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GEMS 5
==== =

Matthew H. Fields

This is the final article in the GEMS series, a set of five essays
of collected ideas from the oral tradition of musical composition
for the thinking composer.

The story so far:

Shortly after the opening of rec.music.compose in June 1992, I posted
a short note offering to write articles regarding some of the "gems of
compositional wisdom" that have been passed down to me over the years,
and I received an enthusiastic response. GEMS 1 (11 August 1992)
dealt with dramatic shape and the expression of climaxes; GEMS 2 (4
September 1992) dealt with the concept of parallel perfect intervals,
and their implications for melodic perception; GEMS 3 (14 September
1992) was a quick list of heuristics for solving tonal harmonization
homework exercises; and GEMS 4 (17 November 1992) dealt with the
relationship between intellectual materials (e.g. fugue) and
expressive composition.

As posted elsewhere, Nathan Torkington has arranged an anonymous FTP site
for these articles. This is not in New Zealand, as erroneously reported
in GEMS 4, but in Saint Louis, Missouri, USA. The specific sites which
currently carry this series are:

/doc/publications/music-gems @ wuarchive.wustl.edu

/pub/gems @ ftp.hyperion.com

/pub/music/composition @ cs.uwp.edu

I understand there is also a site in Denmark which carries these
articles, but I have misplaced the address. For the time being, they
are also available from me via e-mail.

Lately I've been finding partial runs of GEMS on various Gopher services.
For instance, use your Gopher client to connect to gopher.cic.net;
select -> Electronic serials -> alphabetic -> m -> music-gems.

These sites include an introductory article which I call GEMS 0, which
I intend to include with the series whenever it appears on paper. GEMS 0
gives a little bit of backround on and the situation
in which this series arose.

If anybody out there knows of a site carrying a partial run of this series
that should be carrying a complete run, would they please contact me
at fields@eecs.umich.edu.

I have downloaded GEMS 0-4 to my mac, checked their spelling, and
cleaned up some details of grammar, so they are now available from me
in hardcopy. Soon, I expect to have this article available that way as well.

GEMS 4 has appeared here in rec.music.compose already, and GEMS 1-3
have been posted here twice, so I'm not going to spend the bandwidth
reposting them. Anybody wanting GEMS 0-4 can get them from me in
e-mail.

Enough with the preliminary business, and on with the article.

The topic for today is:

SERIAL MATERIALS: WHAT ARE THEY, AND HOW MIGHT THEY BE USED

This article took me much longer to produce than the preceeding
four. The main reason is that I had to really struggle with what to
present and what to leave out. Finally, I decided to dispense with
all but the barest sketches of history, say fairly little on the
musical literature, condense and simplify the discussion of tonality,
atonality, and modality, put very little energy into preaching to
the unconvertable, and concentrate on what fascinates me most about
this topic: the materials themselves.

In writing this article, I am again indebted to my many teachers, and
particularly to certain composers--Dufay, Monteverdi, Bach, Haydn,
Mozart, Beethoven, Brahms, Mahler, Schoenberg, Berg, Dallapiccola, and
Boulez, to name just a few--whose explorations of compositional
methods have shown the way. My usual disclaimer holds perfectly well
here: theorists may claim to have discovered and copyrighted these
materials by analyzing the works of these composers...but composers
developed them for common use long before published writings explained
them, so they are basically in the public domain.

On the other hand, I worked my @#$)(* off to get these ideas written
out here. So, (c)1993 Matthew H. Fields. Distribution is free, but
don't anybody out there exploit these texts as a commodity without talking
to me. That would be very naughty.

INTRODUCTION

One of the most frustrating aspects of bringing up serial materials is
the way it has been taught in times past. For a brief time, roughly
1954-1963, music-compositional academia gave in to a sort of herd
mentality following the leadership of a few successful serialists.
Many teachers went so far as to require their students to work in
Viennese-style 12-tone serialism exclusively. In the rush to be
academically stylish, "simplified" misrepresentations of the materials
were developed ("First you choose a tone row...."). One particularly
vociferous subculture argued that serial materials were supposedly
new, scientific, rational, and somehow emancipated from traditional
Western culture, which they (the members of this subculture) saw as a
monolith stretching from Gregorian Chant to World Wars I and II. In
fighting a tradition which they associated with Fascism, they enforced
an oppressive approach of their own. Naturally, their students
rebelled, and when they in turn became faculty members (say starting
1965), serialism abruptly became taboo in many corners of
musicianship---or the subject of ridicule. It became associated with
unfeeling intellectualism, disdain for tradition, and the madness of
the artist or scientist who perptrates horrors upon the world out of
"unfeeling curiousity"---and all these associations were, naturally
enough, caricatures of the actual stances of the previous generation.
Gradually, the furor subsided.

