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I think a scale of nine tones to the octave is just about perfect, so I might want to consider what 9-polyhexes do for me. There are 6572
9-polyhexes (see http://mathworld.wolfram.com/Polyhex.html ) so this is a big project. However, I'm most interested in polyhexes with a lot of internal nodes, corresponding to triads, so I start by looking at the paralleogram 9-polyhex.

Placing my polyhex on the 5-limit lattice of classes, I get the scale

Scale 0: 1--16/15--6/5--5/4--4/3--3/2--8/5--5/3--15/8

This is epimorphic from h9, and is a Fokker block from
[128/125, 135/128]. I may rotate it by the transformation
2->2, 3->10/3, 5->16/3, giving me

Scale 1: 1--10/9--6/5--5/4--4/3--3/2--8/5--5/3--9/5

This is *not* epimorphic, and hence not a Fokker block or (by my definition) any other kind of block, despite its block-like appearance. However, though its scale steps are a bit more irregular, it seems a perfectly fine scale, and can in fact also be gotten from the first scale by means of the major<->minor transformation. We therefore see that being a block is *not* invariant under
major<->minor.

Rotating again, we get

Scale 2: 1--25/24--6/5--5/4--4/3--3/2--8/5--5/3--28/25

This now is epimorphic from h9, and a Fokker block from
[128/125, 27/25].

We find no more scales by inversion or major<->minor transformation; this is the complete list of scales deriving from the paralleogram
9-polyhex. We might wonder, however, about the Fokker block
[135/128, 27/25] which would seem to go with the other two. This
is

Scale 4: 1--9/8--6/5--5/4--4/3--3/2--8/5--5/3--16/9

This gives us a different 9-polyhex, with a central line five hexes long surrounded by two two-hex wings. I could go on and look at the other scales belonging to this polyhex, but I wasn't sure the process terminates any time soon.

The first three scales have 16 5-limit intervals, and "Scale 4" 14; all of course are connected. This means that any mapping of these scales by a temperament will give us a 5-connected scale belonging to that temperament with at least 16 5-limit intervals. Below I show the number of 5,7,9 and 11-limit intervals for each of these scales for every "standard" equal temperament from 21 to 100 (starting at 21 insures that we have nine tones to the octave.) We can see from it evidence of interesting scales belonging to 81/80, 225/224, and so forth, which should be worth exploring.

[1, 16/15, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 15/8]
[1, 10/9, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 9/5]
[1, 25/24, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 48/25]
[1, 9/8, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 16/9]

[21, 27, 30, 36]
[17, 24, 27, 31]
[21, 27, 30, 36]
[15, 22, 25, 31]

[16, 24, 29, 30]
[17, 20, 27, 32]
[16, 19, 26, 29]
[14, 21, 27, 30]

[19, 24, 31, 31]
[16, 23, 28, 31]
[16, 21, 24, 27]
[19, 24, 31, 31]

[21, 21, 27, 27]
[20, 20, 27, 27]
[21, 21, 27, 27]
[20, 20, 27, 27]

[16, 21, 23, 30]
[16, 21, 25, 32]
[16, 17, 19, 29]
[15, 24, 28, 30]

[16, 19, 24, 31]
[19, 19, 26, 33]
[16, 18, 23, 28]
[19, 19, 26, 33]

[21, 27, 30, 30]
[17, 23, 27, 31]
[21, 27, 30, 30]
[15, 23, 28, 30]

[16, 19, 22, 31]
[16, 23, 26, 30]
[16, 23, 25, 30]
[14, 19, 22, 29]

[16, 21, 29, 30]
[17, 17, 27, 32]
[16, 16, 22, 27]
[14, 16, 24, 28]

[21, 27, 30, 30]
[18, 26, 32, 32]
[21, 27, 30, 30]
[16, 26, 30, 30]

[16, 22, 30, 30]
[19, 22, 30, 30]
[16, 21, 27, 28]
[19, 22, 30, 30]

[16, 24, 29, 29]
[16, 19, 24, 28]
[16, 17, 22, 24]
[14, 21, 27, 29]

[21, 27, 30, 30]
[17, 24, 27, 28]
[21, 27, 30, 30]
[15, 22, 25, 26]

[16, 17, 23, 23]
[16, 19, 26, 29]
[16, 21, 26, 27]
[14, 15, 22, 22]

[16, 20, 25, 31]
[16, 19, 27, 29]
[16, 21, 24, 27]
[14, 15, 22, 28]

