RE: [tuning] 12 note scales [versions]

I wrote,

>> Back at the end of the Mills era we were discussing 12-tone subsets
>> of 7-limit JI and how many 7-limit-consonant intervals they could
>> have. Paul Hahn was the winner -- he found 24 scales with 32 7-limit-
>> consonant intervals each. They are the 24 possible
>> rotations/reflections of this:
>>
>>
>> 5:3---------5:4
>> /|\ /|\
>> / | \ / | \
>> / | \ / | \
>> / 7:6---------7:4 \
>> /.-'/ \'-.\ /.-'/ \'-.\
>> 4:3--/---\--1:1--/---\--3:2
>> |\ / \ /|\ / \ /|
>> | / \ | / \ |
>> |/ \ / \|/ \ / \|
>> 28:15--------7:5--------21:20
>> '-.\ /.-' '-.\ /.-'
>> 8:5---------6:5
>>
>> As with most of the JI scales in this thread, these each have two
>> hexanies and four tetrads. But they are all fairly uneven -- probably
>> none of them have the Constant Structures property.
>>

>Hi Paul...
>Thanks for latticing that 12-note scale I have been working with. I had
>a question concerning the Paul Hahn scales you mention. Are you
>suggesting they are variations on the ratios in the diagram above?? And
>if so, how are they "rotated/reflected??"

This figure must be seen in three dimensions, as a piece of the
octahedral-tetrahedral lattice. Here is an example of a rotation (180º about
a horizontal axis):

7:6---------7:4
.-'/ \'-. .-'/ \'-.
4:3--/---\--1:1--/---\--3:2
|\ / \ /|\ / \ /|
| / \ | / \ |
|/ \ / \|/ \ / \|
28:15--------7:5--------21:20
\'-.\ /.-'/ \'-.\ /.-'/
\ 8:5--/---\--6:5 /
\ | / \ | /
\ | / \ | /
\|/ \|/
28:25-------42:25

(you might prefer to transpose the ratios so that 1:1 appears in the center,
but transpositions are not counted as distinct scales).

Here is an example of a reflection of the original figure (about a plane
perpendicular to the line connecting 1:1 with 7:5):

7:6---------7:4
.-'/|\'-. .-'/|\'-.
4:3--/-|-\--1:1--/-|-\--3:2
|\ / | \ /|\ / | \ /|
| / 49:30-------49:40 \ |
|/.-' '-.\|/.-' '-.\|
28:15--------7:5--------21:20
'-.\ /.-' '-.\ /.-'
8:5---------6:5

There are 24 possibilities in all. However, none of them are even enough so
that the best approximations to familiar chords will necessarily appear
where you expect them to on the keyboard.

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