source file: mills2.txt Date: Thu, 3 Jul 1997 23:13:48 +0200 Subject: Aliquot parts, lattices, etc. From: John Chalmers re "9-Limit" I believe Ervin Wilson was the first to construct lattices in which odd composite factors had their own axes back in the late 60's when he invented or discovered the CPS and similar types of scales. As far as I know, he was also the first to use polygonal lattices higher than the triangular (the tetrahedral-octahedral lattice Paul Erlich mentioned may be drawn as a centered triangular lattice). Personally, I find both representations useful. As for "aliquot parts," Marion's usage is clear enough, but I prefer to use the term only in reference to the linear division of a string into equal parts (what aliquot parts mean) to generate utonal scales such as Schlesinger's harmoniai. Thus the division of 1/2 of a string into 7 aliquot parts (or the whole into 14) generates KS's Mixolydian harmonia (14 13 12 11 10 9 8 7 or 1/1 14/13 7/6 14/11 7/5 14/9 7/4 2/1). In the logarithmic domain (as opposed to the "acoustic," the result of dividing an interval into aliquot parts is an equal temperament, unless the interval is a product of equal just intervals (9/4 into 3/2 x 3/2, etc), in which case one has an equal division, but not a temperament. Geometric division or equal division would seem to cover these concepts adequately. I usually use the term "submultiple" to refer to the notes other than the "Guiding Tone" (Fokker's term) and the 1/1 of an Euler-Fokker genus musicum. The logic is that each note is both a multiple of the 1/1 and a divisor of the guiding tone, the whole genus being simultaneously an otonality of 1/1 and a utonality of the GT (at the GT limit, though it usually is not complete). Since E-F genera are classified by their GT's, the tones are conveniently described as submultiples of them. Marion's LCM when expressed as a product of factors is equivalent to the GT, though one may disagree about the necessity of explicitely writing the powers of 2 (Euler did, as I recall, Fokker usually did not). --John Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Thu, 3 Jul 1997 23:14 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA04377; Thu, 3 Jul 1997 23:15:08 +0200 Date: Thu, 3 Jul 1997 23:15:08 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA04407 Received: (qmail 26791 invoked from network); 3 Jul 1997 21:14:59 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 3 Jul 1997 21:14:59 -0000 Message-Id: <34212fc7.258127516@kcbbs.gen.nz> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu