If we take (10/9)^2 (15/14)^2 (16/15)^2 (21/20)^3 = 2 as scale steps, and simplify the scale-finding problem by assuming 4/3 and 3/2 both belong to the scale, we obtain four scales, the third and fourth of which are the inverted forms of the first and second. A version of the major/minor transformation, exchanging 10/9 with 16/15, which is equivalent to saying 2-->2, 3-->3, 5-->24/5, 7-->168/25, exchanges the first and second, as well as the third and fourth. The first "decaa", and fourth, "decad", are major versions, having two major tetrads and a minor tetrad, while "decab" and "decac" have two minor and one major tetrad. In any system where 50/49~1 the exchange transform sends tetrads to tetrads and can be considered major/minor. In 22-et in particular, each scale becomes the symmetrical decatonic. All of the scales have 23 intervals, 17 triads and 3 tetrads. ! decad.scl ! [15/14, 10/9, 21/20, 16/15, 15/14, 21/20, 10/9, 15/14, 16/15, 21/20] inversion of decab 10 ! 15/14 25/21 5/4 4/3 10/7 3/2 5/3 25/14 40/21 2/1 ! decab.scl ! [21/20, 16/15, 15/14, 10/9, 21/20, 15/14, 16/15, 21/20, 10/9, 15/14] (10/9) <==> (16/15) transform of decaa 10 ! 21/20 28/25 6/5 4/3 7/5 3/2 8/5 42/25 28/15 2/1 ! decac.scl ! [15/14, 16/15, 21/20, 10/9, 15/14, 21/20, 16/15, 15/14, 10/9, 21/20] inversion of decaa 10 ! 15/14 8/7 6/5 4/3 10/7 3/2 8/5 12/7 40/21 2/1 ! decad.scl ! [15/14, 10/9, 21/20, 16/15, 15/14, 21/20, 10/9, 15/14, 16/15, 21/20] inversion of decab 10 ! 15/14 25/21 5/4 4/3 10/7 3/2 5/3 25/14 40/21 2/1 ------------------------ Yahoo! Groups Sponsor ---------------------~--> Free $5 Love Reading Risk Free! http://us.click.yahoo.com/NsdPZD/PfREAA/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: < //www.freelists.org/list/tuning-math > .