Meanwhile, a fairly small number of people continued working on and
passing on a concept of serialism from the 1920's, a concept closely
bound with the traditional objective of matching fascinating
intellectual patterns with passionate expression. It is this approach
I wish to talk about here.

WHERE SERIALISM COMES FROM
As many of us know, serialism was Arnold Schoenberg's 1921 answer to the
question of how to structure atonal music. So what is atonality, and
where did it come from?

To answer this question coherently, we must first ask what we mean by
tonality, in order to ponder what the absense of tonality could possibly
be. More to the point, we will have to ask what musicians in the 1920's
understood by tonality. Now, many of us tend to use the phrase "tonal music"
interchangeably with "music that I like", and when pressed for an explanation,
say that it's music that is restricted to seven-note scales. There are
several reasons why those are NOT the explanations we will use
in this article:

1) Many of us know a lot of music we like that is all for unpitched percussion,
or is some special kind of folk music; in either case the terminology of
tonality never arises.

2) The meaning of "tonality" that was current in the 1920's referred primarily
to 18th-century classical style as exemplified by Haydn, Mozart, Bach, and
others; use of more than 7 pitches was more the rule than the exception in
this style, and in fact was a fairly common though not constant feature of
that musical tradition for the preceding 500 years. Composers like Gesualdo
and Monteverdi cultivated chromatic styles of modal practice that, in many
ways, sound very much like the late-nineteenth-century and early-20th-century
romantic styles of Schoenberg, Strauss, and Debussy---and used 12 or more
families of pitch in the course of a single work.

What qualities of 18th-century style can we point to as defining
tonality? This is quite a technical question, but to give a flavor of
the answer: tonal music was built out of a fairly small number of
standard melodic shapes and patterns of chords (CADENCES), each of
which was treated in a manner roughly approximating a piece of
sentence structure (clause, phrase, subordinating clause,
sentence-completion, etc....). And here's the catch: these formulas
could be heirarchically nested. So a C chord could be decorated by
motion to and from a G chord, and the same G chord could be decorated
by motion to and from a D chord...and each melodic shape in each of
the several melodic strands expressing these chords could be decorated
by various phrases that could stand in place of either a single note
or a pair of adjacent notes...and all these complications were further
subject to considerations of counterpoint like I spoke of back in GEMS
2, so all the melodic strands would make themselves manifest to the
listener...

Like I said, it gets quite technical when you really sit down to try
to understand it. So what did musicians starting in 1907 mean when they
spoke of "atonal" music? Well, any music NOT organized around the fairly
narrow set of concepts present in the music of Haydn and Mozart.

What led musicians to stray from the practices of Haydn and Mozart?

To reflect on this it helps to get just a little bit technical. In
tonal (in our narrow sense) music, while a core major or minor scale
reigned, a key part of standard rhetoric was MODULATION, a calculated
shift to a DIFFERENT major or minor scale. Modulation functioned as
part of the heirarchy: once a C chord had been elaborated into the
chord sequence C-G-C, this could be further elaborated by replacing
each chord with a whole segment of music in the KEYS of C, G, and C.
The move to G involved the substitution of of F# for F in the scale.
So the appearance of this F# was potentially an important event, since
it marked a turning point in the grammar and rhetoric of the music.

As musicians worked with this grammar in the 19th century, they
gradually extended it in all directions, first by applying all the
available transforms to every possible moment, then by adding some
phrases from folk musics (which remained true to earlier traditions)
to the set of possible transforms...then adding more transforms. Each
such extension brought with it more and more frequent use of notes
outside the basic seven-note scale. Finally, the act of expanding a
single pitch into a chord, and a chord into a key, and thence into an
audible heirarchy of keys, became more of a post-hoc explanation for
expressive musical practices. New pitches occur often enough in,
e.g., the prelude of Wagner's Tristan und Isolde, that they no longer
have the specialness, the markedness, the rhetorical power that the
turning-point F# had in a C-major composition 100 years before. Many
musicians were using other traditional means of organizing their
works:

a) around the rhetoric and poetic images in a sung text;
b) around a story or drama;
c) around a surge to a climax, without reference to a specific story;
d) around motifs---short bits of melody, harmony, rhythm, and tone color
which were repeated and endlessly varied throughout their compositions,
so any given piece would continuously evolve and at the same time
continuously state its identity.

Organization principle d) above was known as "organicism", from the concept
that an entire composition grew "organically" from the seedling of one or
two simple, memorable motifs.

These principles were also actively used by the composers of Mozart's
days... and for hundreds of years before (Mozart's generation used
them in conjunction with the narrow grammar and conventions of
tonality). But the language evolved idiosyncratically until some of
its organizing principles were no longer recognizable, while others
came to dominate.

Since Arnold Schoenberg (1874-1951) first expressed (in his Suite op.25)
the concepts we're talking about here, let's look at the values Schoenberg
wished to preserve:

1) Organicism, the building of compositions from repetition and recognizable
variation of small, cellular, distinctive segments.

