ET, JT, etc.

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