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.