Plugging the holes in the 9-limit tonality diamond

Before we attempt to plug them we should first find them. The 9-limit diamond has a 5-limit diamond sitting inside it: {1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3}. In a symmetrical 5-limit pitch-class lattice, this is the hexagon surrounding the central 1/1. Off to one side we find {9/8, 7/6, 5/4, 7/5, 3/2, 14/9}. This is approximately a hollowed-out 5-limit diamond, meaning minus the central 1/1, transposed down a secor, in the sense that half the notes are transposed 15/14 down and half 16/15 down. Similarly, {9/7, 4/3, 10/7. 8/5, 12/7, 16/9} is an approximate hollow 5-limit diamond, or hollow hexagon, transposed up a secor.

Hence, we can fill in the holes by adding a secor (for which we can use sqrt(8/7)) up and down to the diamond, obtaining the following:

! diamond9plus.scl
9-limit tonality diamond extended with two secors
21
!
115.587
10/9
9/8
8/7
7/6
6/5
5/4
9/7
4/3
7/5
10/7
3/2
14/9
8/5
5/3
12/7
7/4
16/9
9/5
1084.413
2/1

Alternatively, we can use marvel temperament, tempering out 225/224 (and not so incidentally, 385/384):

! diamond9plus-marvel.scl
Marvel tempering of diamond9 plus secors up and down
! Tuning is 1/6 septimal kleisma flat fifth
! 1/3 septimal kleisma flat major third
! Minimax over the 9-limit diamond
21
!
115.58705
182.40371
201.33949
231.17409
268.15616
316.92654
383.74321
432.51359
499.33025
585.08270
614.91730
700.66975
767.48641
816.25679
883.07346
931.84384
968.82591
998.66051
1017.59629
1084.41295
1200.00000

Aside from all the 9-limit harmony, there are distinct 11-limit implications in these scales.