2) Awareness of the push and pull between consonance---the perception of
synergy among several sounding tones---and dissonance---the perception of
disbalance among several sounding tones, with the understanding that these
tones belonged to melodic strands that would soon move into a state of
consonance. The exploration and constant redefinition of consonance and
dissonance and motion between them was an area of continual experimentation
in the previous 8 or so centuries.

3) Constant expression of forward motion or dramatic change through the
constant introduction of "new" pitches, i.e. a continuance of features we might
hear in the Tristan Prelude we looked at a few paragraphs ago, or the
pitch language of e.g. early Baroque-era madrigalists like Gesualdo.

4) Familiar patterns of drama, verse-structure, and other overall forms.

5) A fluid perspective on melody (sequential tones) and harmony
(simultaneous tones). Schoenberg wrote that he felt melody could melt
smoothly into harmony and vice versa, through the persistence of
memory. He was referring, of course, to the concepts of arpeggiation
(playing of the tones of a chord sequentially), compound melody
(timesharing between two separately-perceiveable melodies played or
sung by one sound source), and similar devices which had been
developed over the preceding 600 years. What was perhaps new about
Schoenberg's attitude, as we shall see, was an interest in using
patterns of arpeggiation of a small number of chords as the melodies
in a work--- a new attitude towards organicism that he hoped might
make non-tonal music stick better in the listener's memory. The
several melodies in a contrapuntal texture might each be an
arpeggiation of a chord similar to each of the chords arising in the
music, for instance. Or, a tune could be presented with each tone
sustained somehow, so the final effect would be of a ringing chord.

6) Some sense of markedness, of something special announcing a turning
point or structural point in a piece. Since his style now called for
using all possible tones most of the time, the classical idea of the
New Pitch (e.g. F# in a piece otherwise in C) wouldn't be very
effective. Schoenberg took a backwards approach and suggested that
the return of an Old Pitch Class that had been momentarily absent
might sound like a milestone or marker. In 12-tone equal temperment,
an arrangement of all twelve pitch classes would be simply the longest
phrase you could build before having to return to an old pitch. So even
years before he started working with 12-tone rows, Schoenberg noticed a
tendancy for his phrasings to be clumpy, with each clump containing
ten, or eleven, or twelve different pitch classes.

Before we watch a 12-tone row at work, it will be illuminating to see some
of the ways Schoenberg approached these matters in the decades BEFORE he
formulated his "system". But to do that, we'll need some technical
terminology.


DEFINITIONS
In the past my Definition section has been pretty short and minor; this
time I'm loading it with some bulky, nutritious ideas, so if you're skimming,
please don't skip this section.

By "pitch" I mean a single (percieved) tone; for the acoustically minded,
that's a single fundamental frequency.

For the bulk of this article, I will be talking about Schoenberg's
approach to serialism, which assumed the use of 12-tone equal
temperment, the division of the octave (acoustically, the 2:1
frequency ratio) into twelve equal semitones (ratios of the 12th root of
two = ~1.059463094359). So in this parlance, middle c = B#=dbb. Most
grand pianos are constructed to play 88 pitches.

By "pitch class", I mean the closure of a pitch under octave
transposition. A pitch class can be identified by a representative.
So, the pitch class of g"-flat is the collection of all g flats,
whether written as g flat or f#, without regards to octave. Most
grand pianos are constructed to convey 12 pitch classes.

The grouping of phenomena into classes like this isn't new to musical
thought. In fact, the idea of referring to all octaves of g-flat as
g-flat is very old.

In most of what follows, I will be speaking of pitch classes rather than
actual pitches, and I may informally slip into using the term "pitch" to
mean "pitch class"...but my meaning will be clear from context.

By "pitch collection" I mean an unordered set of pitches. The act of
collecting them might, in the course of a composition, be expressed by
playing them together as a chord, by playing them sequentially as a
tune, by assigning them all to the same instrument, by grouping them
all in one register while other pitches might be sounded --- all much
higher or all much lower than these --- or by many other means that
the composer finds expressive. All that is implied a priori by
"collection" is that the composer is somehow going to group these
pitches. So, for instance, the open-position triad C-G-e is a pitch
collection --- the same collection as BB#-G-fb.

At this point in the basic definition process, it becomes handy to
introduce numerical names for pitch classes. I will be using a bit
of simple arithmetic to help formulate some of the ideas in this paper.
Before I do so, let me point out that calling a pitch class "zero" instead
of "C natural" does not in any way denigrate it or subvert its expressive
potential beneath a mad scientist's algebra. It's merely a naming
convention that proves expedient. In fact, I think this system of numbers
is slightly simpler than the numbers used to describe Mozart's practice.
Consider a typical statement from classical theory:

^ ^ ^ ^
2 1 7 1

6 -- 5
6 4 -- 3
ii V I

This series of symbols describes four chords, specifying their melody
notes, bass notes, and providing enough information to formulate the
middle notes, while at the same time stating their function relative
to the rhetoric of a major key...without identifying the key. It uses
carat-decorated Arabic numerals to indicate scale steps of individual
tones, lower-case Roman numerals to indicate scale steps of root notes
of minor and diminished chords, upper-case Roman numerals to indicate
scale steps of root notes of major chords, and unmarked Arabic
numerals to indicate displacements of chord tones relative to
whichever chord tone happens to be being played lowest.
Numerical names for things is really nothing new.