[21, 21, 27, 27]
[20, 20, 27, 27]
[21, 21, 27, 27]
[20, 20, 27, 27]

[16, 21, 23, 27]
[17, 20, 26, 32]
[16, 19, 23, 29]
[14, 21, 26, 30]

[16, 16, 21, 24]
[19, 19, 26, 27]
[16, 16, 19, 24]
[19, 19, 26, 27]

[21, 21, 27, 27]
[17, 17, 24, 24]
[21, 21, 27, 27]
[15, 15, 22, 22]

[16, 21, 29, 29]
[16, 16, 25, 25]
[16, 16, 25, 25]
[14, 16, 24, 25]

[16, 21, 29, 30]
[16, 16, 24, 30]
[16, 18, 26, 29]
[14, 16, 24, 26]

[21, 27, 30, 30]
[17, 23, 27, 31]
[21, 27, 30, 30]
[15, 23, 28, 30]

[16, 22, 30, 30]
[19, 22, 30, 30]
[16, 21, 27, 28]
[19, 22, 30, 30]

[16, 24, 29, 30]
[17, 20, 27, 32]
[16, 19, 26, 29]
[14, 21, 27, 30]

[16, 19, 24, 31]
[19, 19, 26, 33]
[16, 18, 23, 28]
[19, 19, 26, 33]

[16, 17, 23, 26]
[16, 19, 26, 27]
[16, 21, 24, 28]
[14, 15, 22, 24]

[16, 17, 22, 28]
[16, 19, 27, 29]
[16, 21, 24, 26]
[14, 15, 22, 28]

[16, 16, 21, 22]
[16, 16, 23, 29]
[16, 18, 23, 26]
[14, 14, 21, 22]

[16, 21, 23, 27]
[16, 19, 23, 30]
[16, 17, 19, 26]
[14, 21, 26, 28]

[16, 22, 30, 30]
[19, 22, 30, 30]
[16, 21, 27, 28]
[19, 22, 30, 30]

[16, 21, 29, 30]
[17, 17, 27, 32]
[16, 16, 22, 27]
[14, 16, 24, 28]

[16, 21, 29, 29]
[16, 16, 25, 25]
[16, 16, 25, 25]
[14, 16, 24, 25]

[16, 21, 29, 29]
[16, 16, 24, 24]
[16, 16, 22, 22]
[14, 16, 24, 24]

[16, 24, 29, 29]
[16, 19, 24, 28]
[16, 17, 22, 24]
[14, 21, 27, 29]

[16, 16, 21, 24]
[19, 19, 26, 27]
[16, 16, 19, 22]
[19, 19, 26, 27]

[16, 16, 22, 23]
[16, 16, 23, 29]
[16, 18, 23, 26]
[14, 14, 21, 22]

[16, 16, 21, 21]
[19, 19, 26, 26]
[16, 16, 19, 19]
[19, 19, 26, 26]

[16, 17, 23, 26]
[16, 19, 26, 27]
[16, 21, 24, 28]
[14, 15, 22, 24]

[16, 21, 23, 27]
[17, 20, 26, 32]
[16, 19, 23, 29]
[14, 21, 26, 30]

[16, 21, 29, 30]
[16, 16, 24, 30]
[16, 18, 26, 29]
[14, 16, 24, 26]

[16, 17, 22, 25]
[16, 19, 26, 27]
[16, 21, 24, 28]
[14, 15, 22, 23]

[16, 22, 30, 30]
[19, 22, 30, 30]
[16, 21, 27, 28]
[19, 22, 30, 30]

[16, 21, 29, 30]
[16, 16, 24, 30]
[16, 18, 26, 29]
[14, 16, 24, 26]

[16, 21, 29, 29]
[16, 16, 25, 25]
[16, 16, 25, 25]
[14, 16, 24, 24]

[16, 17, 22, 22]
[16, 19, 26, 26]
[16, 21, 24, 25]
[14, 15, 22, 23]

[16, 16, 22, 23]
[17, 17, 26, 31]
[16, 16, 19, 22]
[14, 14, 21, 24]

[16, 16, 21, 24]
[19, 19, 26, 27]
[16, 16, 19, 22]
[19, 19, 26, 27]

[16, 16, 22, 25]
[16, 16, 23, 24]
[16, 16, 19, 22]
[14, 14, 21, 23]

[16, 16, 21, 21]
[19, 19, 26, 26]
[16, 16, 19, 20]
[19, 19, 26, 26]

[16, 17, 23, 23]
[16, 19, 26, 26]
[16, 21, 24, 24]
[14, 15, 22, 22]