source file: m1407.txt Date: Wed, 6 May 1998 16:53:36 -0500 (CDT) Subject: RE: Reply to Paul Hahn From: Paul Hahn On Tue, 5 May 1998, Paul H. Erlich wrote: > >(a) I wish to go beyond basic (level 1) consistency because it is > >possible for an ET to be level 1 consistent and still err from a given > >just ratio by nearly half the stepsize of the ET. An extreme example: > >18TET is consistent to the 7-limit, but its 11-step "fifth" is only > >barely better as an approximation to the 3/2 than its 10-step interval. > >One is over 31 cents high, the other more than 35 cents low; the > >difference between the two errors is less than four cents! > > That's a good reason, but it would seem highly unlikely that the most > appropriate consistency level for this purpose would turn out to be an > integer, unless you had particular composite ratios in mind that you > care about representing consistently. That's fair--you could, say, choose tunings where the roundoff error was at most half the distance to the other alternative, which would result in a cutoff consistency level of--uh--3/2, I think. Still, by and large I think consistency levels make much more sense as integers, since they correspond to secondary, tertiary, etc. intervals being represented consistently. Why I care that they are, well, see below. > It was the ratios of large > composite numbers I was objecting to, since treating them more carefully > than ratios of numbers with larger prime factors, even if the latter > numbers are smaller than the former ones, is something that those who > believe in prime characteristics will advocate, but I find no > justification for. Okay, so the question is, why would I care, for example if I was working in 5-limit harmony, about how 9/8s or 10/9s are approximated, but not any intervals involving 7? Short answer: voice-leading. Medium-length answer: I wish to approximate just intervals not merely in isolated chords, but in the progression of one chord to another. Representing the movement of one consonant identity to another without producing (what are to me) bizarre perceptual artifacts requires level-2 consistency. Long answer, with example: let's take this forum's favorite whipping boy, 12TET. It's consistent to the 9-limit, so it ought to be okay for some septimal harmony, yes? Let's take a 4:5:6:7 tetrad and move to another 4:5:6:7 tetrad so that the 6 of the first tetrad becomes the 7 of the second. If we voice everything so that all voices move as little as possible, the 7 of the first moves down by a 64/63, the 6 holds, the 5 moves up by a 36/35, and the 4 moves up by a 15/14. (Okay, so the second tetrad is really in "first inversion" becoming 5:6:7:8.) How is this represented in 12TET? A 4:5:6:7 tetrad has 4, 3, 3, and 2 steps between the members. If the first tetrad is rooted on 0 and the second keeps the common tone, we have 0 4 7 10 moving to 1 4 7 9. IOW, the 64/63 is represented by 1 step, the 36/35 by 0 steps, and the 15/14 by 1 step. To me, this doesn't make sense--since there's a common tone, we should be able to hear the 8/7, 6/5, and two 7/6s relative to it fairly accurately. Why should the 36/35 vanish when the much smaller 64/63 doesn't? But while I'm worrying about these rather large odd numbers, ratios involving (say) 11 never entered into it. I realize full well that these sorts of oddities are rich ground to explore; indeed, a major part of Easley Blackwood's _12 Microtonal Etudes_ springs from exploiting them. However, personally, my goals are different. --pH http://library.wustl.edu/~manynote O /\ "Churchill? Can he run a hundred balls?" -\-\-- o NOTE: dehyphenate node to remove spamblock. <*>