source file: mills2.txt
Date: Sat, 5 Jul 1997 23:20:38 +0200

Subject: Re: Octave equivalence

From: DFinnamore@aol.com

If this was dealt with in digest 1122, I apologize - I haven't received that
one yet.

In digest 1124, refering to something I wrote in 1119, I write:

> ... However, there is still a
> unique relationship if you consider that the lower one will cross the zero
>  point only at points at which the higher one is also crossing _and 
> completing
>  an even number of cycles_.  It turns out that odd-number multiples cross
in
>  mid-cycle where the lower one crosses, and even ones cross at completed
>  cycles.

Ack!  I did it again!  Sometimes I feel like Andy Griffith: "Barn, can you
believe that I can fit this gre't big foot - all of it - in my mouth?!"

Actually, any number which is [a power of two greater than one] of any whole
number (x*2^y, where x any whole number, and y is any whole number > 1 --
12, for example) will fit my definition (I think).  There seems to be no
especially unique relationship for powers of two.  I see now that I was
originally visualizing waveforms with the following series of relationships:
1/1, 2/1, 3/2, 4/2, 5/4 ... - in other words, I was assuming octave
invariance in my attempt to prove octave invariance - bad reasoning all
through.  Sorry, everyone.  Ah, the power of a false assumption!  Having
discovered that, I was still clinging desparately to Marion's observation:

>On octave invariance, it has just occurred to me that, in LCM
>terms, the octave has a property that no other ratio shares. 
>That is two notes that differ only by octaves can be mixed
>without making the length of the resulting "interference" pattern
>longer than the waveform of the lowest note.

But wouldn't that be true of x:1 where x any whole number, not just powers
of two?  So it appears to be an argument for otonal ratio invariance, of
which octave invariance is merely the simplest example.  Am I on the right
track yet?

Well, maybe psycho-physical evidence will prove to be a good arrow in our
quiver, Paul E.

David J. Finnamore

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