[tuning-math] Three generator step scales

I want to look a little more closely at some of the scales I discussed in
the previous article, focusing on the ones with three generator steps. By
this I mean the scales deriving from [-2 0 0]-[-1 0 0]-[0 0 0]-[1 0 0], 
[0 -2 0]-[0 -1 0]-[0 0 0]-[0 1 0], and [0 0 -2]-[0 0 -1]-[0 0 0]-[0 0 1].
As JI scales, these are isomorphic sets of chords in the lattice, and
hence have isomorphic 7-limit graphs. We have:

From [0 0 1] as generator

[1, 15/14, 7/6, 5/4, 7/5, 3/2, 49/30, 7/4, 28/15, 15/8] 

72-et version [0, 7, 16, 23, 35, 42, 51, 58, 65]

From [0 1 0] as generator

[1, 21/20, 25/21, 5/4, 21/16, 10/7, 3/2, 5/3, 7/4, 40/21]

68-et version [0, 5, 17, 22, 27, 35, 40, 50, 55, 63]

From [1 0 0] as generator

[1, 36/35, 35/32, 6/5, 5/4, 48/35, 35/24, 3/2, 12/7, 7/4]

84-et version [0, 3, 11, 22, 27, 38, 46, 49, 65, 68]

All of the JI scales are graph-isomorphic, each having 21 intervals, 16
triads and 4 tetrads in the 7-limit.
The first scale makes the least sense as a JI scale, having a step of
225/224; two notes are conflated in the 72-et
version, giving an excellent 9-note scale. The second scale makes the
most sense as a JI scale, and the least sense as a tempered scale, since
the tempered version is isomorphic to the non-tempered version. The last
scale is a little irregular so far as step size goes, but does gain from
tempering via Orwell, as the tempered version has four more intervals and
two more triads.



------------------------ Yahoo! Groups Sponsor ---------------------~-->
Save on REALTOR Fees
http://us.click.yahoo.com/Xw80LD/h1ZEAA/Ey.GAA/wHYolB/TM
---------------------------------------------------------------------~->

To unsubscribe from this group, send an email to:
tuning-math-unsubscribe@xxxxxxxxxxxxxxx

 

Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/ 


____________________________________________________________

To learn how to configure this list via e-mail (subscribe,
unsubscribe, etc.), send a message to listar@xxxxxxxxxxxxx
with the subject line "info tuning-math".  Or visit the
website:  < http://www.freelists.org/list/tuning-math > .

Other related posts: