source file: mills2.txt Date: Mon, 18 Nov 1996 12:30:09 -0800 Subject: Reply to Gary Morrison; Meeting in NYC From: PAULE >One of Ivor Darreg's observations about 17TET is that it reverses one of >the >usual premises of harmony. In particular, the major third is such a dreadful >dissonance that it must be resolved outward to fourths. That then makes it a >good candidate for quartal and quintal harmony (Ivor didn't say that >specifically though). So it's not necessarily a case of putting any kind of >harmony behind a 17TET melody sounding "YUCK!", but triadic harmony in >particular. Very true! Or you could resolve the major third outward to a fifth, as was done by Medieval polyphonists working in Pythagorean tuning. >> I just happened to note that theorists who derive the diatonic scale from >> three triads are perpetrating a historico-geographic fallacy. >Is that the topic of your up-coming Xenharmonikon paper? It sounds >interesting, since that's one of two ways that come to mind immediately as the >basis for the construction of the traditional major scale. The other of >course >is tetrachord. Since there are big dissimilarities between the scales >constructed by those means, and since the scale itself came about before >triadic >harmony, I can imagine that there is fertile ground for a paper there. I just added this to my paper as an aside, but it is a good reflection of the philosophy of the paper in general. Of course the major mode was not one of the favorite modes of the diatonic scale until triadic harmony had been in use for some time -- the tritone resolves to the tonic triad only in the major and minor modes. My paper is an attempt to find a set of pitches which, like the diatonic scale, stands up on a melodic basis alone, but leads as naturally to 7-limit harmony and 7-limit tonality, as the diatonic scale leads to 5-limit harmony and tonality, and (if you believe Yasser was on the right track) as the pentatonic scale leads to 3-limit harmony and tonality. The attempt succeeds even though I look no farther than the first 34 equal temperaments (the solution is found in 22-equal). This brings me to recount the events of this weekend. Johnny Reinhard, Richard Kassell, Adam Silverman, and I met in Manhattan on Saturday for a Chinese-Latin lunch. Much enlightening discussion ensued. I spent much of the rest of the day with Johnny. His wife is a world-class microtonalist in her own right; we brought in her defective Kurzweil K2000 only to find out that the model is no longer in production -- it has been superseded by, if I recall correctly, the VX-24, which has additional expansion ports or something. Anyway, Johnny practically drowned me in bliss with his collection of music and written materials. Our dinner discussion over sushi found him confirming my view that the melodic properties of the diatonic scale are best reflected by Pythagorean intonation, its harmonic properties by just intonation, and the integration of the two by temperament (this could mean meantone temperament, not necessarily 12-equal). He associates these aspects with the left-brain, right-brain, and whole-brain functions, respectively. (I may have reversed the first two.) Gary: I believe you are right in pointing to the tetrachord as the basis of the diatonic scale. The reason this favors Pythagorean tuning is that only in this tuning does each and every octave species contain two identical tetrachords. (The chromatic and enharmonic tetrachords, of course, fail to produce a scale with this property). The pentatonic scale and my new scale have this property as well. It is worth pointing out, however, that many theorists have instead found the basis in a property one might call "maximal evenness," or spacing two unequally-sized steps as uniformly as possible around the octave. Although this can lead to the diatonic and pentatonic scales if 12-tet is presumed (another historical fallacy), it leads to a scale in 22-tet that is slightly different from the one that is constructed according to "tetrachords." Both scales are so interesting harmonically that I have allowed for either derivation in my paper. I do think the "tetrachordal" one is superior melodically, though. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Tue, 19 Nov 1996 01:16 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA13899; Tue, 19 Nov 1996 01:18:10 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA13916 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id QAA08735; Mon, 18 Nov 1996 16:18:08 -0800 Date: Mon, 18 Nov 1996 16:18:08 -0800 Message-Id: <199611190015.TAA23161@freenet5.carleton.ca> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu