source file: m1412.txt Date: Mon, 11 May 1998 02:58:54 -0700 Subject: Odd vs. Prime From: Carl Lumma The debate is being pushed to the limit. What do I think? It depends on how you're using "limit"... 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. ~~~~~ (a) ~~~~~ In Just Intonation, no interval with a given prime factor in it will ever hit an interval without that prime factor, no matter how many times you stack the intervals. As Gary Morrison points out... >>Admittedly, 2 is a very special prime number, >>in that it is the only one which is even (not odd), so >>most likely, this fact can go a long way to explain >>why we hear these similarities. > >You think so? 2 is the only prime that's even, but is that really all that >important? 3 is the only prime that's a multiple of 3, and 5 is the only >prime that's a multiple of 5. ..this even applies to factors of 2. That we consider intervals related by factors of 2 to be more or less equivalent most of the time supports the idea that prime-limit interval groups (not odd-limit, since 2 is even) share subjective qualities. Against this, Paul E. argues... >"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... 2 - sameness 3 - steely coldness 5 - sweetness 7 - flourescent lightingness [etc] Now since we've brought stacking up, I've said I agree with his first paragraph (taken as writen ) in that equivalence does fall off as you stack. Paul offers... >There seems to be a pervasive (mis)conception that, for >example, an 81/64 pythagorean major third has some of the >"strong/steely" characteristics of the perfect fifth and fourth, while >the 5/4 just major third has a different, "sweet/emotional" character. >Supposedly, this is due to the "character" of the prime numbers 3 and 5, >respectively. ..except the particular example of 81/64 does not involve enough stackings to take away the "3-ness" of the interval. I don't know how many stacking ya need, and it's probably different for every listener, for every timbre, for every limit and for every musical context. But if I'm breathing, I'm not at a loss to hear the "3-ness" in the 81/64 or the "5-ness" in the 5/4. Although Paul E. adds... >To me, this sounds preposterous, as there seems to be no >mechanism or reason for the auditory system to be performing prime >factorizations. ..I doubt the current understanding of the auditory system (or, more likely to be needed, the cognitive system that interprets the data) is sufficient to declare the existence or non-existence of such a mechanism. Talk of chaos theory notconvincing. So, with the "ness" of the prime, non-odd 2 firmly in hand, we can get back to the original point, shunning interval acoustics in favor of describing music. We have the fact that no stacking (powers) of an interval will change the prime factors of the interval. It will change the odd factors, however. That's the point. Why is it musically significant? Because in music, we stack intervals. And here's how it works. 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), 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 :~) And that's the case for prime limit. It's good because it describes music made in JI. ~~~~~ (b) ~~~~~ In root of two equal temperaments, stacking CHANGES THE INTERVALS. Except when dealing with 2's, becuase they're just. This means odds are all fair play. Witness Paul Hahn's measurement of level 2 and greater consistency: far from meaning he's on the "other side", it only makes sense at an odd limit. Because ET's make every interval in all the modes available to the 1/1 (like a tonality diamond does in Just Intonation), you can measure lots of intervals just by looking at the primary intervals in each limit. Say you've got good 3/2's and good 7/4's, you can bet you'll have good 7/6's too. But that doesn't make the 6-7-9 triads consistant; there could be a better 9/7 that isn't 3/2 minus 7/6. So you need seperate consistency measurements at both the 3 and 9 limits. The reason, I suspect, that Paul Erlich fights for odd limits: He is a man of equal temperaments :~) 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. I use lots of 9 in my 12-tone compositions because it's so well represented in 12. Listen to my compositions on my web page and see if the 9's don't work. Carl