Kraehenbuehl & Schmidt

To be fair to these guys, let me summarize their theory. Kraehenbuehl and
Schmidt, inspired by Yasser, proposed an evolution proceeding from 5 to 7 to
12 to 22 to 41 tones per octave, corresponding with a harmonic limit that
increased from 3 to 5 to 7 to 11. They defined all the tones with just
ratios and defined the harmonic limit as the highest prime number used in
these ratios (thus they skipped 9). The evolution proceeds by interpolating
one new note into each of the larger steps of the previous system, first
using only the older harmonic limit. Then, once the newer system begins to
be used in its entirety, the next harmonic limit begins to cause new
"inflections" in the tuning of the older system to occur. Once the newer
harmonic limit takes effect, the process begins again. They saw jazz (this
was 1962) as approaching a 12-tone totality, with lots of 7-limit
inflections. There may be some truth to this. If they were right, we may
expect jazz musicians to begin to realize that the inflected 12-tone scale
has 10 "holes" in it, which they will begin to fill in. The problem with
this is, if you look at the table in my previous post, you see that two
different notes of the hyperchromatic scale are represented by ratios (4/3
and 27/20, for example) that are separated by a syntonic comma (81/80).
Standard jazz harmonic formulae are still 5-limit in origin and so assume
that the syntonic comma occurs only as inflections of a single note. So
evidently this linear type of "evolution" is not going to get us to a
22-tone system. We need to "start from scratch," or at least "back up a
step" and then "leap forwards," in order to find a good 7-limit/22-tone
system.

I've been dropping hints here in case some lurker has been following my
posts. That was the biggest one yet. Tomorrow is my last day reading this
list for a while, and the day I send out my paper for publication. I
discovered the 7-limit/22-tone system that is the topic of the paper back in
'91, having only read Helmholtz, Yasser, and Partch. Then I got really
interested and started reading everything. Soon, thanks to John Starrett,
I'll have a nice 22-tone guitar, and all this theorizing will be supplanted
by an even more enjoyable activity. Anyway, I hope someone out there chooses
to contemplate these ideas over the next month, as it's always more exciting
to figure something out for yourself than to be told outright. Happy
Thanksgiving/Hanukkah/Christmas/Kwanzaa/New Year/etc. . . !


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