Block definition example

Let's do this for the classic example--the JI diatonic. If we denote
by (16/15, 25/24, 81/80) the matrix whose rows are the three
intervals expressed in prime factored notation, and by
[h7, h5, h3] the matrix whose columns are the 7, 5, and 3 ets, then
(16/15, 25/24, 81/80)^(-1) = [h7, h5, h3]. We need instead the
inverse of (2, 25/24, 81/80), and it is easily checked that
(2, 25/24, 81/80)^(-1) = [1/7 h7, h5 - 5/7 h7, h3 - 3/7 h7]; if g_i
is the ith column and e_j is the jth row, then g_i(e_j) = d_ij (the
Kronecker delta, meaning 1 if i=j and 0 if i<>j.) If therefore
[a,b,c] is a note in the [h7,h5,h3] notation, then

||[a,b,c]|| = 1/7 max( |a|, |7b - 5a|, |7c -3a|).

In this notation we have 16/15 = [1 0 0], 10/9 = 16/15 * 25/24 =
[1 1 0] and 9/8 = 10/9 * 81/80 = [1 1 1]. Using these scale steps we
see that the JI diatonic scale is

[0 0 0], [1 1 1], [2 2 1], [3 2 1], [4 3 2], [5 4 2], [6 5 3].

If we transform [a,b,c] to [a, 7b - 5a, 7c - 3a] we get

[0 0 0], [1 2 4], [2 4 1], [3 -1 -2], [4 1 2], [5 3 -1], [6 5 3].

We note that 5/3, represented by [5 3 -1] is closer to 1 than 15/8,
represented by [6 5 3], which may make Paul happy until he notices
how much closer 9/8 is to 1 than either. We also notice that the
biggest difference is between 15/8 and 4/3 or 1; we have
[6 5 3] - [3 -1 -2] = [3 6 5], so that the distance
d(15/8, 4/3) = 6/7, and [6 5 3] - [0 0 0] = [6 5 3] and
d(15/8, 1) = 6/7; the diameter of the diatonic scale is therefore
6/7, less than 1. To show it is a block under the definition we must
show it is maximal; this we may do by looking at the matter modulo 7.
The [a, 7b - 5a, 7c - 3a] transformed notation applied to our scale
steps gives us 16/15 = [1 -5 -3], 10/9 = 16/15 * 25/24 =
[1 -5 -3] + [0 7 0] = [1 2 -3], 9/8 = 10/9 * 81/80 =
[1 2 -3] + [0 0 7] = [1 2 4]. The three scale steps are therefore all
[1 2 4] mod 7, and so the scale is [n, 2n, 4n] mod 7. Hence we have
one representative mod 7 for each coordinate, and adding another note
will entail that at least one coordinate will differ by a least 7, so
that the diameter will be at least 1/7 7 = 1.

We might note also that if we consider instead meantone tempered
scales, we simply drop the last coordinate from the definition of the
norm, so that any just block leads to a meantone block.