meant was, "there are two integer constants a and b, and if you keep adding
a to the numerator and b to the denominator, you'll get a bunch of fractions
that line up with one another" . . . it sounded like you may have been only
thinking of the case a=b.

-----Original Message-----
From: daeron@... [mailto:daeron@...]
Sent: Friday, March 23, 2001 2:59 PM
To: harmonic_entropy@yahoogroups.com
Subject: [harmonic_entropy] Re: Digest Number 66

--- In harmonic_entropy@y..., "Paul H. Erlich" <PERLICH@A...> wrote:
> But can you see what exactly lines up? A set of fractions falls on a
> straight line if . . .

The same amount gets added to (or subtracted from the other
direction) both the numerator and denominator?

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--- In harmonic_entropy@y..., "Paul H. Erlich" <PERLICH@A...> wrote:
> Well, that's one set of lines, but there's plenty more . . . unless
what you
> meant was, "there are two integer constants a and b, and if you
keep adding
> a to the numerator and b to the denominator, you'll get a bunch of
fractions
> that line up with one another" . . . it sounded like you may have
been only
> thinking of the case a=b.

I was. No extra credit points this time. Do they line up as a by-
product of the formula you're using, or does it indicate something
about the way humans really hear? Or is it possible to separate the
two in this case?

The fact that they line up really surprised me -- in fact this would be a
great way to diagram the various ratios apart from any harmonic entropy
considerations. The reason they line up this way is that the height on this
graph, for the ratios indicated, is proportional to numerator times
denominator (good old Benedetti!). Why this relationship came out of the
formula I'm using . . . beats me. Anyhow, the pattern clearly falls apart if
you try to extend it to more complex ratios; the harmonic entropy curve
imposed a sort of 'ceiling' to perceptual complexity. So to answer your
question in a strange sense, it is the _failure_ of the pattern to continue
for more complex ratios that I think indicates something interesting about
the way humans really hear.

But really you're right, you can't separate the two -- the harmonic entropy
model is a _model_ of how we hear, and the nice patterns don't have any
bearing on whether the model itself is valid or not (or partially valid).

-----Original Message-----
From: daeron@... [mailto:daeron@...]
Sent: Friday, March 23, 2001 7:15 PM
To: harmonic_entropy@yahoogroups.com
Subject: [harmonic_entropy] Re: Digest Number 66

--- In harmonic_entropy@y..., "Paul H. Erlich" <PERLICH@A...> wrote:
> Well, that's one set of lines, but there's plenty more . . . unless
what you
> meant was, "there are two integer constants a and b, and if you
keep adding
> a to the numerator and b to the denominator, you'll get a bunch of
fractions
> that line up with one another" . . . it sounded like you may have
been only
> thinking of the case a=b.

I was. No extra credit points this time. Do they line up as a by-
product of the formula you're using, or does it indicate something
about the way humans really hear? Or is it possible to separate the
two in this case?

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