I posted a while back on the geometric relations between 9-limit tetrads, using a system of coordinates defined by the vector sum of the exponents, or in other words, the product of the notes of the chord in exponent-vector form. This form of the coordinates allows one to find the vector form easily from the JI form, is generalizable to any p-limit, and works in the 5-limit also. However, there is a coordinate transformation which is special to the 7-limit which is more or less essential to understanding the how the 7-limit chord geometry works. The coordinates I've used may be related to the exponent coordinates of the notes by the observation that it is simply the centroid of the notes, times a scalar product of four. Since the notes themselves have a symmetric Euclidean metric given by the quadratic form Q(3^a 5^b 7^c) = a^2+b^2+c^2+ab+ac+bc, the chords inherit this metric. If we look at the interval adjacency vectors for major/minor, minor/major connections, namely [-2,2,2],[2,-2,2],[2,2,-2],[2,-2,-2],[-2,2,-2],[-2,-2,2] we find that these are either of opposite sign or are orthogonal. We may therefore select three of these to give us an orthogonal basis; if we also shift the coordinate center to the major tetrad [1,5/4,3/2,7/4] we may define a coordinate transformation which takes [a,b,c] to [(b+c-2)/4, (a+c-2)/4, (a+b-2)/4], the inverse of which takes [a,b,c] to [-2a+2b+2c+1, 2a-2b+2c+1, 2a+2b-2c+1]. This now makes the major and minor tetrads represented as the points of a cubic lattice, a nice feature which is unique to the 7-limit (the 5-limit gives a hexagonal tiling, but not a lattice; the lattice appears because of a special property of the 7-limit note lattice, which belongs to two different classes of lattices at once.) The pumps are particularly easy to find and understand in this coordinate system; for example [3 4 1][3 3 1][3 2 1][2 2 1][2 1 1][1 1 1][0 1 1][0 0 0] is a 1029/1024 pump I gave previously; in this form it is easy to see how we can obtain other such pumps. We again can represent subminor and supermajor tetrads, which transform to non-lattice points. In particular, 1-7/6-3/2-5/3 is represented by [0 -1/2 -1/2] and 1-9/7-3/2-9/5 by [-1 1/2 1/2] We can also represent complete 9-limit harmonies, basing ourselves on the major quintad 1-9/8-5/4-3/2-7/4 and its minor quintad inverse; the same coordinates serve for these. Formerly, given a 4-et value (that is, the value h4(q) for h4 = [4,6,9,11], which reduces mod 4 to [0 2 1 3], a complete set of representatives mod 4) and a chord, we could reconstruct the note. Now we may similarly use a 5-et value, based on h5 = [5,8,12,14], and because h5(1)-h5(9/8)-h5(5/4)-h5(3/2)-h5(7/4) is 0-1-2-3-4, a complete set of mod 5 residues, we can again recover the note from a chord and h5 value. (Similar comments apply to triadic harmony using h3 and 13-limit harmony using h7.) In terms of these chords, there are 12 interval adjacencies from major to minor, and 12 from minor to major. ------------------------ Yahoo! Groups Sponsor ---------------------~--> Free $5 Love Reading Risk Free! http://us.click.yahoo.com/TPvn8A/PfREAA/Ey.GAA/wHYolB/TM ---------------------------------------------------------------------~-> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxxxxxx Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/ ____________________________________________________________ To learn how to configure this list via e-mail (subscribe, unsubscribe, etc.), send a message to listar@xxxxxxxxxxxxx with the subject line "info tuning-math". Or visit the website: < //www.freelists.org/list/tuning-math > .