Reply to William Annis

>The 18:21:24:28 (makes a square in
>the chart above, say 7/4, 21/16, 3/2, 9/8) acts as a fine dominant if
>you need such a thing.

This chord, which is Kami Rousseau's favorite tuning for a minor seventh
chord (aka a 6th chord), is discussed at
http://www.cix.co.uk/~gbreed/erlichs.htm and
http://www.cix.co.uk/~gbreed/ass.htm. It is more consonant that the
appearance of the number 21 might suggest, since each of the six intervals
in the chord is within an odd limit of 9.

>I've also found 12:14:21 (1/1, 7/6, 7/4) is
>very useful if prepared.

A subset of the chord above (expressed as 12:14:18:21), and this triad is
the third chord in my piece _TIBIA_. I then exploit the fact that in
22-equal, 4/3 and 21/16 are the same, by adding this note in the bass.

>You can even get a useful I-IV-V-I out of
>it, though I find the progression V-IV-I more effective if you keep
>the relative voicing such that the top voices go 7/4 - 14/9 - 3/2
>(intervals: 8/9 * 27/28).

Can you elaborate?

>Do other composers take this approach to exploring the
>possibilities of some system of ratios?

Sure -- the most similar example I've seen might be LaMonte Young who used a
grid with 3 on one axis and 7 on the other axis. However, about the duodene:

(a) it seems particularly geared toward the 12-tone periodicity block that
results on the 3-5 plane, so some other number of notes other than 12 might
be more appropriate when using other constructing intervals (I'd be happy to
pursue this if you're interested); and
(b) one should remember that it fails to allow even some of the simplest of
Western triadic music to be performed.

>From: "Paul H. Erlich" <PERLICH@ACADIAN-ASSET.COM>
>
>>I've also found 12:14:21 (1/1, 7/6, 7/4) is
>>very useful if prepared.
>
>A subset of the chord above (expressed as 12:14:18:21), and this triad is
>the third chord in my piece _TIBIA_. I then exploit the fact that in
>22-equal, 4/3 and 21/16 are the same, by adding this note in the bass.

Ah. I'll have to play with that and see what comes out.

>>You can even get a useful I-IV-V-I out of
>>it, though I find the progression V-IV-I more effective if you keep
>>the relative voicing such that the top voices go 7/4 - 14/9 - 3/2
>>(intervals: 8/9 * 27/28).
>
>Can you elaborate?

I can try, although I think this is one of my own little
musical idiosyncrasies.

Although I listed various chords in my previous mail, I have a
strong tendency toward counterpoint, or at least simple two voice
conterpoint above some harmonic background or a drone with a complex
timbre. So, I think the sequence of pitches 7/4 14/9 3/2 in the same
octave have a very satisfying sound. Something about that small shift
to the last note (about 62 cents) says "we've reached a resting point"
to me. Those notes fit conveniently in an analog of IV-V-I if you
take the septimal minor triad on 1/1 as I.

On the other hand, I tend to avoid a small leading tone from
below. So, the common practice practice [ :) ] of raising the 7th in
the various flavors of the minor is something I generally avoid when
making music in 12tet (or however it is my mandolin frets are really
tuned).

>>Do other composers take this approach to exploring the
>>possibilities of some system of ratios?
>
>Sure -- the most similar example I've seen might be LaMonte Young who used a
>grid with 3 on one axis and 7 on the other axis. However, about the duodene:
>
>(a) it seems particularly geared toward the 12-tone periodicity block that
>results on the 3-5 plane, so some other number of notes other than 12 might
>be more appropriate when using other constructing intervals (I'd be happy to
>pursue this if you're interested); and

Baby steps, baby steps! I have to start somewhere, and a
little set of 12 pitchs is a fine starting place for now. When I'm
feeling more adventurous and confident, I can do this:

*** Duodene(vrat=Ratio(7, 6), hn=5, vn=5)
<98:81> <49:27> <49:36> <49:48> <49:32>
<28:27> <14:9> <7:6> <7:4> <21:16>
<16:9> <4:3> <1:1> <3:2> <9:8>
<32:21> <8:7> <12:7> <9:7> <27:14>
<64:49> <96:49> <72:49> <54:49> <81:49>

Or, I suppose I can try to find some convenient way to represent
multi-dimensional grids. In fact, since reading the JIN Primer I've
wanted to toss together a quick program to generate PostScript of that
7-limit 3d diagram, but generalized so you can choose your own
intervals for each axis. Some day...

>(b) one should remember that it fails to allow even some of the simplest of
>Western triadic music to be performed.

That's fine. I'm not trying to do western triadic music,
though of course that will be an influence in my music, since I've
steeped myself in it for 15 years or so now. :)

--
William S. Annis wsannis@execpc.com
Mi parolas Esperanton - La Internacian Lingvon www.esperanto.org

William, you might want to check out the customizable EPS files I have at:
http://www.mbay.net/~anne/david/eps4ji/index.htm
I think you use them to do what you suggest...

> -----Original Message-----
> Message: 7
> Date: Fri, 31 Mar 2000 07:59:43 -0600 (CST)
> From: "William S. Annis" <wsannis@execpc.com>
> Subject: Re: Reply to William Annis
>
[snip]
>
> Or, I suppose I can try to find some convenient way to represent
> multi-dimensional grids. In fact, since reading the JIN Primer I've
> wanted to toss together a quick program to generate PostScript of that
> 7-limit 3d diagram, but generalized so you can choose your own
> intervals for each axis. Some day...
>
> --
> William S. Annis wsannis@execpc.com
> Mi parolas Esperanton - La Internacian Lingvon www.esperanto.org
>
David Canright (831) 656-2782 (or -2206)
Math. Dept., Code MA/Ca (831) 656-2355 (FAX)
Naval Postgraduate School DCanright@NPS.Navy.mil
Monterey, CA 93943 USA http://www.mbay.net/~anne/david/