source file: mills2.txt Date: Wed, 30 Apr 1997 18:06:41 -0700 Subject: 22 etc. From: Paul Rapoport re PAULE's comments on 22 and my original: > >whether you mean just (as in 81:80) or not. The context usually > >determines; in any ET, it's obvious. > NO, IT'S NOT OBVIOUS AT ALL! All I said and meant was that there is no just 81:80 in any ET. Terribly obvious. But your elaboration is more interesting: In any ET, we would agree that the perfect > fifth is the closest approximation to the 3:2. We might not, but let's for the moment. We would also agree that the > syntonic comma is NOT necessarily the closest approximation to the 81:80 -- > for example, in 22-tET the closest step size to an 81:80 is 0 steps, and yet > we would both call 1 step the syntonic comma. Why? Because the syntonic > comma is defined as interval obtained by tuning several successive CONSONANT > intervals. I agree that we may define crucial consonances as closest to just in an ET (usually representing the pure harmonics, with a power of 2 in the denominator, in case anyone else isn't sure what we are talking about). After that, I would start (not necessarily conclude) by defining such things as commas in structural terms. They are left over as differences between multiples of harmonics. The syntonic comma has a standard definition; often it comes out positive or zero, sometimes it is negative. To say that it may not be the closest to 21.5 cents is certainly true but is not affected by the structural definition. If you don't keep the same definition across tunings, you can't compare what is being defined, as your subsequent demonstration shows. [two usually equivalent definitions omitted, one for m3s, one for M3s] > So the syntonic comma is 1 step in 22-tET, even though 0 steps would be > closer in size to the JI syntonic comma. This situation would never occur > for a basic consonace like the perfect fifth. Perhaps not relevant, but it would. There are plenty of tunings where there may be a choice of P5, and we may prefer the one which is farther from just for one of several reasons. > Thus the "pseudo" terminology > is warranted. If you don't try to make dissonances define themselves like consonances, it's not necessary. > Now there are tunings where (1) and (2) give different answers! For example, > in 21-tET, definition (1) gives a syntonic comma of 0 steps, while > definition (2) gives an syntonic comma of -1 (that's minus 1) steps. Another > example is 20-tET, where definition (1) gives a syntonic comma of 1 step, > while definition (2) gives a syntonic comma of 2 steps. So I would argue > that if an ET is not consistent within the 5-limit, the syntonic comma is > undefined in that tuning. This I find most interesting, and I must get back to your notion of consistency when I have time to return to the theory. Of course, if a m3 is defined in terms of a M3 and P5, this discrepancy will never come up. Still, the m3 is valuable enough that the problem you mention is worth consideration. I tried to do so, in fact, in my 25-tET piece, because it has two viable m3s, at 288 cents and 336 cents. Obviously [!], if a syntonic comma is defined only one way (the difference between four P5s and a M3), there will be only one result. Paul R Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Thu, 1 May 1997 03:09 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA01599; Thu, 1 May 1997 03:09:33 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA01585 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id SAA21948; Wed, 30 Apr 1997 18:07:58 -0700 Date: Wed, 30 Apr 1997 18:07:58 -0700 Message-Id: <199704302242.PAA10310@ella.mills.edu> Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu