An example

Let's put the definitions I just gave to use by looking at how the
Blackjack scales might be derived. We may start from either ets or
commas, but ets are easier to find and so it probably makes the most
sense to began there. If we look at 7-limit ets, we find
10,12,15,19,22,27,31,41,68,72,99 as ets h_n with n between 10 and 100
and cons(7,n)<1. If we pick h_{31} and h_{41}, we generate a rank 2
group M which also contains h_{10} = h_{41}-h_{31},
h_{72}=h_{41}+h_{31}, etc. Then K=null(M) is generated by the notes
[-5, 2, 2, -1] and [-5,-1,-2,4], which correspond to the tones
225/224 and 2401/2400.

If we look for ets contained in M we find h_{10}, h_{11}, h_{20},
h_{21}, ... and so forth. If we select h_{21}, we find ker(h_{21}) is
generated by [2,2,-1,-1] (corresponding to 35/35) and K. If we make
[2,2,-1,-1] a chroma and {[-5,2,2,-1], [-5,-1,-2,4]} commas then K is
the commatic kernel and L=N_7/K is a note group of rank 2.

If we choose a tuning for L in a reasonable way we now should have a
good tone system for the 7-limit. "Reasonable" might for instance
mean tuning octaves pure and picking a good value for the remaining
generator. A particularly practical form of "reasonable" is to tune
another et in M; thus we could have 21 notes out of 31 with 36/35 one
step, 21 notes out of 41 with 36/35 two steps, or 21 notes out of 72
with 36/35 three steps.

Suppose r^3-r-1=0, and we have steps of size A, B, C in sizes
proportional to 1:r:r^2. Then we can preserve this by sending
A->C-A, B->A, C->B and get a larger scale.