RE:A catalog of chords [reply to Gene, Part A]

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Date: Thu, 10 Mar 2005 16:43:32 +0000
From: Gene Ward Smith
Subject: Re: Re: A catalog of chords

[Yahya]
Then PMPP, the product_j p_j^m_j is well defined for all chords
a_1:a_2:a_3:...
with integer a_i. (I'm sure there's a name for this product in the
theory of numbers - didn't Euler use it?)

[Gene]
Yes, it's the least common multiple, denoted LCM(a1, ..., an).

[Yahya]
Oh, so you _were_ paying attention! :-)
Of course, the way we choose to express something may serve to
either clarify or obscure the picture. Sorry for playing a silly joke.
And thank you for being a gentle man!

[Gene]
I thought about using the LCM somehow, but I figured I'd need to
do more than just use it as a metric.

[Yahya]
But this to me is the interesting point! What happens if you do?
Which metric - of the many possible - best accords with our intuitive
notions of harmonic distance? And which best accords with our
psychoacoustic makeup?

I'm going to divide this reply to you, Gene, in two parts; Part A is
the easy part, while Part B will take me a little longer to work out.

Part A
-------

It seems evident to me that any theory of harmonic distance that
considers only frequency ratios must inevitably miss out on the
contribution that timbre or spectrum makes. Given that limitation,
I would still hope objective grounds exist - that we might find - for
preferring one metric to another, all other factors being held
constant.

We can justify using the form a^2 + b^2 + c^2 on the grounds
that it's equivalent to using the Euclidean metric, which has
proven useful in many other contexts. The same argument applies to
the "city-block" metric |a| + |b| + |c|. Both these metrics have the
general "hyper-Euclidean" form dk = (|a|^k + |b|^k + |c|^k)^ (1/k);
or, since a, b, c are > 0, dk(a:b:c) = (a^k + b^k + c^k)^(1/k); for
some natural k.

Obviously our metric should deal as well with chords of one, two,
three, four or any other natural number of components.
More generally, we would want to associate the chord
(a1:a2: ... :an) with the forms
dk(a1:a2: ... :an) = (a1^k + a2^k + ... + an^k)^(1/k)
for any n and our chosen k.

Another point: I had hoped we could treat unison as the origin
of the space of chords with our metric. But obviously the form
dk(1:1) = (1^k + 1^k)^(1/k) = 2^(1/k), which is not zero.
It would be strange indeed to talk of a unison chord as being
somehow distant from the point of complete harmony, which is
identity.

Perhaps we should be looking at this form instead:
Given two chords A = (a1:a2: ... :an) and B = (b1:b2: ... :bn),
and some number k, define the distance between the two chords
dk(A, B) = dk((a1:a2: ... :an), (b1:b2: ... :bn))
= ((a1-b1)^k + (a2-b2)^k + ... + (an-bn)^k)^(1/k).

Then we have dk (A, A) = 0 for all chords A. That is, any
chord is perfectly consonant with itself.

What other notions "make sense" for harmonic distances?

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Part B to follow ...
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