Planar Temperaments

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 start from a lattice of pitch classes for just intonation, and orthogonally project onto the subspace perpendicular to the space determined by the commas of the temperament. For instance, 7-limit just intonation has a symmetrical lattice structure and a 7-limit planar temperament is defined by a single comma. If u = |* a b c> is the octave class of this comma, then v = u/||u|| is the corresponding unit vector. Then if g1 and g2 are the two generators, j1 = g1 - (g1.v)v and j2 = g2 - (g2.v)v will be projected versions of the generators, and (j1.j1)a^2 + 2(j1.j2)ab + (j2.j2)b^2 the projected quadratic form defining the metric stucture of the planar temperament lattice. Here the dot product is defined by the bilinear form giving the metric structure. One good, and canonical, choice for generators are the generators found by using Hermite reduction with the proviso that if the generators so obtained are less than 1, we take their reciprocal.

The Hermite generators for marvel (225/224 planar) temperament are {2,3,5}, and the projected lattice strucuture is defined by the norm sqrt(11a^2-14ab+11b^2), where "a" is the exponent of 3 and "b" of five. The Hermite generators for breed (2401/2400 planar) temperament are {2,49/40,10/7}, and the projected lattice structure has 49/40 perpendicular to 10/7, so it has orthogonal axes, with a norm given by sqrt(11a^2+8b^2), where now "a" is the exponent of 49/40, and "b" the exponent of 10/7.