source file: m1422.txt Date: Wed, 20 May 1998 15:01:01 -0400 Subject: Reply to Joe Monzo From: "Paul H. Erlich" >> This whole argument smacks of what I object to in the prime-limit >> theoreticians' approach. Harmonically speaking, an interval is never >> more likely to be interpreted as a higher-odd-limit, lower-prime-limit >> ratio than a higher-prime-limit, lower odd-limit ratio, holding the >> approximation error constant. [Monzo:] >I'd be interested in seeing some quantifiable info about this. >Have there been experiments which prove this statement? >I think this is an important and overlooked aspect of the prime/ odd >debate in this forum. Lemme see some numbers. The experiments that Ken Wauchope and I both did (independently) are one bit of tangential evidence. The problem is that it's hard to come up with an experimental test of what ratio is implied, acoustically, by a given interval. But there are mathematical ways of defining such things, such as which pair of overtones is beating most slowly, etc., and anything reasonable along those lines will favor something like an integer-limit rule. Superimposing octave equivalence then brings you from interger-limits to odd-limits. >Well, Paul, the reason you say these things is because you favor >ETs, where you must be concerned with consistency, approximations, >etc. No! >In real just tuning, the differences are quite audible. I agree that the differences are audible >The >5-limit >tritones provide a biting dissonance in the "dominant 7th" chord >that *demand* resolution onto the major or minor triad on the "tonic". I agree that you wouldn't want the 7-limit approximations to be too smooth -- that would take away from the need to resolve. However, the tritones are not, in any relevant harmonic sense, 5-limit. >A perfectly tuned 4:5:6:7 chord is only a tiny bit less consonant >than a plain old 4:5:6 triad. There's a big difference in the sound >and feel from the 3- and 5-limit "dominant 7th"s, provided they are >in perfect tune also. And we're not talking about the tritone dyad >by itself, we're talking about a 4-part chord. Right. I think that the dominant 7th arises because it can be formed from diatonic scale steps. The diatonic scale admits a wide range of tunings, and the dominant 7th "works" in virtually all of them. Pinning its tritone down to some complex ratio has no acoustical relevance. Even contextual relevance is suspect since the diatonic scale in common-practice music requires the vanishing of the syntonic comma, so whatever ratio you give, I can probably give you another one, differing by 81:80, that has as much contextual meaning.