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).