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.