Microtonal serialism

Although I am posting this to the whole tuning-list, given the evident (to
me) slant of the majority toward scale theory and acoustics, the following
is probably more appropriate for a small sublist. So, if there are any
serialist composers out there who are playing with microtones (or who
would like to try), please send me a quick note with your name & email
address so I can organize an informal discussion group. This is also fair
warning to those who believe serialism is moribund to hit your delete key
now.

Last week I sent the following note to a few friends. It is a description
of a "hyper-12T operation" that I probably won't have time to work up into
publishable form for some time, but I wanted as many comments on
compositional possibilities as possible.

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

It seems I've stumbled onto a "hyper-atonal" version of a Cohn function
(sans the requirement that the transformations have to always produce a
TTO version of the same chord). After moving recently I've misplaced my
copy of David Clampitt's paper on his "Q-relation" which also might have
some bearing here. Whether or not it's far-fetched to relate the
following to the Cohn function is your call, but the operation (P for
parity) sure is parsimonious.

One of my primary theory motivations lately has been the idea of
transformations between spaces of different sizes, thus the recent foray
into scale theory resulting in a "warp" function to generate hypertonal
structures. But in the back of my mind has always been the generalization
of 12-tone and other atonal structures and systems into the microtonal
sphere, i.e., the development of "hyperatonal" compositional
strategies.

This past weekend I was doing some pre-compositional exercises and came
across what I think is a very interesting operation. It started in
quartertones, but then I tried it in the more manageable 12-space, and
some startling things began to fall out. The results are obviously
applicable to any n-TET space where n is even.

I started by constructing a "warped" row of 12 notes in 24-space using the
well-explored and exploited 014-trichord as a generator. I can't remember
my original 12T row for sure this morning, but I think it was this:

0 4 3 8 7 e 6 9 t 2 5 1

I then divided the row into 3 successive tetras and raised the middle one
by a quartertone:

S: (0 4 3) (8 7+ e+) (6+ 9+ t) (2 5 1)

This effectively leaves the outer trichords untouched in an ambivalent
12-space while warping the inner trichords unambivalently into 24-space.
The outer tetras on the other hand remain in "even parity" 12-space
compared to the inner tetra in "odd parity" 12-space (all this using the
partition of the chromatic into two WT-scales as a model). In Z/24, then,
I translated S to:

S': 0 8 6 16 15 23 13 19 20 4 10 2

I then dutifully (and laboriously) generated the usual 4 (hyper)TTO lists
(call inv & transp in any space HTTOs -- conveys bad 12-TET bias though),
and proceded to write some brief exercises for a scordatura string quartet
(1st & va tuned 1/4-tone up).

But before reaching a reasonable compositional facility here, I decided to
see what the S' complement looked like. O fortuna! -- a z-related twin
that looks like some sort of skewed-inversion of S'--

T*(unordered): 1 3 5 7 9 11 12 14 17 18 21 22

$({S'}) = 2 2 2 2 2 3 2 1 3 1 3 1
$(T*) = 2 2 2 2 2 1 2 3 1 3 1 3

Since I know of no HTTO to get from elements of S' to elements of T*, I
decided to see how much I could get T* to "look like" S' and came up with
the row

T': 0 8 6 16 17 11 13 21 20 4 10 2

and worked out the HTTO's.

Now, 24 integers ain't easy to work with, and I had to admit that I wasn't
working in a totally "free" 24-space system since it was highly dependent
on 12-space as a reference -- even my notation was bound by inflections of
12-space notation -- the curse of the microtonalist. So I ended up with
HTTO charts that were 12-tone with pluses (as I started out with). This
quickly made something clear even to me when I spotted the S look-alike
complement Almost-T:

S: 0 4 3 8 7+ e+ 6+ 9+ t 2 5 1
Almost-T: 0+ 4+ 3+ 8+ 6 7 9 e t+ 2+ 5+ 1+

So I corrected my "mistake" by rearranging T to conform to the operation
"keep the integers but exchange their parities" making:

T: 0+ 4+ 3+ 8+ 7 e 6 9 t+ 2+ 5+ 1+

Obviously, in 24-space notation, this operation translates (and quickly
generalizes) to something like "evens become odd and odds become even."
To play with it a little, I tried this definition of "parity
transposition" Pn:

If element e is even, P/n/e = e+n;
if e is odd, P/n/e = e-n
(all mod space cardinality)

This would relate S and T above as T = P/1/S with the inverse
S = P'/1/T = P/-1/T.

