source file: m1429.txt Date: Wed, 27 May 1998 13:37:30 -0400 Subject: 4D lattices From: "Paul H. Erlich" Paul Hahn wrote, > 10/9 --- 5/3 --- 5/4 > / \ / \10/7 / \ > / \ / \ / \ > /14/9 \ / 7/6 \ / 7/4 \ >16/9 --- 4/3 --- 1/1 --- 3/2 --- 9/8 > \ 8/7 / \12/7 / \ 9/7 / > \ / \ / \ / > \ / 7/5 \ / \ / > 8/5 --- 6/5 --- 9/5 > >This is a projection of part of the 3D 7-limit lattice into a plane. >The triangles are in the 3-5 plane, and the ratios inside the triads are >either one layer above or below depending on whether they're inside >major or minor triads. This shows all pitches separated from the 1/1 by >intervals I consider consonant or primary within the 9-limit; however, >without drawing lines which intersect and become confusing I can't >draw lines for all those intervals, even when I'm not restricted to >ASCII art. Several of the 9-limit intervals look like secondaries >instead of primaries when represented this way. In 4 dimensions it >would look like this: > > 5/3 --- 5/4 > / \10/7 / \ > 10/9 / \ / \ 3/2 --- 9/8 > / \ / 7/6 \ / 7/4 \ \ 9/7 / > / \ 4/3 --- 1/1 --- 3/2 \ / > /14/9 \ \ 8/7 / \12/7 / \ / >16/9 --- 4/3 \ / \ / 9/5 > \ / 7/5 \ / > 8/5 --- 6/5 This pitch set is just the 9-limit Partch diamond, by the way. There are better ways of visualizing 4- and higher dimensional lattices, though, but you'll have to use your imagination since ASCII isn't up to the task. John Chalmers sent me some lattice diagrams of pitch sets where there are four axes representing the prime numbers 3, 5, 7, and 11. The diagrams were developed by Erv Wilson. Essentially, these are projections of 4-d space onto two dimensions along a very special direction. The 3-d diagrams above (which are also used by Wilson) show the basic tetrad with three pitches around an equilateral triangle and the fourth pitch in the center. The fact that its six intervals are represented by lines in six different directions helps one construct fairly complex structures without pitches overlapping in the two-dimensional projection. However, three steps of 7/4 (343/256) will lead to a pitch at the same 2-d location as one step of 3/2 and one step of 5/4 (15/8). So certain structures, like Wilson's stellated hexany, are better represented with slightly irregular triangles. The 4-d diagrams show the basic pentad with the five pitches around an equilateral pentagon. Its ten intervals are represented by lines in only five different directions. Complex structures will have many overlapping lines, and it will become difficult to see at a glance which pitches are really connected and which just happen to coincide with a pre-existing line. However, since the diagonal of an equilateral pentagon is the golden ratio times the side, new pitches along a given line will always divide the distance between existing pitches on that line in the golden ratio. The golden ratio is of course the only ratio that has this iterative property. So the magic of the pentagon ensures that one can construct very complex lattices in 4-d without worrying about the pitches overlapping in 2-d. If one doesn't want the lines to overlap, one can use slightly irregular pentagons. When John sent me the diagrams, I started drawing my own to determine whether the 4-d CPS scales fit together to tile 4-d space. They do. I had a very intersting e-mail exchange with John and/or the Tuning list at that time. Unfortunately, my e-mail rcords from 3/96 to 6/96, which includes that stuff, got deleted. For 5-d, Wilson uses an equilateral pentagon with one pitch in the center . . .