Octave equivalence

>>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.

Actually, David, I think Marion is making the same mistake as you, so it
may not be as obvious as I thought :-)

Taking the LCM as a function of two integers, there is nothing special
about 2, other than it being the smallest prime. LCM(1,2) but
LCM(1,3)and LCM(1,15547)574. There is something special about even
number multiples if you look at half-wavelength traces.

If you allow for the octave equivalence of intervals, all triadic
inversions contain the same intervals. However, if you disallow octave
equivalence of intervals, and you define harmony in terms of intervals,
you must also reject the octave equivalence of harmony. Inversional
invariance follows from octave invariance, unless you have a
privileged root to lift the otonal/utonal degeneracy.

That notes separated by an octave are the same note in different registers
is a truism. That notes separated by an octave are more similar than
notes separated by other integer multiples is a psychoacoustic question
open to proof. The experiments I know of are inconclusive on this.
Even if it were proved, however, it would not prove octave invariance of
harmony. My own ears testify to the contrary. You may need to contract
the octave dimension of harmonic space, but that doesn't mean it has to
be removed completely.

To specifically mention the ratio 15/1. To my ears, this sounds like
two independent notes. It doesn't have the roughness associated with
dissonance, so I would call it a consonance. 15/8, however, is
a dissonance as is 16/15.

In some cases, in order to represent a lattice on a two dimensional
page, the octave dimension is removed. This is a good idea. In
some cases, chords are simplified by pretending they lie within an
octave. This is also a good idea in some circumstances. Some rules
of harmony are also octave invariant. This is fair enough, as no
quantitative system of harmony is good enough to distinguish all chords.
In many cases, notes are named within an octave, and extended beyond
it. With most tempered systems, just octaves are retained. This is
a very good idea under most circumstances. However, none of this
implies that octave transpositions are harmonically negligible. Many
people state this, and I believe it to be a result of sloppy thinking.
That is my manifesto, and I welcome comments on it.

On a slightly different thread, despite favouring the LCM, I am fully
aware of its faults. Primarily, the major tetrad 4:5:6:8 and
the major 7th 8:10:12:15 both have an LCM of 120. To my ears, and
those of tradition, they are not equally consonant. To borrow a
phrase: the LCM is the worst measure of dissonance, except for all
the others.

Graham

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>So far, all of the limits I have seen discussed have been _upper_ limits.
> Has anyone explored the idea of lower limits - say, a lower 3-limit, which
>would not allow a power-of-two to stand alone in a numerator or denominator?

Oh yes! Up to a point anyway. That's an essential element of, or
approach to perhaps, a nonoctave tuning, which I have long found
interesting. I however have been working mostly with two nonoctave equal
temperaments.

But either way, you're absolutely right that the complexity of the
harmony increases more rapidly when you omit two (or more). And so also
does the variety.

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