I've been ranging n = 0,1,2,..., for exploration of examples, but of
course if we set P = P/1/, say, (and retaining I = P/0/ for an identity)
then the ranging can be expressed as composition; e.g., P(PX) = P/2/X or
P^2^(X) and so on.

For manageable tests I took P back into 12-space and found some
interesting characteristics, which I think are far from exhausted yet, so
the following assume 12-space.

~~~P partitions all of Z/12's subsets -- e.g., every trichord can be
placed into one (and only one) of 6 P-partition classes.

~~~The members of any given P-set (all sets generated by P/n/, n ranging
over 0,...,11) are themselves often (always? - haven't tested yet) related
by other operations -- e.g. trichords again: starting with X = {0,1,2}
which produces the following P-set:
set $ vector
a P/0/ 0 1 2 1 1 t 210000
b P/1/ 1 0 3 1 2 9 111000
c P/2/ 2 e 4 3 2 7 011010
d P/3/ 3 t 5 2 5 5 010020
e P/4/ 4 9 6 2 3 7 011010
f P/5/ 5 8 7 2 1 9 111000
g P/6/* ... ... ...
etc.
* P/6/ begins repetition of the string series since $P/+6/=$P/-6/
~~~ g etc. are all T6-related to a through f
~~~ b,f and c,e are TnI-related
~~~ a,d; b,c; e,f are TnM5-related

~~~Since the evens and odds retain their respective EVEN intervallic
relations throughout the consecutive operations, all even vector entries
remain the same. But, since you are adding +n or -n, the odd entries
(which represent the vector between the even subset and odd subset) trace
a careening diagonal path as the P-related sets are generated. Thus,
displaying only the odd vector components from the previous example:

component: 1 3 5

2 0 0
( \
1 1 0
\ \
0 1 1
\ )
0 0 2
/ )
0 1 1
/ /
1 1 0
( /
2 0 0
( \
...

~~~ The behavior of the odd vector components in a partition class may
explain why m5- and z-related sets (in 12-space) are often members of the
same P-set. Testing the usual suspects:

set $ V
02367 21315 212221
13276 11316 311221
24185 12135 213211 <-----> inverse P-set generated by 01457
35094 33114 212221
46et3 12414 211231
57te2 23133 113221
(1 z/m5-pair; 2 m5-pairs)

01247 11235 222121
10356 12216 |
2e465 32115 | <-----> no unique inverse
3t574 11235 |
49683 12216 |
58792 32115 V
(multiple z/m5 pairs)

01348 12144 212320
10259 31134 |
2e16t 12144 | <-----> no unique inverse
3t07e 31134 |
49e80 12144 |
58t91 31134 V
(multiple z/m5 pairs)

0137 1245 111111
1026 1146 210111
2e15 2136 112101 <-----> inverse
3t04 2316 111111
49e3 2415 110121
58t2 3324 012111
(again: 1 z/m5 & 2 m5s)

~~~ Strictly diatonic moves are not out of reach here by fooling with the
parity definition, thereby getting even closer to the "classic" Cohn
function. Generating the P-set based on 037 produces this consecutive
pair: P/5/(037) = 5t2 and P/6/(037) = 691. This suggests the move
"subtract/add 1 to the tonic and mediant; add/subtract one from the
'dominant'":

0 3 7 C -
e 2 8 G#o
t 3 7 Eb+
e 2 6 B -
t 1 7 G o
.
.
.
e 4 8 E +
0 3 7 C -

36 moves to close is a pretty hefty cycle, but does the literature support
segments larger than one move? I don't remember seeing a mention of this
possibility, but have a feeling I've missed it or forgotten it.

~~~ Another interesting diatonic sidebar: starting with a C-maj/a-min
scale, P will take you through four "exotic" scales (actually 2 with
inversions) before you arrive at F#-maj/d#-min.

