source file: m1533.txt Date: Wed, 23 Sep 1998 06:18:24 -0400 Subject: Re: scale derived by intersection of sets From: "Paul H. Erlich" >> 1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 12/7 7/4 9/5 15/8 2/1 Bob Lee wrote, >That leaves the 12/7 and 7/4. Can you form beatless harmonies with these >and other notes of the scale? I'm having a hard time imaging their use, >maybe because I'm basically a 5-limit player. 7/4 of course figures in the "barbershop" dominant seventh chord or "otonal tetrad" 1/1:5/4:3/2:7/4 (or 4:5:6:7). This is a relatively beatless harmony (but even a just 5-limit triad has some beats among the higher partials) and unequivocally implies a fundamental two octaves below the root. If the 12/7 is transposed down an octave to 6/7, it combines with 1/1:6/5:3/2 to form a sort of half-diminished seventh chord or "utonal tetrad" (1/7:1/6:1/5:1/4) which at low volumes is as beatless as the otonal one but is more ambiguous as to its fundamental. However, in this latter chord all tones have a clear common overtone two octaves above the 3/2, which, if attention is drawn to it, should be easily audible (for tuning purposes, if nothing else) unless the intonation is too exact and phase cancellation occurs. Paul Hahn wrote, >The scale can also be described as all the 5-limit pitches that form >(5-limit) consonances with either 1/1 [or] 3/2, plus the two 7-limit >pitches that form (7-limit) consonances with _both_ 1/1 and 3/2. In other words, this is the scale of the most consonant notes against a 1/1-3/2 drone. I once tried tuning several guitar strings to such a drone and found that I could generally tune a remaining string to any pitch that formed a 7-limit consonance with either 1/1 or 3/2, or an 11-limit consonance with both 1/1 and 3/2. (This is sort of mentioned in my 22-tET paper.) The resulting scale: 1/1 21/20 15/14 12/11 9/8 8/7 7/6 6/5 5/4 9/7 21/16 4/3 11/8 7/5 10/7 3/2 8/5 5/3 12/7 7/4 9/5 15/8 (2/1) Coincidentally, this scale has 22 notes while Robin Perry's has 12.