More on the comma experiment

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...)


The "bad" version [ii "regular" minor triad]:

20
17
13 13
7 9 7
0 0 4 4 0
8va ----------------------------------------------------
16
10 13
^
|
comma shifts


I IV ii V I


The "good" version [ii "diminished" triad with a subminor third]:

20
15
13 13
7 9 7
0 0 4 4 0
8va ----------------------------------------------------
16
9 13
^
| deg. 15 is borrowed tone


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

Paul wrote:

''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