Octave equivalence

In digest #1119 I wrote:

> Have you never seen pictures of
> waveforms? E.g., given two waves, one of which has a frequency at a
> power-of-two of the other, the lower one will cross the zero point
(volage-
> or SPL-wise) only at points at which the higher one is crossing as well.
> This is a unique property of frequencies related by powers-of-two; they
> have, then a special relationship of some kind, no? The higher one is
> repeating at the same frequency as the lower one, as well as at its own
> nominal frequency. They can, then, in a very real and physical sense, be
> considered to be the same note; i.e., the same "pitch" but in different
> registers.

I was wrong. 8-)> Graham Breed was gracious enough to point out my error to
me off-list. The above definition would apply to any whole-number multiple
of the lower frequency, not just powers of two. 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. I acknowlege that this weakens my argument considerably.

Much more interesting was 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.

That's what I was picturing in my mind but stated rather poorly, using the
wrong point of reference. Thanks, Marion. So for now I still maintain my
conclusion that two pitches separated by the interval 2:1 are the same note
in different registers. That triad inversions sound different is the result
of a different set of intervals within each inversion of the chord, and so
does not seem to require throwing out the idea of octave equivalence.

As to Gary M.'s comment that this argument would be distroyed by simply
mistuning the octaves slightly, I think that since its proximity to 2:1
causes the ear to treat it (musically) as if it were exactly 2:1, the theory
is still valid.

David

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>To check against paper, using the ratio for the 5th or 3/2, or 1.5,
>and using A220, (a) for conveince, four fifths gives a frequency of
>220*1.5^4 (as entered on the W95 calculator, OK so ^ is the x^y key)
>or 1113.75. So two octaves down or /4 gives 278.4375 for c# ' .
>Hmm already that is sharper than 277.183 called for by ET.

That's because you used 3:2 exactly rather than tempering it downward.
Quarter-comma meantone's fifth is tempered downward by a quarter of a
(syntonic) comma, 81:80. That makes a stack of four of them land you on an
exact 5:4 ratio.

Mathematically, QC meantone's fifth represents a frequency multiplier of
1.5 divided by the fourth root of 81/80, which works out to about 1.49535,
or a pitch difference of about 696.6 cents.

Using an exact 3:2 gives you Pythagorean tuning, which as you correctly
pointed out has that very sharp 81:64 M3.

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