source file: mills2.txt Subject: Post from McLaren From: John Chalmers From: mclaren Subject: Paul Erlich's innovative & insightful ideas in Digest 749 -- In Tuning Digest 749, Paul Erlich pointed out that "A typical way of evaluating equal temperaments is to measure the smallest possible tuning deviation of several just intervals of interest. The problem with this approach is that it does not guarantee that the approximating intervals are consistent with one another." -- Paul Erlich This is an excellent point, and it's not clear that anyone else has made it so cogently. In fact this could serve as the basis of a fascinating article, were any of the 12-TET-obsessed music theory journals interested in publishing such an article. (You never know. Sometimes they relent.) The earliest reference to an idea similar to this one appears to be found in Mersenne's Harmonicorum Libri... (1636), pp. 126-127 (last pagination of 2nd ed., 1648). Mersenne defines each interval as equal to a certain number of commas, and some of the commas are incompatible with one another. This is in the same ballpark as, but clearly not identical to, Erlich's insight. -- Gary Morrison's conceptual difficulties in working with multiple simultaneous tunings are probably what Ivor Darreg used to call a "pseudo-problem." In other words, something that conceptually seems like an unimaginably difficult issue but which, in real life, turns out to be largely irrelevant or trivial. (Playing 19-TET or 22-TET on a 7-white-5-black keyboard is such a "pseudo-problem." In both cases, the answer is to use open-voiced chord with third and root played by the left hand, fifth by the right hand. Simple. Trivial. What's the problem?) In combining 12-TET and 19-TET, for instance, we in the Southern California Microtonal Group have found it useful to view 19-TET as a gapped nondiatonic scale with some very xenharmonic ornamental tones. As I've pointed out before (and as Paul Erlich's experiments with his roomate demonstrate), listeners hear the whole-tone portion of a 19-TET diatonic scale as very reasonable and familiar-sounding, but the 2/19 of on octave "semitones" (a misnomer, since this term presupposes that whole tones are divided into 2 parts whereas in 19-TET they are divided into 3 parts) always make folks squirm. They sound...weird. "Out-of-tune" is the typical description. The solution, in my experience, is to fracture the mislabelled "semitone" intervals in the 19-TET approximation of the 12-TET diatonic scale into 1/19s of a tone. Thus a typical melody in 19-TET will proceed by whole steps which sound very reaonable in combination with 12-TET. Then a constellation of 1/19-octave steps are used to bridge the gap twixt the next "island" of whole steps in the 19-TET approximation of the diatonic scale. This seems to work reasonably well *provided* that 12-TET semitones and 19-TET 1/19s of an octave don't play simultaneously. If, instead, these intervals "trade off" between different players (the way good jazz players will trade off playing flatted blues thirds), the 12 + 19 combo seems to work well. Another trick that seems to work is to play melodies full of familiar 12-like intervals in 19-TET, but to go above or below the last note of the melody by 1/19 of an octave. This works well, by the way, for many incommensurable ETs: 15 and 12, 14 & 17, 15 & 17, 19 & 22, etc. Many ETs are, however, commensurable-- that is, one forms a subset of the other, or both form gapped subsets of a much higher division of the octave. When combining 5-TET and 7-TET, for instance, it is often useful to throw in 35-TET chromatic passages. This emphasizes that the tiny intervals heard between 5-TET and 7-TET pitches that happen to occur vertically are not accidental, but part of the coherent 35-TET scale. The issue of combining non-just non- equal-tempered scales with xen ETs, or combining, say, all 3 different classes of tuning simultaneously, is more complex. I'm still feeling my way along in this regard. --mclaren