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the 6th chord and odd-limit theory (Paul E)

🔗Manuel.Op.de.Coul@ezh.nl (Manuel Op de Coul)

6/17/1997 9:32:34 AM
From: PErlich

George Kahrimanis and I have been discussing tunings of the 6th chord (aka
the minor seventh chord). The two alternatives are 12:15:18:20
(15:1/12:1/10:1/9), and 14:18:21:24 (36:1/28:1/24:1/21). Kami Rousseau
suggested the latter on this list some time ago.

>>213

> In my experience, only "2" is (usually) an allowed factor in this
> context, not "3"; that is, only inversions are "free"

I would agree with you here. This motivates my adherence to the odd-number
definition of limits. However, as I demonstrated, every interval in Kami's
chord is within the 9-limit, which is also true of 12:15:18:20. So the 21 is
not really that bad. If we want to look at the full chord, Kami's is a
21-limit utonality or 21-limit otonality, which I agree is more complex than
the 12:15:18:20, which is only a 15-limit utonality or otonality.

But psychoacoustically, and this should be evident to a listener, these
chords are no more dissonant that a 9-limit utonality. I believe that this is
because THE ONLY REASON UTONALITIES ARE CONSONANT IS BECAUSE THE INDIVIDUAL
INTERVALS ARE CONSONANT. The common overtone is there but doesn't make the
chord more consonant. Once again, I don't believe in dualism. So even though
the 6th chords above may seem to demand explanation in terms of the 15- and
21-limit, I would say, if you admit utonalities at all, then both those
chords are 9-limit chords, no higher.

I wonder if there are any 7-limit analogues to these chords (i.e., chords in
which each interval is within the 7-limit but the chord as a whole is in a
higher limit). I think the answer is no. Anyone care to come up with a
counterexample?

BTW, I don't think the 13-limit is inherently less consonant than the 15- or
17-limits. I think the 15-limit shows up a lot incidentally, due to combining
5- and 3-limit intervals, and I think the 17-limit was discovered in the
diatonic scale and so is very familiar. Ironically, my use of 22-equal
prevents me from exploring the 13-limit; leaving 13 out, everything through
the 17-limit is consistently expressed in 22-equal.

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