The porcupine-hemithirds complex

If we take a 7-note porcupine (250/243) MOS, we have a corresponding
5-limit block

1-10/9-6/5-4/3-40/27-8/5-16/9

with step sizes of 10/9 and 27/25, ending with 9/8. In terms of the
22-et, it is
3333334; whereas 37-et has it as 5555557. If we add another step,
we have instead 33333331 and 55555552 respectively; with the block
becoming 1-10/9-6/5-4/3-40/27-8/5-16/9-48/25.

6/5
1
8/5
4/3
10/9
16/9
40/27

is the 5-limit lattice, with 40/27~36/25 connecting the top and bottom of
the scale; if we go to the 8-note scale, we get the 48/25-6/5-36/25
triad, filling the connection out with a chord.

This 5-limit (or <2,3,5>-group) scale can be transformed to a
corresponding "no 3s"
<2,5,7>-group scale by means of 2-->2, 3-->20/7, 5-->32/7, which can be
extended to a complete 7-limit transformation by the addition of
7-->48/7. This transformation sends
250/243 to 3136/3125, and 5-limit porcupine to what I've called
hemithirds no-threes.
The map sends 10/9-->28/25, 27/25-->125/112, and the 7-note scale to

1-28/25-5/4-7/5-196/125-7/4-49/25

the addition of 48/25 is then sent to 35/16. In terms of the 31-et, which
does the hemithirds no-threes quite nicely, we now have 5555551 for the
7-note scale and
41555551 for the 8-note scale; the 37-et, which does both of these, has
it as
6666661 and 51666661 respectively.

Finally, we may also send 2-->2, 3-->10/3, 5-->14/3, 7-->32/3; this sends
250/243 to
(1/2)(3087/3125), and the 5-limit tempered by 250/243 to the <2,5/3,7/3>
group tempered by 3125/3087. The 45-et does a good job on this group and
temperament, and in terms of it we have steps of size 25/21 and 147/125
both being sent to 11, and
scales [10,1,10,1,11,11,1] and [10,1,10,1,10,1,11,1]. The amazing 37-et,
which covers
250/243, 3136/3125 and 3125/3087, and hence this entire transformational
complex,
has it as 8181991 and 81818191 respectively.