Generators for temperaments

I was discussing this a little on the tuning group:

/tuning/topicId_59689.html#60228

I didn't want to go into it more deeply there, but in case it wasn't
clear, here is a more mathematical look.

Paul mentioned the use of the generator pairs 10;9 and 16;15 or 9;8
and 16;15 for meantone, where the semicolon indicates we are using
tempered intervals. By adding the comma 81/80 to the mix we can shift
to pure intervals, and consider triples such as [10/9 16/15 81/80] or
[9/8 16/15 81/80]. Taking the matrix of monzos for these and
inverting, we get matrices with columns of vals.

[9/8 16/15 81/80]^(-1) = [<5 8 12|, <2 3 4|, -<2 3 5|]

[10/9 16/15 81/80]^(-1) = [<5 8 12|, <2 3 4|, <3 5 7|]

[16/15 25/24 81/80]^(-1) = [<7 11 16|, <5 8 12|, <3 5 7|]

[25/24 128/125 81/80]^(-1) = [<12 19 28|, <7 11 16|, <3 5 7|]

You might compare the above to octave-generator:

[2 3/2 81/80]^(-1) = [<1 1 0|, <0 1 4|, <0 0 -1|]

You can also look at higher prime limits like this:

[10/9 16/15 81/80 126/125]^(-1) = [<5 8 12 15|, <2 3 4 4|, <3 5 7 8|,
<0 0 0 1|]

And of course other temperaments:

[16/15 135/128 225/224 1029/1024]^(-1) = [<11 17 26 31|, -<1 1 3 3|,
<6 9 15 17|, <2 3 5 6|]

etc etc.