12 note scales [versions]

> Let me lattice this scale so that we can see it harmonically in the 7-
> limit (9-limit is ignored for now):
>
> 35/32----105/64
> ,'/ \`. ,'/ \`.
> 5/4-/---\15/8-/---\45/32
> /|\/ \/|\/ \/|\
> / |/\ /\|/\ /\| \
> / 7/4------21/16-----63/32\
> /,' `.\ /,' `.\ /,' `.\
> 1/1-------3/2-------9/8------27/16
>
> So only one note is different from either of Paul Hahn's scales that
> I posted earlier, up to a transposition (which is arbitrary anyway).
> Again, there are not one but _two_ 1.3.5.7 hexanies.
>
> 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??"

__________ _____ ___ __ _
Joseph Pehrson