~~~ Finally, returning to where I started: iff the generating set is a
hexachord AND can be reduced to a warped 6-space notation with each
element in Z/6 appearing once (e.g., 0,3,5,6,8,t --> 0,1+,2+,3,4,5), then
its P/1/ partner will also be its complement (0+,1,2,3+,4+,5+ -->
1,2,4,7,9,e)

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

A few days later.....

Combinatoriality Post Script

At first I thought the P-operation would be most useful in navigating the
large universe of possibilities in quartertone space (which it does of
course), and that it would have only limited use in the more familiar
12-space. Not so, and the following 12-space discovery is easily
generalized to even microtonal spaces....

If X in Z/12 is a "parity warp" of Z/6, and m+n = 1 mod 12, then

{P/m/X} U {P/n/X} = Aggregate

Example:
0,1+,2,3,4+,5+ --> 0,3,4,6,9,11
The full P-set (letting n range 0,...,11) is:

set $ P-combinatorial
0 3 4 6 9 e 131232 \
1 2 5 7 8 t 212323 /
2 1 6 8 7 9 141114 --------+
3 0 7 9 6 8 111333 ------+ |
4 e 8 t 5 7 121215 ----+ | |
5 t 9 e 4 6 113115 --+ | | |
6 9 t 0 3 5 | \ | | | |
7 8 e 1 2 4 | / | | | |
8 7 0 2 1 3 (rpt) --+ | | |
9 6 1 3 0 2 | ----+ | |
t 5 2 4 e 1 | ------+ |
e 4 3 5 t 0 V --------+

The M5/Z/Complement-relationships between the P-subsets (using the initial
digit as a set name) trace two (disjunct) paths:

0--(M)--3--(C)--t--(M)--7 2--(C)--e
| | | | | |
(Z/C) (Z) (Z) (Z/C) <~~~P-only~~~> (M/Z) (M/Z)
| | | | | |
1--(M)--4--(C)--9--(M)--6 5--(C)--8

NB: T6-related diagonals (0&6, t&4, 5&e, etc.) overlay yet another path
which was a little hard to display in ASCII.

So, in (12-tone) pre-composition, you could plan numerous paths of
P-combinatorial hexachord pairs expressing T, I, M5, C, Z, and P *without
ever leaving* the 12 hexachords in the P-set. Transposition by 1 through
5 would then be a kind of "atonal modulation" (I love that one) which
would effectively scramble the appearances of the basic single trichordal
building block, the 026-SC (every one of which is contained in the
P-set-class & partitions each hexachord due to the construction of the
source set), and hence 026 might profitably be employed in modulation as
a common chord (though other common-chord possibilities are interesting as
well -- e.g., SC-016 appears in every hexachord).

Serialism is dead. Long live Serialism!

On Tue, 9 Feb 1999, Stephen Soderberg wrote:

> Although I am posting this to the whole tuning-list, given the evident (to
> me) slant of the majority toward scale theory and acoustics, the following
> is probably more appropriate for a small sublist. So, if there are any
> serialist composers out there who are playing with microtones (or who
> would like to try), please send me a quick note with your name & email
> address so I can organize an informal discussion group. This is also fair
> warning to those who believe serialism is moribund to hit your delete key
> now.

What's serialism?

-Bram

> What's serialism?

Now there's a big question!

