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.