source file: mills2.txt Date: Tue, 26 Nov 1996 17:10:05 -0800 Subject: Kraehenbuehl & Schmidt From: PAULE 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. . . ! Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Wed, 27 Nov 1996 05:20 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA09163; Wed, 27 Nov 1996 05:22:01 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA09008 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id UAA15436; Tue, 26 Nov 1996 20:21:59 -0800 Date: Tue, 26 Nov 1996 20:21:59 -0800 Message-Id: <199611270418.UAA29425@sunatg1> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu