set-theory in xeno-ETs

> Steven Kallstrom wrote:

> I know that this is a hot topic in the music theory world
>now. There are a
>few recent articles in which the theory of scales, harmonic progressions,
>and set-theory are expanded into the microtonal world. I could post a
>short
>bibliography if anyone would like.

I'd be interested in that---especially set-theory as applied to non12tet.

I am experimenting myself these days in set-theory applied to various ETs,
lately 19tET. One of the joys of this, is to take my favorite
pitch-class set(s) from 12tET hear the various "warped" versions of
it/them in a different ET.

For example, one of my old favorites was 013, (I especially enjoy it
melodically expressed in close position as 1-3-0 (i.e. Db-Eb-C), but all
of its registrations and melodicizations have affective resonances for
me). Now, in 19tET of course, minor thirds are mighty close to Just, so
in a way it's the perfect trichord to futz with in 19tET.

But you have different versions:

250(19tet) is the one sounding most like 12tET's 130.

150(19tet) is still recognizable as 130(12tet), but poignantly warped
with the "thin" 2nd 0-1.

140(19tet) is still more dissonant and painful contraction of the
trichord, and finally

130(19tet) I find almost "too dissonant" for my ears---an
almost-chromatic "compression."

Just playing these four "versions" of this simple sound are enough to keep
me happy as a proverbial pig in a vat of microtonal @#$% for a good while.

Of course, I am totally coming at this from a 12t-ET perspective, still in
the "honeymoon" period of seeing things as "different from" that
perspective in various ways. This I decided to fully admit early on in my
microtonal explorations, and to work and compose from this point of view
un-ashamedly, seeing if I couldn't make some good music from it.
--Also, from the perspective of a "closet" serialist guy (we're not
generally respected in the open these days).

In case any one's interested, here's the 19-tone "row" I'm working with
these days, which is highly saturated with "013(12tet)" variants:

6 1 4 3 8 5 18 2 16 0 14 17 13 9 12 7 10 15 11

Saturation:

6 1 4 (025)
1 4 3 (013)
4 3 8 (014)
3 8 5 (025)
18 2 16 (025)
2 16 0 (025)
16 0 14 (025)
0 14 17 (025)
14 17 13 (014)
13 9 12 (014)
9 12 7 (025)
12 7 10 (025)
10 15 11 (015)

note that the % of saturation of the different variants of 013(12tet) is
weighted towards 025(19tet), the most "familiar" and/or "consonant" of
the variants.

This "row" can then in turn be placed into an "array" a la Milton
Babbitt's constructs----these "arrays" are formed of "lynes" which are
based on transformations (R, I, RI, T) of the row, and columns which are
made up of "aggregates"--that is, each column contains all of the 19
pitch-classes of 19t-ET once each exactly.

Here is one such "array" that I'm thinking of utilizing, using that
"013(12)-variant-saturated" 19-tone row:

http://music.columbia.edu/~chris/base.array.html

Anyway, I'll leave this report at that for now. . . . might as well see if
anyone else is interested in this @#$%@#$ before I spend more energy
typing.

g'day,

Christopher Bailey

***From: Christopher Bailey******************

http://music.columbia.edu/~chris

**********************************************

--- In tuning@egroups.com, Christopher Bailey <cb202@c...> wrote:

http://www.egroups.com/message/tuning/13868

>
> I am experimenting myself these days in set-theory applied to
various ETs, lately 19tET. One of the joys of this, is to take my
favorite pitch-class set(s) from 12tET hear the various "warped"
versions of it/them in a different ET.
>

> http://music.columbia.edu/~chris/base.array.html
>
>
> Anyway, I'll leave this report at that for now. . . . might as well
see if anyone else is interested in this @#$%@#$ before I spend more
energy typing.
>

Well, Christopher, somebody HADDA do it! As somebody who only VERY
OCCASIONALLY dips into a bit of serialism, I assume that eventually I
might "dip into" xenharmonic patternings as well. I believe Neil
Haverstick has also done a bit of work in this direction...

so there *are* people interested in your explorations!
____________ ___ __ __
Joseph Pehrson

Christopher Bailey wrote:

> I'd be interested in that---especially set-theory as applied to
> non12tet.
>
> I am experimenting myself these days in set-theory applied to various
> ETs, lately 19tET.

Hi - I just dug up something I did a bit of work on quite a few years ago,
with some results that might answer Christopher Bailey's question. I
haven't seen most of these anywhere else - sorry if they're well-known to
you.

First, a table showing how many set-classes there are of each cardinality
in 19tET, with the figures for 12tET included for the sake of
comparison:

card. 12tET Z 19tET Z

3 12 0 30 0
4 29 1 120 0
5 38 3 324 0
6 50 15 756 21
7 1368 57
8 2052 90
9 2494 156

( The 'Z' column tells you how many Z-related pairs there are of each -
for example, of the 29 12tET tetrachords, only one pair, [0137] and
[0146], have an identical interval vector. There are no Z-related
triples either in 12tET or 19tET - there are some in 16tET iirc
though)

When I worked this out I remember finding the sheer number of available
set-classes overwhelming in the 19tET universe - just the number of
tetrachords alone is daunting. Mandelbaum gives a couple of numbers like
these, but as I remember it he doesn't define his chord-types like
set-classes, so they're different from mine.

I also worked out the deep scales, which are the 9- or 10-note sets that
have every interval a different number of times, like the 7-out-of-12
diatonic. There are 9 of them, and I've since found them listed in an
article by Carlton Gamer, in the journal of music theory in the 60s.
They've probably been mentioned on this list before too, but here goes:

0 2 4 6 8 10 12 14 16
0 2 4 5 7 9 12 14 16
0 2 3 5 8 10 11 13 16
0 1 3 4 6 7 10 13 16
0 1 4 5 8 9 12 13 16
0 1 4 5 6 9 10 14 15
0 1 2 6 7 8 12 13 14
0 1 2 3 4 10 11 12 13
0 1 2 3 4 5 6 7 8

(these are all in prime-form, any of their rotations or inversions is also
good of course)

Since 19 is prime, you get whole families of chords inter-related by
multiplicative transforms (unlike in 12tET where only M5 can produce a
different set-class). The deep scales above are one such family. Here's
another example:

0 2 3 7 -- 0 2 3 8 -- 0 3 4 9 -- 0 2 6 9 -- 0 1 7 9 -- 0 2 7 10 --
0 1 4 11 -- 0 4 5 11 -- 0 2 6 11

Any of these, under any multiplicative transform mod 19, will give another
member of the set (once you put it into prime-form). The 120 tetrachords
divide into 13 such families of 9 chords, plus one family of 3 chords:

0 1 3 8 -- 0 3 5 9 -- 0 1 7 11

The M-transforms, as they're called, are fairly bogus musically, if you
ask me.

Kind of the "opposite" of the deep scales is the one set (and its
complement) in the whole 19-tone universe that has _equal_ numbers of each
interval. The set is:

0 1 4 7 8 9 10 12 14

and it has the interval vector <444444444>.

Aren't you pleased you asked?

--- In tuning@egroups.com, jon wild <wild@f...> wrote:

http://www.egroups.com/message/tuning/14154

Sarn... Jon Wild is the person you should post concerning work with
xenharmonic serial ordering systems -- about which you were
inquiring. It looks like he has done some work in this direction...
_______ ____ __ _ _
Joseph Pehrson