More definitions

Let N be a note group. The dual group to N, N` = Hom(N, Z) had
elements I called "ets". I think a better name would be *vals*, since
these are generalized valuations. An et is a val which is close to
being the real valuation, or valuation at infinity, on some p-limit
group N_p. The dual group of a note group is its *val group*, it is a
finitely generated free group isomorphic (non-canonically) to its
dual. The dual of a val group is its note group. If N_p is the
p-limit note group we will call the dual group to this, N_p`, the
p-limit val group.

Let p be a prime and N be a note group representing (exactly or
approximately) notes in the p-limit, and let N_p` be the p-limit val
group. If the number of primes up to p is n, then N has rank k <= n.
Suppose {g_1, g_2, ... g_k} are k vals in N_p` which generate a
vector space of dimension k and such that any Q-linear combination,
(that is, any e1*g1 + ... + ek*gk where the coefficients are rational
numbers) which belongs to N_p` is a Z-linear combination (that is,
the coefficients are integers.) Then we define the k-tuple of vals
[g_1, g_2, ..., g_k] to be a *notation* for N. The elements of N are
thereby given musical meaning in a different way than by giving a
tuning for N; we can give N a tuning, a notation, or both in order to
make it musically meaningful.

If q is a rational number which when factored into primes has p^e as
the factor corresponding to the prime p, then a standard definition
of number theory defines the p-adic valuation v_p by

v_p(q) = e.

If we apply this to N_p, where 2^e2 * 3^e3 * 5^e5 ... corresponds to
[e2, e3, e5, ...] then v_2([e2, e3, e5, ...] = e2,
v_3([e2, e3, e5, ...] = e3 and so forth. These v_p are therefore
vals, and it seems reasonable to use this same standard notation. We
therefore obtain a notation for N_p which consists of
[v_2, v_3, ..., v_p], which as a matrix is the n by n identity
matrix. This notation is the identity notation. In general, if the
notation consists of k vals in an N_p` of rank n (meaning there are n
primes <= p) then it may be written explicitly as a matrix with n
rows and k columns.

To take a more interesting example, suppose we define vals g_2, g_3
in N_5` by

g_2([a, b, c]) = a-4c,
g_3([a, b, c]) = b+4c.

If u is a note in N_p, then g_2(u) defines how many octaves and g_3
(u) how many twelfths in a meantone system we require to get to the
note associated to u; [g_2, g_3] is therefore a notation for the
meantone note group. As a matrix, this notation is

[ 1 0]
[ 0 1]
[-4 4].

We get a different notation for the same group from [h_7, h_12],
which tells us how many steps respectively in the 7-et and in the
12-et the note u is represented by; as a matrix this notation is

[ 7 12]
[11 19]
[16 28].

The matrix

[ 7 12]
T = [11 19]

transforms a note in the first notation to the same note in the
second notation; the fact that the matrix T is unimodular (that is,
that |det(T)| = 1) shows that T is inevitable and the two notations
define the same meantone note group. The inverse matrix

[ 19 -12]
T^(-1) = [-11 7]

transforms something in the second notation back to the first. Hence,
for instance if [1, 2] is one step in the 7-et and two steps in the
12-et, [1, 2] T^(-1) = [-3, 2], which corresponds to 9/8 in a
meantone system (so 10/9 and 9/8 are represented by the same note)--
in other words, a full tone. A tone (in this sense!) is, as we might
expect, [1, 2] in the [h_7, h_12] notation.