Block recurrences

I was thinking of Dan's question, and something occurred to me which
is interesting in its own right, as well as for this business, namely
that the blocks themselves have recurrence relationships.

If we start from the JI diatonic scale, with steps of size 16/15,
10/9 and 9/8, we can write the steps as a matrix and invert it:

[ 4 -1 -1]^(-1) [2 2 3]
[ 1 -2 1] = [3 3 5]
[-3 2 0] [4 5 7]

Applying the Tribonacci recurrence to the columns of the inverted
matrix gives:

2 2 3 7 12 22 ...
3 3 5 11 19 35 ...
4 5 7 16 28 51 ...

The starting matrix is unimodular, and the property is preserved by
the transformation, which can be viewed as multiplication by a
unimodular matrix. Hence each of the 3x3 matrices we get is
unimodular, and each therefore defines a block. We have as inverse
matricies ones which represent <16/15,10/9,9/8>,
<25/24/135/128,16/15>, <81/80,128/125,25/24> ... and so forth. The
rule to go from one to the next is <r1,r2,r3>--><r2/r1,r3/r1,r1>,
which is the Tribonacci transpose operation. We get in this way 5-
limit blocks with 7,12,22,41 ... notes to the octave, which
approximate to the Tribonacci recurrence scales we get by starting
from r1=1/d, r2=(t^2-t)/d, r3=t/d where d=2+2(t^2-t)+3t and applying
the same rule.

This can be regarded as a generalization of what we would get by
inverting the Pythagorean pentatonic intervals of 256/243 and 9/8,
obtaining

[2 5]
[3 8]

and extending this to

2 5 7 12 19 31 50 ...
3 8 11 19 30 49 79 ...

The Pythagorean scales are turned into meantone versions by using the
meantone 3 of (8 phi + 3)/(5 phi + 2) = (19 - phi)/11.

One can also attempt a generalization of the basis change from two
vals to octave plus generator, which might be from three vals to 2,3
and generator, but this no longer is canonical, and the above seems
more interesting.