source file: mills2.txt Date: Thu, 22 May 1997 04:54:36 +0200 Subject: ET, JT, etc. From: James Kukula If by ET we mean a scale made of frequencies 1, x, x^2, x^3, ... - where the ^ means exponentiation or powers, and if by JT we mean a scale composed of rational number freqencies, then ET and JT are not mutually exclusive. Just pick x rational. Of course the only rational where x^ncan happen is x So any nontrivial scale that's both ET and JT will miss the perfect octave. Still, it might be fun territory to explore. Such an ET scale is not the only kind of scale with "modulation symmetry", where every note has the same intervals available. One can move to higher dimensions, for example allowing x^n y^m for every pair of integers (m,n) for some fixed pair of generating intervals x and y. Surely 2^n 3^m deserves to be called a classic Pythagorean scale? If x and y are incommensurable, i.e. x^n y^m 1 only for mand n then such a scale will have infinite notes in any pitch range. This is not really the same as opening up to the entire continuum of pitches. Cantor's set theory shows that integer pairs (m,n) are still countable, but the continuum is not. Discussions have drifted through here on two dimensional keyboards. Given the electronic generation of tones, the concept of a keyboard can be opened way up. I've seen a little of this around. For example, some kind of body suit could follow the state of flexion/extension/rotation of some set of joints in the body. This raw data would determine a point in some quite high dimensional space, which could then be mapped down to some smaller dimension before getting read as a pitch. At a very elementary level, one could have two surfaces covered with keys. The right hand picks a key on the right keyboard, the left picks a key on the left keyboard. This determines a point in a four dimensional space. One might thus provide all pitches 2^n 3^m 5^p 7^q. Jim Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Thu, 22 May 1997 05:16 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA07426; Thu, 22 May 1997 05:16:17 +0200 Date: Thu, 22 May 1997 05:16:17 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA07429 Received: (qmail 18274 invoked from network); 22 May 1997 03:16:14 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 22 May 1997 03:16:14 -0000 Message-Id: <199705220309.UAA22360@sunatg3> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu