source file: mills2.txt Date: Thu, 27 Mar 1997 13:59:12 -0800 Subject: RE: Rowell article (Paul E) From: Manuel.Op.de.Coul@ezh.nl (Manuel Op de Coul) From: PAULE to John Clough John, For a simpler example, take the definition of 2nd-order maximal evenness with respect to the series 12->7->3. By your definition, diminished triads are 2nd-order-ME as well as major and minor triads. This is because, after deriving the diatonic scale as the 7-deg ME scale in 12-tet (which is historically fallacious, but never mind for now) you pretend that the 7 scale degrees are actually 7-tet when determining the 3-deg ME scale. Then you remember the actual 12-tet positions of the 7-tet scale, and using them you translate the 3-deg scale back to 12-tet. In my definition, one keeps the 7 scale degrees always in 12-tet, even when determining the closest approximation to 3-tet. Then only major and minor triads are obtained. In case that wasn't clear, imagine a clock. First you take a disk with seven equidistant arrows pointing away from the center. No matter how you rotate this disk, the numbers on the clock face that come closest to the arrows form a diatonic scale. That's just 1st order maximal evenness. Now, on the SAME clock, put a disk with three equidistant arrows. No matter how you rotate it, the diatonic numbers derived previously that come closest to the three arrows will form either a major or a minor triad. (If two arrows each fall exactly between two notes, which happens with probability zero if you "spin" the disk randomly, and you decide to round one arrow clockwise and the other counterclockwise, you can get either a diminished or a suspended triad. But such cases should be disallowed, or else you can conjure up some funny results of 1st-order ME, such as a 12-out-of-22 scale with intervals 1-2-2-2-2-1-2-2-2-2-2-2, which I'm sure you'd disallow as well, although there are other reasons for liking this scale!). Your procedure requires two clocks, one with 12 hours and one with 7 hours. You put the 7-arrow disk on the 12-hour clock, the 3-arrow disk on the 7-hour clock, and then translate from one clock to the other. So your procedure requires more construction and more abstraction (the translation process) than mine. Anyway, stepping up to 22->12->7, you'll see that my definition does not require the additional rule that there be only one tritone, which apparently you derived from Jay Rahn (what possible justification is there for this rule?), in order to isolate the four gramas. I view this as little more than a curiosity since, if you buy this theory as an explanation, you still have to explain where the three numbers 22, 12, and 7 come from. But I thought this would be interesting to you. I have a paper on 22-tone equal temperament in the upcoming Xenharmonikon. I have two new scales (one ME, one not) that have more of the important properties of the diatonic scale than any other scale ever invented, save perhaps the pentatonic scale. A 7-limit (tetratic) TONAL system is the goal and this goal is attained. I also discuss plausible derivations of the 22-tone quantification of the Indian system. Finally, I propose an unequal 22-tone tuning in which several transpositions of my new scales as well as well-tuned (close to JI) Indian scales are available. I'm having my guitar refretted to 22-equal . . . I look forward to hearing from you. -Paul E. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Thu, 27 Mar 1997 23:04 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA19907; Thu, 27 Mar 1997 23:04:15 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA19903 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id OAA08326; Thu, 27 Mar 1997 14:02:38 -0800 Date: Thu, 27 Mar 1997 14:02:38 -0800 Message-Id: <009B1E8ACE1DCBED.8F69@vbv40.ezh.nl> Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu