Octave invariance revisited

I said,

>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.

This is clearly wrong in that the property is also shared by
ratios like 1:3. My thanks to the people who pointed this out.

When I wrote this I was thinking of ratios as restricted to a
single octave, including both endpoints, even though I cited an
example of a trans-octave ratio in the next paragraph--clearly an
example of muddy writing and probably muddy thinking.

I think that it remains true that there is an important acoustic
division between repeating patterns of sound pressure. Some have
zero energy in the fundamental, and will thus mostly be heard as
chords, and those with an appreciable amount of energy in the
fundamental will mostly be heard as a single note.

Possible exceptions would be the examples like 15:1 where there
is a wide gap between the fundamental and a higher harmonic.

I have been reading Partch again a little. I still find myself
put off by the man's style. It seems to me that making a big
deal of whether the numerator or denominator of an interval is
odd is wallowing in irrelevancy.

I also find his practice of representing scales and chords with
the 1/1 ratio in the middle somewhere to be strange.

All, in all, I have found his book to be useful source of
historical information, but I still question the usefulness of
his theories.

Marion

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David Finnamore wrote:
>BTW, I like a concept that Graham Breed introduced me to: harmonic
>dimensionality using prime limits. As I understand it, using all octaves
>(like P. D. Q. Bach's Don Octave) would be 1-dimensional, Pythagorean
scales
>(3-limit in practice) would be 2-dimensional, 5-limit would be
3-dimensional.

Although this is the most familar scheme (Riemann, Tanaka, Johnston,
Tenney), it is often useful not to lock in the identity of a given
dimension with a given generating element. For example, I might - like LaMonte Young - want to use sevens but no fives. Why should I have to speakin terms of four dimensions when I am only using three? Or, in the other
direction, I might choose to treat nine as a separate dimension from three,
in which case, Partch's diamond is viewed as six-dimensional rather than
the five yielded by counting primes. Let's reserve ''dimension'' for thenumber of dimensions on the graph independent of the particular intervalsto which they are assigned.

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In response to my mention of prime number-based scale dimensionality, Daniel
Wolf suggests:

>... Let's reserve ''dimension'' for the
>number of dimensions on the graph independent of the particular intervals>to which they are assigned.

I appreciate your pointing out the terminology problem posed by that use of
that term. It sure would be nice to come to a general agreement on terms for
the various kinds of "dimensionality" that one can visualize from various
perspectives. I find use for all of the kinds that you mentioned, each in
its own way. I suppose it could be a bit incongruous to speak of using,
e.g., the 2nd, 3rd, and 5th dimensions only. Anyone have any suggestions?

David J. Finnamore

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