Or: musicians are accustomed to doing (or faking their way through) arithmetic
to make sure the notes they've written add up to the length of a measure.
We won't be looking at anything harder than that here.

As you may have guessed from the fact that we're (for the moment) using
12-tone equal temperment, I need 12 different numbers. For reasons of
convenience that will become obvious later, the symbols I choose are
not one through twelve, but zero through eleven. To save space, I will
write "t" for ten, and "e" for eleven, so all my numerals are single
digits, and can be written without spaces and without confusion between
1,1 and eleven. The set of names: {0123456789te}

I adopt a system which I call "fixed zero" in which 0 always represents
the pitch *CLASS* c natural, 1 always represents the pitch class db/c#,
2 always represents the pitch class d natural, etc. Some other authors
use a "moveable zero" system, in which the meaning of the number 0 is
assigned on a per-composition basis, and might typically be some important
pitch of a composition, like the first pitch sounded. Each system
is convenient for describing certain composers' works, much as moveable-
and fixed-do solfege systems each have their advantages.

The twelve pitch classes form a cycle which I like to diagram using the
twelve-tone clock face, to help express the concept of modular arithmetic:

Fig.1. Z-12. 0=all C naturals, B#, and Dbb. 1=all C#,B##, and Db.
2=D nat.,C##, Ebb.
0 3=D#,Eb, Fbb. etc.
e 1

t 2
* *


9 3
* *
* *
* *
8 *********** 4

7 5
6

I define the INTERVAL BETWEEN TWO PITCHES as the distance between
them, in the conventional way; the INTERVAL CLASS between two pitches
or two pitch classes is the distance between the numbers on the
circle, *the* *short* *way* *around*. This means that, for instance,
minor 3rds and major sixths are grouped under one big family heading,
IC 3 (distance of 3 semitones the short way around). So I'm ignoring
octave placement for *both* tones, and considering them in terms of
their pitch classes. Notice that (unisons and octaves aside) there
are only 5 interval classes. Once an interval gets larger than IC 6
(a tritone) its octave complement becomes the shorter way around the
circle. So, for instance, IC 5 groups together all perfect 4ths,
perfect fifths, perfect 11ths, perfect 12ths, perfect 18ths, perfect
19ths, etc. Notice also that if we impose an order on a pair of
pitches, we can speak of ascending and descending intervals. To
capture equivalent information regarding pitch CLASSES, we define
DISPLACEMENT CLASS as the modulo twelve DIFFERENCE between the
numbers--which depends on their order. So between middle c and the
second A natural below it, the directed INTERVAL is a descending minor
tenth, or minus 15 half steps; the INTERVAL CLASS is plus 3, and the
DISPLACEMENT CLASS is plus 9 (which means descending minor third or
ascending major sixth or some compoundment).

Remember: INTERVAL gives information about the number of octaves compounding
an interval; directed interval gives the same interval plus a direction.
Interval class gives the smallest distance between the notes without
regard to octave. Displacement class gives either interval class or
its twelve's-compliment, and thus gives information about order without
information about octave.

info about octave
yes no
+-----------+------------+
yes| directed |displacement|
| interval | class |
info about direction+-----------+------------+
| interval | interval |
no| | class |
+-----------+------------+

If you take an ordered pair of notes and reverse their order, the
interval between them is the same, but the directed interval has the
opposite direction. The interval class remains the same. But the
displacement class is replaced by its octave complement, that is,
twelve minus the old displacement class. So if instead of descending
from middle c to the second A natural below it, we play the same two
notes in revers order, the interval is still a minor tenth, but the
directed interval is an ASCENDING minor tenth or PLUS 15 half steps;
the interval class is still 3, and the displacement class is now 3.

