source file: mills2.txt Date: Fri, 31 Jan 1997 10:32:37 -0800 Subject: Quadratic Corners From: John Chalmers Franck: Last year I started writing some simple BASIC programs to factor and plot scales on various types of tonal lattices as I needed to make some new graphics for some talks I intended to give, and I am a slow and not very competent draftsman. The lattices included both Cartesian types (powers of 3 on the horizontal, 5 on the vertical, 7 on a 30 degree diagonal, etc.) and polygonal ones similar to those invented by Ervin Wilson. Once I had the programs running on my old monochrome Mac, I started generating scales which had interesting visual appearances. A Quadratic Corner is an incomplete Euler-Fokker genus consisting for the factors 3.5.7 of 1 3 3^2 5 5^2 3x5 7 7^2 3x7 and 5x7. One can add the inversion of each of these terms and reduce the whole to a common octave. The concept may be extended to include higher prime factors, such as 11 and 13. In the 3.5.7 rectilinear lattice, this collection has a nice spikey appearance as my program connects only nearest neighbors (tones differing by plus or minus 1 step along any axis). Some versions of the polygonal lattice plotters allowed diagonally related notes to be connected as in the familiar triangular lattice where major and minor triads appear as equilateral triangles. I make no claims for the musical utility of these scales nor of the related Cubic Corners, but visually they are interesting in some representations. --John Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 31 Jan 1997 20:57 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA21282; Fri, 31 Jan 1997 20:57:11 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA21315 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id LAA20376; Fri, 31 Jan 1997 11:54:57 -0800 Date: Fri, 31 Jan 1997 11:54:57 -0800 Message-Id: <32F24E14.142F@sprynet.com> Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu