Suppose ha and hb are two vals defining an icon [ha, hb] for a linear temperament. We define two corresponding vectors in R2 as follows:
let a = ha(2), b = hb(2), d = sqrt(a^2 - ab + b^2). Then we set u = [(2a - b)/2d, sqrt(3)b/2d] and v = [(2b - a)/2d,
-sqrt(3)a/2d]. We may check that the dot products of these two vectors are (u, u) = 1, (v, v) = 1, and (u, v) =
-1/2. They are therefore unit vectors 120 degrees apart. We also may easily check that a u + b v = [d, 0].
We now define a mapping from the p-limit for which the icon is defined to R2 by
B(q) = ha(q) u + hb(q) v
for any q in Np. The image under B of Np in R2 is an equilateral triangular lattice (A2 lattice) with the distance between nearby lattice points equal to one; every note of the linear temperament corresponds to a unique lattice point, and vice-versa. Since a u + b v = [d, 0], we have that B(2^n) = [n d, 0] and the octaves are all mapped along the x-axis. We call such a lattice a Bosanquet lattice. It may be regarded as a mathematical abstraction of a keyboard for use with the temperament in question; if we surround each lattice point with a circle or hexagon, we can see the keyboard design emerge.
The Bosanquet lattice, and hence the keyboard design, depend on the chosen icon, so we would like a way of finding reasonable candidates for the icons. One method is to make use of Dave Keenan's spreadsheet which was designed to produce a keyboard layout given the generator and period of a linear temperament. We will illustrate by using orwell as an example.
If we put the orwell generator of 19/84 = 271.4 cents into Dave's spreadsheet, we find 114 and 157 cents at the bottom of the chart next to 0, which we can choose as keyboard generators. (We could also choose 43 and 114.) We now select two equal temperament vals generating orwell up to our chosen prime limit; in this case that could be 22 and 31, and the 11-limit. We now find that [114, 157] * (22/1200) = [2.09, 2.88] and [114, 157]*(31/1200) = [2.945, 4.06]. Rounding, we see that [2, 3] are the generators in 22-et, and [3, 4] in 31-et. The matrix
is unimodular, inverting it gives us
[ 3 -2]
so that l4 = 3 h22 - 2 h31 = 3 [22, 35, 51, 62, 76] - 2 [31, 49, 72, 87, 107] = [4, 7, 9, 12, 14] and l5 = -4 h22 + 3 h31 = [5, 7, 12, 13, 17].
Below are links to jpgs of various Bosanquet lattices; the verticies are labeled either by an appropriate choice of equal temperament, or by p-limit intervals which map to the verticies under the Bosanquet mapping B.
31 equal labels
72 equal labels
37 equal labels
114 equal labels
[3, 6, 8]&[4, 7, 9] kleismic
53 equal labels
[4, 7, 9, 12, 14]&[5, 7, 12, 13, 17] orwell
84 equal labels