In a few minutes I will define a concept called COLLECTION CLASS.
Given a pitch collection---say, for the duration of this paragraph
only, we call it P---we may assess the way intervals lie in it, and
find all other pitch collections V(P) that have the same interval
classes lying in it in more or less the same way. This is interesting
to an organicist composer because, if the composer has in mind some
motif M where all the notes of M are members of P, this composer might
want to look at exactly the set of variants V(M) that are suggested by
V(P), as a source of materials both different from and at the same
time closely related to M. In so doing, the composer will look at
motif M abstractly in terms of collection P and collection class V(P)
(we'll introduce some simpler notation in a few paragraphs) in order
to help concentrate on the properties they wish to work with (this
abstract approach may be initially uncomfortable to many musicians but
will be especially attractive to those who also indulge in theoretical
mathematics; again, it's no more atypical of musical thought than the
concept of a tonic and dominant, which are, of course, abstractions of
specific chords). Now, suppose I plot the members of P on the clock
face by making a mark around the numbers of each pitch. It should be
intuitively clear that the set of interval classes (distances between
marks) in this set of markings is determined by the exact SHAPE of the
set of markings, NOT by the particular numbers marked. The intervals
involved (and thus the melodic properties, or, in some sense, the level
of consonance or dissonance implied by the chord) remain the same if I
pick up the set of marks and rotate it AS A WHOLE with respect to the
bunch of numbers: the distances between marks remains the same. What's
more, if I pick the set of marks up and flip it over so it shows it's
mirror image to us, the set of intervals is STILL the same...and the
way they present themselves to us differs only very subtly.

Now, if I take my motif M that lies in P and carefully transpose it so
that the resulting transposition preserves, to the semitone, the size
of the original intervals, I get a motive T(M) that preserves some
properties of M but is higher or lower. If I plot the notes of T(M)
on my circle, I will see that I still have the same shape as P, but it
has been rotated. So, rotation on the circle is an abstract kind of
transposition.

Furthermore, if I take my motif M and invert it about some center or axis,
so all it's ascending intervals become descending intervals and vice-versa,
I get a motif I(M) that preserves the size and proxmity of intervals, but
reverses their directions. In Western language, this amounts to swapping
questions for answers and answers for questions...or creating a response
to a call or a call for a response. If I plot the notes of I(M) on my
circle, it may come as no surprise that they now form a mirror image of P.
So mirror-reflection is an abstract kind of inversion.

This is getting a bit heavy, so let's take time out for a story. Richard
Hoffmann, professor of composition at Oberlin College, and co-editor of
the Schoenberg Collected Works, explains the idea of collection class this
way.

HOFFMANN: (holds up a Swiss Army Knife): All right, class, what have we
here?

CLASS (ALL EXCEPT FOR FIELDS): A collection class.

FIELDS: A pen-knife.

HOFFMANN: (turning to Fields) All right, smart-aleck, NOW what
do we have? (turns his pen-knife on it's side)

FIELDS: Um, it's still
a pen-knife?

HOFFMANN: (grinning so nobody can tell whether he's happy or has just caught
Fields in a major boo-boo) You are co-RECT! Now, class, who wants
to tell me, (turns his pen-knife upside down) what have we here?

CLASS (ALL): A pen-knife!

I really don't know what we would have said or done if he had ever
unfolded the knife. But apparently he didn't think we'd ever encounter
THAT serial operation.

As I write this I'm nagged by the thought that many of you might not
realize just how often and for how long composers have turned to these
two concepts---transposition and inversion---to create musics varied
within unity. Consider that when in 1750 Bach wrote Art of Fugue, the
consequences of using these tools had been explored and catalogued for
over 500 years. If you're really unfamiliar with these ideas, you
might want to go back and listen to Art of Fugue now (I recommend
Musica Antigua Koln's CD) and become aware of how Bach takes his short
opening tune and subjects it to transpositions, inversions, changes of
meter, changes of tempo, changes of ornamentation, etc. while always
keeping it recognizable---and strings together all these variants into
expressive, dramatic shapes.

Ok, back to work.

Now, right here in the definition section, come the two main tools of
serial thought: the serial concepts of transposition and inversion, which are
abstractions based on the classical concepts with the same names, but with
this reductionist octave-ignoring attitude in place. These are usually
considered the main serial operations because they are the only operations
which maintain the shape of ANY collection of pitch classes. Sly serial
composers sometimes match special collections of pitch classes with
other special operations because those operations maintain the shape of
those particular bunch of notes (linear algebraicists: eigenvalue alert!).
It is my opinion that anybody who explores serial materials can find these
special operations when they are needed and useful, so I'm going to
leave them out of my subsequent discussion.

AND WHAT ABOUT RETROGRADE?

Well, you're getting ahead of me here. I've been talking about
unordered sets, and retrograde is an operation on ordered sequences.
All in due time.

A BIT OF NOTATION
I'm about to start using some notation, so let me give you some
idea what I'm talking about. By example:

PITCH CLASSES
PC0 Pitch Class 0

COLLECTIONS OF PITCH CLASSES
{014} The unordered collection of 3 pitch classes: PC0, PC1, PC4
{401} Same as previous

CLASSES OF UNORDERED COLLECTIONS OF PITCH CLASSES
(014) The collection class (shortly to be defined) having {014} as its
canonical representative. Also called CC014.

ORDERED SEQUENCES OF PITCH CLASSES
[014] The ordered sequence of three elements, where the first element
is PC0, the second element is PC1, and the third element is PC4.
[401] The ordered sequence of three elements, where the first element
is PC4, the second element is PC0, and the third element is PC1.

