This is a complete post Paul Erlich sent to the Tuning List on 17 Jun 1997. Not all is relevant here but you may as well see the whole thing. -- GJB

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 (=1/15:1/12:1/10:1/9), and 14:18:21:24 (=1/36:1/28:1/24:1/21). Kami Rousseau suggested the latter on this list some time ago.

>>21=7*3

> 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?

[The answer is no -- GJB]

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.