A rank three temperament is a regular temperament with three generators. If one of the generators can be an octave, it is called a planar temperament, though the word is sometimes applied to any rank three temperament. The octave classes of notes of a planar temperament can be embedded in a plane as a lattice, hence the name. The most elegant way to put a Euclidean metric, and hence a lattice structure, on the pitch classes of a planar temperament is to orthogonally project onto the subspace perpendicular to the space determined by 2 and the commas of the temperament. To do this we need a Euclidean metric on the space in which p-limit intervals reside as a lattice, and the most expeditious and theoretically justifiable choice of such a metric seems to be Euclidean interval space.
7-limit marvel temperament is defined by tempering out a single comma, 225/224. If we convert that to a weighted monzo m = |-5 3.17 4.64 -2.81> and call the weighted monzo |1 0 0 0> for 2 "t", then the two-dimensional subspace perpendicular in the four-dimensional 7-limit Euclidean interval space is the space onto which we propose to orthogonally project all 7-limit intervals. One way to do this is by forming a 2x4 matrix U = [t, m]. If U` denotes the Moore-Penrose pseudoinverse of U, then letting Q = U`U take P = I - Q, where I is the identity matrix. P is the projection map from weighted monzos onto the two-dimensional lattice of tempered pitch classes. We have that mP and tP are the zero vector |0 0 0 0> representing the unison pitch class, which is to say octaves, and other intervals are mapped elsewhere. We find in this way that the lattice point closest to the origin is the secor, 16/15 and 15/14, and the second closest independent point the fifth (or alternatively, fourth). The secor and the fifth give a Minkowski basis for the lattice, but we could also use the major third and fifth as a basis. The secor and fifth are at an angle of 106.96, and the major third is angled 129.84 to the fifth.
If we list 2 first in the list of commas, the matrix P for any planar temperament will always have a first row and first column with coefficients of 0. We may also change coordinates for P, by monzo-weighting the columns of P, which is to say, scalar multiplying the successive rows by log2(q) for each of the primes q up to p, which allows us to project unweighted monzos without first transforming coordinates.
Rank three but not planar
An earlier version of this article may be found at http://lumma.org/tuning/gws/planar.htm