Bosanquet Lattices

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

[2 3]

[3 4]

is unimodular, inverting it gives us

[-4 3]

[ 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.

5&7 meantone

10&11 miracle

7&8 porcupine

10&12 diaschismic

[3, 6, 8]&[4, 7, 9] kleismic

[4, 7, 9, 12, 14]&[5, 7, 12, 13, 17] orwell