Serialism is a very involved topic, but an extreme oversimplification would
be that it's based upon a tone row. A tone row is an ordering of the 12 (or
however many) notes within the octave, not reusing any one before you've
mentioned all of them. To that row you apply one or more rhythms, run the
result in retrograde or inversion or both, run the rhythm in retrograde but the
row in forward order, or half-speeding, double-speeding, and all kinds of other
such variations.

Arnold Schoenberg was probably the foremost pioneer of serial techniques.

> Serialism is a very involved topic, but an extreme oversimplification would
> be that it's based upon a tone row.

Actually, best I understand serialism at least (I certainly don't claim to be
an expert on the topic), other quantifiable aspects of sound can also be
serialized. That would include volume, duration, timbral brightness, and such.
So, saying that serialism is based upon tone rows is not only an "extreme
oversimplification", but a heavily pitch-oriented oversimplification.

On Tue, 9 Feb 1999, Gary Morrison wrote:

> > What's serialism?
>
> Now there's a big question!
>
> Serialism is a very involved topic, but an extreme oversimplification
> would be that it's based upon a tone row. A tone row is an ordering of
> the 12 (or however many) notes within the octave, not reusing any one
> before you've mentioned all of them. To that row you apply one or more
> rhythms, run the result in retrograde or inversion or both, run the
> rhythm in retrograde but the row in forward order, or half-speeding,
> double-speeding, and all kinds of other such variations.

That sounds a lot like other descriptions I've heard of atonal music. The
thing I don't get is, it seems like an awful lot of rules to follow, and
I've never heard an explanation of the reasoning behind them.

There's an interesting rhythmic effect where instead of having an obvious
repeating rhythm music enters into what sound like an endless state of
collapse. I'm not sure if this is related to atonality or is just an
effect of complicated synchopated polyrhythms - at least, I can make
rhythms which have that effect in 5/4 or 9/4, but I'm not sure if that's
an artificially enforced order.

-Bram

<< The thing I don't get is, it seems like an awful lot of rules to follow,
and
I've never heard an explanation of the reasoning behind them. >>

The most basic explanation for the common rules of serial composition was
given by Schoenberg, when he referred to his then new method of composing,
which he called "A Method for Composing with Twelve Tones Related Only to Each
Other". (Actually, I don't think I have his title quite right, but you get the
idea.) The idea is that the functions commonly applied to the row (the
original ordering of the chromatic) in serial music take the place of harmonic
functions in tonal music. There's a lot more to it than that, of course, and
if you're interested in learning a little more about classical serialism I
suggest reading Words About Music by Milton Babbitt.

Drew Wheeler

At 09:25 PM 2/9/99 +0000, you wrote:
>From: Gary Morrison <mr88cet@texas.net>
>
>> What's serialism?
>
> Now there's a big question!
>
> Serialism is a very involved topic, but an extreme oversimplification
would
>be that it's based upon a tone row. A tone row is an ordering of the 12 (or
>however many) notes within the octave, not reusing any one before you've
>mentioned all of them. To that row you apply one or more rhythms, run the
>result in retrograde or inversion or both, run the rhythm in retrograde
but the
>row in forward order, or half-speeding, double-speeding, and all kinds of
other
>such variations.
>
> Arnold Schoenberg was probably the foremost pioneer of serial techniques.
>
>
>
Yes -- Schoenberg discovered the row around 1923 - but Charles Ives used a
row in _Tone Roads_ in 1911. Schoenberg's use of the row was restricted to
pitch, although Ives applies the row to rhythm and other parameters - and
_Tone
Roads_ sounds like it was written by a 1950's serialist. Around 1948 Pierre
Boulez
and Milton Babbitt independently discovered "total serialism", which means the
row is used for as many parameters as the composer wants to use. Karlheinz
Stockhausen applied the row not only to individual notes (Punkte - points),
but to
groups (Gruppen) of notes, and even to large scale form (Momente).

