**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.`

` `