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RE: Rowell article (Paul E)

🔗Manuel.Op.de.Coul@ezh.nl (Manuel Op de Coul)

3/27/1997 1:59:12 PM
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.

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