An interesting class of notations

Recall that "jacks", the ratios between adjacent elements in the half
(octave) of the Farey sequence from 1/2 to 1, plus those derived from
approaching 1/2 from below, began as superparticular ratios, or
epimorios, whose denomiators are either triangular or square numbers.
If t(n) = n(n+1)/2 is the nth triangular number, we may denote these
by T(n) = t(n)/(t(n)-1) = and S(n) = n^2/(n^2-1). We then have

T(n) = S(n)S(n+1),

S(n) = T(2n-1)T(2n),

as we may verify by a simple calculation.

Let us began from T(2)S(2) = (3/2)(4/3) = 2.

If we take the ratio between them, 9/8, and add to it 4/3 then
anything which is a product of powers of T(2) and S(2) (which is the
full 3-limit) will likewise be a product of powers of 4/3 and 9/8;
this is therefore a note-basis for a notation for the 3-limit. If we
write the matrix as

[ 2 -1]
B(1) = [-3 2]

then the inverse matrix

[2 1]
N(1) = [3 2]

gives us the notation itself.

We now apply the splitting rule for the larger element, T(2), and get
T(2) = S(2)S(3) = (4/3)(9/8). If we take the interval from 9/8 to 4/3
we have 32/27, and we get [9/8, 32/27] or

[-3 2] [3 2]
B(2) = [5 -3], B(2)^(-1) = N(2) = [5 3]

Continuing on in this way we get <T(3),T(4),S(4)> leading to
[10/9,16/15,81/80], so that

[ 1 -2 1]
B(3) = [ 4 -1 -1], [ 5 2 3]
[-4 4 -1] N(3) = [ 8 3 5]
[12 4 7]

Since (16/15)^5(81/80)^5 < 2, we can compute a pentatonic scale whose
nth note is given by
(10/9)^n * (16/15)^ceil(-1/2+2n/5) * (81/80)^ceil(-1/2+3n/5), giving
us 1-9/8-4/3-3/2-16/9-(2) as our PB scale.

Continuing on in this way, we get <S(3),T(4),S(4)> leading to

[ 4 -1 -1] [ 7 5 3]
B(4) = [-3 -1 2] N(4) = [11 8 5]
[-4 4 -1] [16 12 7]

and scale 1-10/9-4/3-3/2-5/3-9/5-(2); then <T(4),T(5),S(4),T(6)> and

[-2 1 -1 1] [10 2 7 5]
B(5) = [ 2 -3 0 1] N(5) = [16 3 11 8]
[ 6 -2 0 -1] [23 5 16 12]
[-5 2 2 -1] [28 6 19 14]

with scale 1-16/15-9/8-56/45-4/3-7/5-3/2-8/5-16/9-28/15-(2), then
<T(5),S(4),T(6),S(5)> and

[-3 -1 2 0] [12 7 10 3]
B(6) = [ 6 -2 0 -1] N(6) = [19 11 16 5]
[ 1 2 -3 1] [28 16 23 7]
[-5 2 2 -1] [34 19 28 8]

with scale 1-16/15-9/8-56/45-4/3-3/2-8/5-16/9-28/15-(2).

People may not think much of these PBs, but there is actually a lot
of flexibility in these ratios and we could even build scales with
only epimoric steps out of them for those who are fans of that sort
of thing. We do finally run into two factors which make the process
no longer automatic--36/35, which is simultaneously T(8) and S(6),
and so which breaks down both as S(9)S(10) and as T(11)T(12), and
50/49, the first jack appearing with a denominator which is neither
triangular nor square.

--- In tuning-math@y..., genewardsmith@j... wrote:

> [-3 -1 2 0] [12 7 10 3]
> B(6) = [ 6 -2 0 -1] N(6) = [19 11 16 5]
> [ 1 2 -3 1] [28 16 23 7]
> [-5 2 2 -1] [34 19 28 8]

I wrote the same scale down twice, the one which goes with the above
notation is 1-16/15-28/25-25/21-5/4-4/3-7/5-3/2-8/5-5/3-16/9-15/8-(2).