CLASSES OF ORDERED SEQUENCES OF PITCH CLASSES
401 The sequence class (shortly to be defined) having [401] as its
canonical representative element. This has the least punctuation
on it because I plan to use it a lot.

TRANSPOSITION
If you have any group of pitch classes marked out on the twelve-tone
clock face and you rotate it so the number 0 is now where the number N
(for any N in Z-12) used to be, you will have TRANSPOSED your group of
pitches N steps. The operation you have performed is modulo 12
addition: you added N to all the numbers you started with, and
subtracted 12 from any that went higher than eleven. We write:

T {abc}={a+N b+N c+N}... (modulo 12 operation is implicit)
N

So a B major triad, B-D#-F# or {e36}, could be transposed up a minor third
(IC 3) by this operation:

T3{36e}={269} = D-F#-A, a D major triad (see, it does what we expect it to).

T4 3 = 7, i.e. transposing the note Eb up a major third gives G.

Arithmetic check: we said this rotation should move the number zero
to the number N. TN 0 =0+N =N, so everything we've said is consistent.

Since we are working in modulo 12 arithmetic, it should be clear that
I've defined 12 T operations: T0 (the do-nothing operation), T1,
T2, ... T9, Tt, Te. My choice of the numbers zero though eleven instead
of one through twelve should now be clear: I chose my set of numbers
so I could cheaply steal the existing language of modulo arithmetic
to express myself.

We should notice that the index N of the transposition operation

T is not a pitch name, but rather a measure of the absolute interval
N

through which a pitch class must be rotated clockwise on the clock.

And remember, while there are twelve transposition levels, there are only
five interval classes: zero doesn't count as an interval class, and 6 is
the greatest distance between two points on the clock face.

It may prove handy to get a small disk of transparent material and
mark our chosen set on that while holding it in front of the clock face.
Then we can freely rotate the transparent disk relative to the clock.

INVERSION
Or we can pick up the disk and turn it over so we see the mirror
image. Let's choose an axis on which to flip it over. This axis will
pass through its center, and will either lie on a line connecting two
numbers that are 6 places apart from each other (e.g. a line from 2 to
8), or it will lie on a line that passes between two numbers (e.g. a
line from halfway between 2 and 3 to halfway between 8 and 9). Once
again, it should be clear that there are 12 such axes, and each of them
exchanges position 0 with a different position on the clock.

If we have some set of pitches marked on our clock (or on our transparent
disk which we superimpose on the clock) and we flip them into mirror image
in such a way that the numbers 0 and N would trade places, the new marked
set of pitches is the Nth inversion of the original. We write:


I {a,b,c}={N-a,N-b,N-c} and note that the operation of mirror imaging
N

is accomplished by subtracting from a constant.

Arithmetic check: We said the Nth inversion makes pitch classes 0 and N
swap place.

I 0 = N-0 = N I N = N-N = 0
N N

so again our arithmetic appears to do exactly what we said it does.

COLLECTION CLASS
Now we can, working backwards to get what we want, define collection class.
Given any pitch class collection P, the collection class generated by P
is the closure of {P} under transposition and inversion.

What are these collection classes? Well, for one thing, all members of a
given class have the same number of different pitch classes in them. In
some sense, they all have the same distribution of interval classes within
them...and so in a sense they are all at a single level (or narrow band
of levels) of consonance and dissonance.

Let's look at a typical collection class: (037) This class is named
for its canonical representative, a c-minor triad. It includes ALL
minor triads, by transposition; by inversion, it contains all MAJOR
triads as well. So this class contains 24 different unordered
collections. We choose a standard representative so that we can tell
easily whether two chords belong to the same class (by comparing the
standard representatives of their classes). The canonical form of a
collection is found by plotting it on the circle, finding (inspection
is usually as good a means as any) the shortest bracket which wraps
around all the marks on the circle, rotating the marks so the
counter-clockwise end of the bracket is at zero, and optionally
flipping the marks into mirror image so the counter-clockwise end of
the bracket remains at zero and most of the marks cluster towards the
lower numbers... the formal literature gives a formal definition of
canonical form, and I think it's a bit too much of a technicality to
warrant my dwelling on it much here.