Message text written by Bram

>That sounds a lot like other descriptions I've heard of atonal music. The
thing I don't get is, it seems like an awful lot of rules to follow, and
I've never heard an explanation of the reasoning behind them.
<

Twelve tone and serial musics are varieties within a larger category of
atonal (or, as Schoenberg preferred, 'pantonal') music There is also an
important repertoire of 'free atonal' musics that are not composed
according to a set of strict rules, but rather an ad hoc use of a wide
variety of techniques for organizing musical coherence, including tonal
references. The works of Schoenberg, Webern, Berg composed in the years
before Schoenberg's invention of the twelve-tone technique fall into this
category and the technique was largely conceived as a way of overcoming
some of the perceived deficits of free atonal writing. The microtonalist
Haba used what he called a 'athematic' technique, a very free form of
atonality, with, in my opinion, some notable deficits in overall coherence.

Following the second Viennese School (Schoenberg, Berg, Webern) It is
useful to distinguish between an American twelve-tone technique (i.e.
Babbitt) and European serialism. In retrospect and very broadly
generalizing, Babbitt's technique gave priority to the idea of regulating
the consumption of twelve-tone aggregates based upon a given intervallic
set structure and then generalized the intervallic idea to other parameters
so that the other parameters would cooperate with pitches to project the
same underlying structure. Across the pond, European composers generalized
the concept of the series as applicable to the organization of the division
of any continuum. The notion of underlying interval structures common to
the various parameters is not found here. To a certain extent, Europeans
retained the idea of the series as a melody rather than as an abstract
series of PCs or intervals.

I'm giving a very condensed version here that ignores a lot of important
figures (e.g.. Skryabin and the Skryabinists, Messiaen, Harrison, Carter,
Cage) and ideas (e.g. the roles of logical positivism in the US and
Magister Luci cum Marxian mysticism in Europe) but this is the general
idea, and there is no reason (aside from inertia and the general atonal
fatigue) why these techniques could not be adopted to gamuts other than
12tet. Ben Johnston used twelve tone set techniques in several works, John
Cage/Lejaren Hiller's HPSCHD is a extravagant example of what could be
called chance-driven serialism. The just versions of La Monte Young's
twelve tone _for Guitar_ and _Trio for strings_ are quite elegant. One of
Boulez's earliest works, written in the shadow of Wischnegradsky and later
re-edited with microtones erased, used two intertwined twelve tone rows a
quarter tone apart. Stockhausen has applied his brand of serial techniques
to extremely microtonal woodwind scales in studies for his opera cycle,
Nono applied his brand of serialism to the application of microtonal
(apprx. quartertone) accidentals in several of his late works. Some of the
so-called complexity people work in this direction.

Didn't Ben Johnston use serial techniques in his microtonal
compositions?

Sorry, don't know his work well enough.

--
* D a v i d B e a r d s l e y
* xouoxno@virtulink.com
*
* J u x t a p o s i t i o n E z i n e
* M E L A v i r t u a l d r e a m house monitor
*
* http://www.virtulink.com/immp/lookhere.htm

Ok ... I think I see what people are saying about serialism.

Music based on mathematical structures other than harmonics and regular
rhythm has always struck me as being like writing calculus formulas in
fancy calligraphy - the mathematical beauty of the underlying structure is
just plain not conveyed.

-Bram

>From: bram <bram@gawth.com>
>
>Music based on mathematical structures other than harmonics and regular
>rhythm has always struck me as being like writing calculus formulas in
>fancy calligraphy - the mathematical beauty of the underlying structure is
>just plain not conveyed.

The purpose of musical processes is not always to convey the underlying
structure directly. It is virtually impossible to directly hear (especially
for the non-specialist) all of the "rules" involved in common-practice
counterpoint/harmony, and yet the application of that process creates a
very compelling sort of musical effect -- full tertial harmonies,
resolution of dissonances, independence of lines, and so on.

In such a case, as with most serialism, the purpose is to use musical
processes to create such an overall musical effect, not to expect that the
listener will be able to directly perceive every row transformation like
some sort of theme and variations.

