Gary M asked off-list if I had used V or V7 in my experiment, and what I actually did to deal with the comma problem in 22.
I'm not sure if not using the V7 would alter things so far as the comma goes, but I left things as unadorned triads, and used only 3 voices, mostly to keep things as simple as I could, seeing as how I'm still learning much of this. (So please excuse the less than artful voice leading :)
I've appended some notation below which illustrates how I made the progressions in 22. The source scale is 0-2-4-7-9-11-13-16-18-20, and the borrowed tone (deg 15) comes from the same scale on the dominant (13-15-17...)
Of course since then I think I've grokked a little more, so I'll probably go back and try this or similar exercises using 4 voices (as long as the stretch on the keyboard doesn't get out of hand, that is :)
Gary, I hope this answers your questions well enough. I'm looking forward to your comments.
Steve
Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Wed, 15 Jan 1997 22:45 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA04882; Wed, 15 Jan 1997 22:48:50 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA04989 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id NAA12193; Wed, 15 Jan 1997 13:48:48 -0800 Date: Wed, 15 Jan 1997 13:48:48 -0800 Message-Id: <199701152143.VAA01865@chiswick.globalnet.co.uk> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu
''9-limit constructs are not as pretty geometrically, but the math works out..''
This depends on whether 9 is mapped on the 3 axis (3^2) or given an axis of its own. There are some musical contexts (and some temperaments as well - the tuning of the TX81Z comes to mind) where 9 _has_ a harmonic function distinct from 3^2. Of, course, there are other musical settings -especially long duration sound installations - where distinguishing prime identities is more important.
Wilson has graphed his CPSes with factors of 9, 15, and 21 mapped to independent axes. From what I recall of his notes, he has sketched out CPSes with all combinations, including repeated factors, through 15, and tried out a few promising sets with higher factors. The three Eikosany he has worked with most are 3(1,3,5,7,9,11), 3(1,3,7,9,11,15), and 3(1,3,5,7,11,13) and the two Hebdomekontany are 4(1,3,5,7,9,11,13,15) and 4(1,3,5,7,11,13,17,19), the latter of which I have played with on my Rayna. (It can have a Stravinskian quality due to the everpresent quasi-octotonic scales).
Perhaps John can persuade Erv to publish his ''Letter to Adrian Fokker'' and ''Letter to John Chalmers'' where all of this material - including the graphing that Paul describes - were first set out.
Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Thu, 16 Jan 1997 20:07 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA30947; Thu, 16 Jan 1997 20:10:48 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA30918 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id LAA13221; Thu, 16 Jan 1997 11:10:45 -0800 Date: Thu, 16 Jan 1997 11:10:45 -0800 Message-Id: <32DE79AA.28F@top.monad.net> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu