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Re: [Phys-l] Exotic harmonies



Interesting comment, but "dispersion relation" is a pun. I think John
means the relation between frequency and wavelength and not the
Kramers-Koenig relation.
Regards,
Jack

On Wed, 31 May 2006, John Denker wrote:

1) The discussion of temperaments (equal, just, or otherwise) is
interesting ... but it is at best tangential to the points I was
making, and is actually something of a red herring. I did not
mention temperament because I had no reason to. I spoke of
stretched octaves. The fact is, octaves are stretched. The
octaves will be stretched no matter what temperatment is used.
Indeed, you could have a piano where only every 12th key is
usable -- the rest being completely broken -- and the octaves
would still be stretched, for reasons completely unrelated to
temperament.

2) I stand by what I said about integer ratios. Maybe I'm old
fashioned, but I think of being "almost an integer" as sorta
like being "almost pregnant". Integers are not the controlling
idea. The dispersion relation for the string is the controlling
idea. Maybe an ideal string produces integer harmonics, but the
real string doesn't. They might be close to integers, but they
might not even be close. There may be engineering reasons for
favoring strings with high mass, high tension, and low stiffness,
but there is no cosmic significance to it.

I am aware that all sorts of big shots from Pythagoras to Galileo
(inclusive) espoused the opposite conclusion. Words fail to express
how much I don't care. The evidence is the evidence, as previously
described. The evidence indicates they were wrong.

3) Talking about harmonious relationships between sine waves is risky.
J.R. Pierce in _The Science of Musical Sound_ excluded sine waves
from consideration, on the grounds that they were not _musical_
sounds. When I first read that, I didn't understand, but I have
come to agree with him. Many things that are true for sine waves
are not true for ordinary musical sounds, and vice versa.

4) There is nonlinearity in the ear, in the brain (!), and elsewhere.
There is also dispersion in the piano string. The beats that one hears
involve a somewhat tricky interplay between the dispersion and the
nonlinearity. I didn't mention it because I didn't want to get into
it, and I still don't. Suffice it to say that fixating on either
nonlinearity or dispersion (to the neglect of the other) is a common
mistake.

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