COLLECTION CLASSES IN ACTION---A FEW BARS OF SCHOENBERG OP.16
: : :
_ |\ | : |\ |\ +-------+ | :
3 _/. |\ | : | | |\ | : | | :
8 / |\ | : | / |\ | : | | :
x x : x.. x x : x x :
: \_______/ :
: : :
cellos e f : a g# a : c'# :
clarinet 1 d c# : Bb C Bb : A :
clarinet 2 G F# : Eb E Eb : D :
: : :

Thus (with a scampering motion in the contrabassoon and contrabass
clarinet) begins the first of Arnold Schoenberg's Five Pieces, Op.16,
a work from 1908 (revised 1922), 13 years before his first work of
twelve-tone serialism. It is not at all irrelevant to consider that
Schoenberg had already completed most of his smash hit oratorio,
Gurrelieder, and had completed voluminous amounts of unpublished works
demonstrating his adeptness as a romantic, late-nineteenth-century-
style composer. Late in the working out of the last movement of his
second string quartet, he announced an awareness that while he was working
from organic principles, he was no longer using vestiges of 18th-century
tonality as guiding principles. His settings for mezzo-soprano and piano
of Stefan Georg's Poems from the Book of the Hanging Garden continue a
firmly romantic, lush sound while further exploring the ramifications of
non-tonal organicism. And then we have these five orchestral pieces,
each depicting a different mood while elaborating on a different experimental
approach to organicism. By considering just these first three bars in terms
of collection class, I hope to at once intrigue you to listen to and
explore the entire set (look for performances with, e.g., Pierre Boulez
conducting), and also to shed light on the thinking that preceded the use
of tone rows.

Let's look at that 3 bars again, and see what we observe.

_ |\ | : |\ |\ +-------+ | :
3 _/. |\ | : | | |\ | : | | :
8 / |\ | : | / |\ | : | | :
x x : x.. x x : x x :
: \_______/ :
: : :
cellos e f : a g# a : c'# :
clarinet 1 d c# : Bb c Bb : A :
clarinet 2 G F# : Eb E Eb : D :

Well, the second clarinet seems to be moving in contrary motion to the
cellos, with similar, though not identical, intervals. The first clarinet
is moving in parallel fifths with the second clarinet (as you may recall
from GEMS 2, classical composers either use parallel fifths constantly or
not at all)...but then there's the odd note out, the concert middle c in
the middle of the second bar. Suppose the first clarinet had gone to B
instead, and thus maintained its parallel fifths with the second clarinet.
Then, suddenly, in the middle of the bar, the 3 sounding notes would be
g#, B, E: an E major triad, or a very restful sound in the middle of the
phrase. Schoenberg has apparently adjusted the first clarinet part by a
semitone to keep the phrase moving forward into the third bar.

Another thing that strikes the ear is that the cello line consists of
two statements of a 3-note motive, with the second statement
transposed up a major third (4 half steps). Both statements of the
motive are from (015), as you may verify by plotting the notes on the
twelve-tone clock. But the first and last sustained chords---the
second beat of m.1 and the second beat of m.2---are also from (015),
as again you can verify. It's worthwhile at this moment to sit down
and play those two chords, and also play out the tune. The chords
are derived from the tune, and the tune from the chords.

Fig.2. Trichords of the cello melody, mm.1-3 of Schoenberg op.16 No.1

0 | 0 * |
e 1 | e 1 |
| * |
t 2 | t 2 |
| |
first | next |
| |
*9* three 3 | *9* three 3 |
| |
notes | notes |
* | * |
8 4 | 8 4 |
* * | * |
7 5 | 7 5 |
6 * | 6 |


Fig.3. Two sustained harmonic trichords, mm.1-3 of Schoenberg op.16 No.1

0 * | 0 * |
e 1 | e 1 |
* | * * |
t 2 | t 2 |
| * |
m.1, b.2 | m.3, b.2 |
| |
9 3 | *9* 3 |
| |
| |
| |
8 4 | 8 4 |
* | |
7 * 5 | 7 5 |
6 * | 6 |
*

So, in a way, Schoenberg's construction resembles a crossword puzzle.
Such tightly-woven multidimensional construction is typical of
classical music---it's exactly the kind of thinking that goes into
counterpoint.

Just a couple more observations should suffice to give the aroma of
his thinking. The three-note cello motif that starts the piece is one
of 5 motifs presented in the 25-measure introduction, all of which
saturate the rest of the movement from then on. The form of (015)
that ends the opening 3-bar phrase----the chord c#-A-D---is sustained
as a triple pedal point (drone) from m.26 to the end of the movement
in m.128. So, in a sense, the chord at the end of the phrase
foreshadows the 102-measure drone that ties together the bulk of the
piece. The movement has the programatic title "Vorgefuele"
(fore-sensations, that is, premonitions)... and the opening 25 bars
present all the materials---all the threats---that are realized in the
main drama of the piece.

THE EVOLUTION OF COMPOSITIONAL IMPULSES INTO A SERIES
Ok, so it's Monday morning, and Composer X wakes up shouting this
tune:
| | |\ : _
4 | | | |: /.\
4 | | | /:
O X. X : O
:
: b-flat
e :
B :
F :
_____
ff -----=====/ sffz
\-----

"Blammo. Hmmm." After a sip of coffee, the language centers in
Composer X's brain begin to stir.

"French horns," he mutters, "four french horns. Maybe six. In
unison. Cool."

After another sip, he goes and picks up his cello, and plays the
notes.

