source file: mills2.txt Date: Sun, 5 Jan 1997 13:22:10 -0800 Subject: Primes or Odds From: Gary Morrison <71670.2576@compuserve.com> > > But for whatever it's worth, I don't think that 9 would be considered a > > limit on the grounds that it's not prime. > I, for one, completely disagree. I would define the limit as the largest odd > number that occurs in ratios that are considered consonant. I certainly agree with Dan Wolf that we ought to use the definition from the person who coined the term, that being Harry Partch. I was under the apparently false impression that he defined it in terms of primes rather than odds. I confess that I have only read Genesis of a Music in fragments, so I have every reason to believe that Dan Wolf must be correct and I'm not. But regardless of the definition of "limit", there are definitely two different schools of thought here. The premise that odd numbers rather than primes outline various classes of harmony is based on the well-accepted fact that octaves, or stacks of octaves, produce a sensation of duplication, like that of a major tenth being practically the same harmony as a major third. Clearly then even whole numbers, all of which produce "octave duplicates" of an odd number's harmonics, can have no practically-meaningful effect on harmony. Or to put it another way, putting a single two into the product forming a numerator or denominator is as good as putting in as many as you'd like. The case for prime numbers rather than odds outlining new harmonies is based on the idea that stacks of ANY small whole number produce similar effects. That is to say that, multiplying in one of three, for example, is "as good" as adding in any number of threes. That is not, however, in the sense that threes create a sense of duplication like an octave, but that they create another characteristic sensation. As I understand it anyway, the more or less accepted characteristic for three is that of a cold, logical simplicity, as evidenced in the character of P4s and P5s. Based upon this premise, then clearly the elements of sort of classification are primes. That in the sense that multiplying by a composite is the same as multiplying by a product of primes, bringing in each of their effects into the soup. And the idea continues from there, again, as I understand it: 2: Duplication (i.e., pitch classing) 3: Cold, logical simplicity 5: Sweetness 7: A kind of Bluesy "Zap" 11: Ambiguity, lost-in-space sensation My own personal opinion is that the prime approach is correct at its root, but vastly oversimplified, and even still the biggest factor determining the character of a chord is its width. By that I mean that, for example, M3s of ANY formulation are going to have a more similar musical effect than any of them will have to a m7 for example. I base that statement partly on comparing the sensation of 27:16 and 5:3 M6s. 5:3 definitely has a sweetness that I don't hear in 27:16, regardless of their similar size. The oversimplification in the prime idea appears to be this: The ability to stack up an arbitrary number of any particular prime without fundamentally changing the basic nature of the harmony, decreases rapidly as you go up the series of primes. By that I mean that you can stack up plenty of octaves before you break two's characteristic sensation of duplication, but you can only stack a handful of threes before you lose three's sensation of cold logic. You can probably only stack two or so fives before you lose its sense of sweetness. You can only do one 7 or 11 best I can tell, meaning that 49 and 121 don't carry on the sensation of 7 and 11 (respectively) at all. I personally have very little feel for 13, although certain Mayumi and Johnny Reinhard to. I suspect that this case of (more or less) diminishing returns is a consequence of the size of the numbers in the ratio. Generally speaking, the larger the size of the numbers, the more difficult it is to associate a characteristic feel to that harmony, and the greater the chances that they'll be viewed instead as an out-of-tune rendition of a simpler pitch relationship. From what I've seen, to find a distinct character to a ratio whose numbers are larger than somewhere around 50 starts to become limited to very specialized circumstances, like long sustained notes, harmonic or melodic (but not both) presentation, carefully-chosen pitch ranges, pure timbres, very little vibrato or other forms of fluctuation, and such. I can recall at least one episode where I could not attribute a character to an interval - curiously I don't recall the interval other than that it was fairly complex - wherein I had to resort to something like five minutes of side-by-side comparison to the simpler interval that it sounded like an out-of-tune rendition of. That's obviously far beyond any realistic expectation of an audience get excited about in a truly musical performance situation! That ease also depends on how wide in the big sense the interval is. I find it much more difficult to attribute a clear musical sensation to intervals between a triple- and quadruple-octave than those between a unison and an octave. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Sun, 5 Jan 1997 22:20 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA02046; Sun, 5 Jan 1997 22:23:16 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA02044 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id NAA25890; Sun, 5 Jan 1997 13:23:01 -0800 Date: Sun, 5 Jan 1997 13:23:01 -0800 Message-Id: <199701051618_MC2-E48-35DC@compuserve.com> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu