The Hermite Basis
A regular p-limit temperament of
rank n may be defined
in terms of n p-limit vals, which have the property that
the n-fold wedge product is not identically zero, and the
greatest common denominator of the coefficients is one. Considering these n vals to be n columns of a matrix, we
may reduce to Hermite normal form by means of column operations with coefficients which are integers. We define the Hermite basis
for the temperament as consisting of a list of n p-limit intervals, [b1, ..., bn], each greater than one and Tenney-Minkowski reduced with respect to
the commas of the temperament, having the further property that for the ith val hi, corresponding to the ith column
of the Hermite normal form, hi(bi) = +-1, whereas if j is not equal to i, hi(bj) = 0. If the first basis interval,
b1, is 2, then the octave classes of the temperament form a rank n-1 free group, which can be embedded in an n-1
dimensional space as a lattice. Such a temperament may be described as flat, where a flat rank two temperament
is a linear temperament, a flat rank three temperament is planar, and so forth.
If we add to the end of this list the Tenney-Minkowski comma basis for the temperament, ordered in terms of increasing Tenney height, we obtain a list of pi(p) p-limit intervals, which has the property that the matrix of monzos derived from the list is a square integral unimodular matrix. Hence, the columns of the inverse matrix define how to express any p-limit interval in terms of powers of list elements; if the commas are retuned to vanish, the other list elements are now retuned to give the intervals of the temperament.
For example, for 7-limit meantone, the Hermite basis is [2, 3], and the TM basis is [81/80, 126/125]. If we put this together as [2, 3, 81/80, 126/125] and take the matrix of monzos, we get a 4 x 4 unimodular matrix; any 7-limit interval can be expressed in these terms. We see that meantone expresses the 7-limit by way of a tempered 3-limit, which we already knew, of course. For 11-limit miracle, the Hermite basis is [2, 15/14], and the TM basis is [225/224, 243/242, 385/384], and putting the two together gives a unimodular 5 x 5 matrix, which can express any 11-limit interval. The inverse pair of unimodular matrices taken together can be regarded as defining a basis change, or abstract notation, for the p-limit.
The Hermite basis is particularly interesting for rank greater than two, however. Below I give examples in the form of the Hermite basis for the 7-limit rank three temperaments defined by various significant 7-limit commas.
1029/1000 [2, 10/7, 5]
250/243 [2, 10/9, 7]
36/35 [2, 3, 5]
525/512 [2, 3, 5]
128/125 [5/4, 3, 7]
49/48 [2, 7/4, 5]
50/49 [7/5, 3, 5]
3125/3072 [2, 5/4, 7]
686/675 [2, 3, 15/14]
64/63 [2, 3, 5]
875/864 [2, 3, 5]
81/80 [2, 3, 7]
3125/3087 [2, 3, 25/21]
2430/2401 [2, 3, 18/7]
2048/2025 [45/32, 3, 7]
245/243 [2, 3, 9/7]
126/125 [2, 3, 5]
4000/3969 [2, 3, 63/40]
1728/1715 [2, 3, 12/7]
1029/1024 [2, 8/7, 5]
15625/15552 [2, 6/5, 7]
225/224 [2, 3, 5]
19683/19600 [2, 140/81, 5]
16875/16807 [2, 3, 7/5]
10976/10935 [2, 3, 27/14]
3136/3125 [2, 3, 56/25]
5120/5103 [2, 3, 5]
6144/6125 [2, 3, 35/32]
65625/65536 [2, 3, 5]
32805/32768 [2, 3, 7]
703125/702464 [2, 3, 75/56]
420175/419904 [2, 3, 648/245]
2401/2400 [2, 49/40, 10/7]
4375/4374 [2, 3, 5]
250047/250000 [63/50, 3, 5]
78125000/78121827 [2, 6250/5103, 5]