source file: m1413.txt Date: Mon, 11 May 1998 18:21:28 -0400 Subject: RE: Odd vs. Prime From: "Paul H. Erlich" >a) For describing music tuned in Just Intonation >b) For describing music in root-of-two equal-step tunings >For "a" I prefer prime, and for "b" I prefer odd. Now I say I use "limit" >for "describing music", NOT for creating an acoustical theory of interval >perception. I admit the two are intimately tied up, and I will address the >latter only as the former requires. Here's what I've always said: "a" is preferable for describing the resources of a given system of Just Intonations, and "b" is preferable for describing the maximum interval complexity in simultaneities considered consonant in a given style. The latter is very tied up with an acoustical theory of interval perception. >>"An arbitrary power of two produces an effect of equivalence." This must >>of course include the case where the power is zero, and the equivalence >>is greatest. The equivalence evidently falls off as the power increases. > >>But what about steely coldness? Is it there when the power is zero? Does >>a unison contain all potential interval qualities, to a greater degree >>than the intervals themselves? It would seem hard to argue that way. >..And I wonder if he is talking powers or factors. Factors are involved >in the ratio of frequencies in a given interval, and powers are involved in >the stacking of any interval. The idea is that adding a factor of 2 to an >interval produces a new interval with a certain "sameness" to the first, >because 2 is the "sameness factor". And stacking any interval will produce >a new interval with a certain "sameness" to the first, simply because >stacking doesn't add any new factors of any kind. >If he's means what he says ("powers"), then his first paragraph is correct, >but his second is not, as "steely coldness" is an attribute of the factor >3, not the power 3. >If he means factors instead of "powers", then I suppose his whole thing is >correct. Although I do not see why he insists on making the unison into >some kind of "white light", I am willing to oblige. As far as factor >determines which partials line up, and all of the partials in a unison line >up, the analogy holds... I'm not sure what it is in my wording that was problematic. Perhaps you could suggest an alternative wording for each of the interpretations you mention, and I could endorse one, allowing us to proceed? >Mideval music choral music has ratios of 3 and 9, but not of 5 or 7. So is >it 9-limit? Now you might say that we can just as well have music with 7's >and no 5's, creating an ill-defined prime limit. And true, we can have >such a music, but we don't. Except for isolated experiments with >fixed-pitch instruments (I believe Fokker did work with such tunings), And don't forget LaMonte Young!!! >music has evolved by prime limit. Pull out a CD of English tudor music, as >sung by the King's Singers, for example, and try to add the 7's. You >won't. They didn't, not for hundreds of years. They've got ratios of 25 >and everything else needed to modulate around the 5-limit, throwing commas >around like frisbees, but no 7's. Barbershop's got 7's and 28's, and 63's >a-plenty. And 9's, and 18's, and 27's. But no 11's. Never will you hear >an 11 used harmonically in Barbershop music. And the first time you do, >you'll be hearing them again soon and often :~) Again, the odd-limit definition applies not when considering all intervals present, but when considering which intervals can be considered consonant. In all your examples, the higher composites are dissonant. >The reason, I suspect, that Paul Erlich fights for odd limits: He is a man >of equal temperaments :~) Equal temperaments or not, that has nothing to do with this issue. >The idea that 9's do not become harmonically >significan't until we have 7's does not hold in my experience. Try playing >just 4-5-6-8-9 chords and see if the 9 doesn't serve the same purpose as it >does in a 4-5-6-7-8-9 chord. >That is not at all a valid interpretaion of anything I've tried to say.