What effects is serialism suited for? Well, it is well suited to create
atonal music because it enforces an equal distribution of the tones of the
set, thus ensuring that no one receives special status through repetition.
Serialism of other non-pitch parameters can similarly enforce an equal
distribution of the set of possibilities, often resulting in irregular
rhythm, inconsistent timbre, and so on.

Whether such a process results in music that you like is another question
entirely, but I don't think you can blame processes that are not directly
perceptible. Many times the indirect effects are at least as important.

Bill

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
^ Bill Alves email: alves@hmc.edu ^
^ Harvey Mudd College URL: http://www2.hmc.edu/~alves/ ^
^ 301 E. Twelfth St. (909)607-4170 (office) ^
^ Claremont CA 91711 USA (909)607-7600 (fax) ^
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

> That sounds a lot like other descriptions I've heard of atonal music.

Atonalism just means that there's no clear tonal center. Atonal music
isn't necessarily serial, and I suppose it's catagorically impossible for
serial music to be tonal (although it would probably be a somewhat contrived
construct I suppose - at least for serialized pitch).

> The
> thing I don't get is, it seems like an awful lot of rules to follow, and
> I've never heard an explanation of the reasoning behind them.

It's probably fair to say that you have to listen for some different or
additional things in serial music than, say 18th or 19th century music. I
confess though that I don't have a really strong ear for, or knowledge of,
serialism.

> Atonal music
> isn't necessarily serial, and I suppose it's catagorically impossible for
> serial music to be tonal (although it would probably be a somewhat contrived
> construct I suppose - at least for serialized pitch).

Oops, sorry. I meant to say, "... and I suppose it's *not*
catagorically...".

>What effects is serialism suited for? Well, it is well suited to create
>atonal music because it enforces an equal distribution of the tones of the
>set, thus ensuring that no one receives special status through repetition.

Having written plenty of the stuff, herein is also serialism's biggest
problem -
the equal distribution of pitch classes gets BORING after a while, monochrome,
like a painting with an equal distribution of color.

Hi Gary:
Listen again to the Berg violin concerto.

Gary Morrison wrote:

> Atonalism just means that there's no clear tonal center. Atonal music
> isn't necessarily serial, and I suppose it's catagorically impossible for
> serial music to be tonal (although it would probably be a somewhat contrived
> construct I suppose - at least for serialized pitch).

At 08:53 AM 2/11/99 -0800, you wrote:
>From: Rick Sanford <rsanf@pais.org>
>
>Hi Gary:
>Listen again to the Berg violin concerto.
>
>Gary Morrison wrote:
>
>> Atonalism just means that there's no clear tonal center. Atonal music
>> isn't necessarily serial, and I suppose it's catagorically impossible for
>> serial music to be tonal (although it would probably be a somewhat
contrived
>> construct I suppose - at least for serialized pitch).
>
Oh, for that matter, the Webern Symphony, op. 21 - and other works by Webern.
There is not a tonal "key", but the fact that pitch classes are often
restricted to a
particular octave on a twelve-note chord causes that twelve-note chord to
be itself
a tonal center (cf. also Boulez, Carter, Lutoslawski, Feldman...)

Ken:
I agree, but the term 'tonal' means specifically
supporting a tonic-dominant relationship
in the piece, no? If not, we could call
many drone-type things 'tonal', which
they are not.

Ken Fasano wrote:

> particular octave on a twelve-note chord causes that twelve-note chord to
> be itself
> a tonal center (cf. also Boulez, Carter, Lutoslawski, Feldman...)

> I agree, but the term 'tonal' means specifically supporting a tonic-dominant
> relationship
> in the piece, no?

Schenker would presumably agree with that statement. I'm not sure if
everybody would agree that it's that specific. It might be more appropriate
to suggest that tonality requires that the harmony as well as melody point to
a tonal center. Although an authentic cadence is the most common way for
harmony to enforce tonal center, there certainly are other ways.