[tuning-math] A chord analog to Fokker blocks
- From: Gene W Smith <genewardsmith@xxxxxxxx>
- To: tuning-math@xxxxxxxxxxxxxxx
- Date: Thu, 11 Jul 2002 20:52:46 -0700
As I reported in a revious article, the 7-limit tetrads form a cubic
lattice. I haven't gotten much feedback on this stuff, and wonder if this
is well known, not known, or somewhere in between.) By identifying the
major tetrad with root q, q a 7-limit octave equivalence class, with q
itself, we may represent the note-lattice of classes as a sublattice of
the cubic lattice of chords. The basis is [0 1 1] representing 3/2, [0 1
0] 5/4, and [1 1 0] 7/4, so that 3^a 5^b 7^c is represented by
[b+c,a+c,a+b]. In this form, the *usual* Euclidean metric applies to the
note lattice.
Using this representation, we may define a block in a way entirely
analogous to note-class blocks. If for instance we take
<9/8, 15/14, 25/24>, the TM-reduced basis for the kernel of h4, we obtain
upon transforming to the cubic lattice coordinates <[0 2 2], [0 0 2], [2
-1 -1]>. Taking the adjoint matrix M of the matrix (of determinant +-8 =
2*4) defined by these as rows, we may construct a corresponding block by
requiring that if [p q r] = [a b c]M, then -4<p<5, -4<q<5, -4<r<5. This
gives us the following set of eight (= 2*4) chords:
[0, 0, 0], [0, 0, 1], [0, 1, 1], [0, 1, 2], [1, -1, 0], [1, -1, 1], [1,
0, 1], [1, 0, 2]
The notes of these give the following scale:
[1, 25/24, 15/14, 35/32, 9/8, 75/64, 5/4, 21/16, 75/56, 45/32, 35/24,
3/2, 25/16, 45/28, 5/3, 7/4, 25/14, 15/8]
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