"Mmmm. Not going to work very well as a tonal tune, nooooo....."

A cat appears, rubs his leg, meows, jumps up on his shoulder, and
glowers. As he runs downstairs and feeds the cat, he continues
working.

"I like the assertiveness of that four-note motive. I think I'll call
it the Check-Mark motive because of the melodic shape it takes.

"Eeeeeee, fiiive, four-TEEEEEEEEEEEEEEEE! Hmm. It's from (0167). So
it'll invert onto itself, like this: Foooooour, teeeeee, eee-
FIIIIIIIIIIIIIIIIIIVE. This also reverses the order of the diads [e5]
and [4t], but keeps the notes within each diad in its original order.
Cute. The operation is I--- um, I3 [e5 4t] is [4t e5], but I also have
I9 [e54t], which is [t45e]. And I have T6 [e54t], which is [5et4].
So that gives me 3 operations relating this motif to a permutation of
its pitch classes while retaining the sequence of its intervals. Well,
not really the sequence of intervals, but each either has all the same
displacement classes in order, or it has all the complements of the
displacement classes in the same order. So I get either Checkmarks
within these four notes, or upside-down Checkmarks in the same four
notes."

Kitty meows at him as if to say, why are you blathering at me like
that. He ignores Kitty and goes over to the piano. First he plays
his little motif, sustaining the notes with the pedal, then looks up
in glee and says "Let's try T3." He plays the same notes up a minor third:

c'#
g
d
Ab

The pedal is still down. He thinks he hears something he likes, so he
plays the two tetrachords over again quickly:

a# c'#
e g
b d
F Ab

And then it dawns on him. "It's a @#$ @#$(*& octatonic!" Just to make
sure, he reorders the notes in scale order, to verify that they alternate
whole-step, half-step, whole-step, half-step...

whole steps * * * *
f,g,g#,a#,b,c#,d,e,f...
half steps * * * *


"Miu?"

"Oh, look, kitty, this is simple stuff, but it sure is fun. And I was dreaming
of big natural forces when I got going on this tune, so that'll be the
program for the piece.

"And I like the idea of following up Checkmark with T3 of Checkmark to make
an octatonic. But unless I want to write yet another commentary on
Messiaen's Abyss of the Birds, I'm sooner or later going to have to bring
in the other four pitch classes. Lessee, an agregate, take away an
octatonic, leaves what? A full-diminished seventh. c, e-flat, f#, a, I
don't know what order yet. I think I'll be a bit flexible about the
order of that T3 of Checkmark, too, because I might stumble on some
reason to rearrange it. Ok, so I have a kind of music going on here
that uses a lot of different pitch classes, maybe all of them. So it's
going to organize into little clumps, where the beginning of a new clump
is kinda marked by the return of a tone from the previous clump. All
the clumps are going to have so many tones in them that I really can't
worry anymore about distinguishing one from another based on which tones
they do and don't have in them, like I could with major and minor scales.
About all I have to work with is the order of the tones within each clump."

"Mew."

"Yeah, I know. Big deal. But let's see what I've got now.

[e54t] {1278} {0369}
/ | \
"This is ordered "Unordered, for "Unordered.
in a definite order, now. I'd like it to It's the standard
because I started out with be a recognizable member of (0369).
the Checkmark Motif and I'm variant of the Checkmark. But I don't like
holding on to it. It's a It's another (0167)." it. Uh, oh."
member of (0167)".

"Let's look at these on a clock face, and see what else I learn."

Fig.4. Three Tetrachords.
E,F,Bb,B={45te}=.A. C#,D,G,G#={1278}=.B. C,Eb,F#,A={0369}=.C.(unmarked)

.A. 0
e 1 Vertical and
.A. | .B. horizontal
t \____ | ___/ 2 axes reflect .C.
\___ | ___/ .B. on self, .A. & .B.
\|/ on each other.
9 ----------*---------- 3 Diagonal axes
___/|\___ reflect all on
.B. ___/ | \___ selves.
8 / | \ 4 Rotate .A. 6 stations
.B. | .A. to get .A., 3 or 9 to
7 5 get .B. Rotate .C.
6 .A. 3,6, or 9 stations
to get .C.

"My first four notes sound so strong, and so do my next four notes. But
my last four notes are a full-diminished seventh. They sound so wimpy. How
could I fix that? Well, I can think of two ways right off the bat. I could
change my choice for the middle four notes so I'd get different notes for
the last four notes...but then I'd be giving up that lovely octatonic. Or
I could promise myself that I'd always sound an additional note or two
from the middle four notes when sounding the last four notes. Let's see."

Composer X dabbles around at the piano, playing a full-diminished seventh
with his left hand, while adding tones with his right hand. He soon realizes
that he gets the same 5-note collection class no matter what one note he
adds to the (0369). Any way he looks at it, it's (0147t). "So what."

He starts picking notes from his middle tetrachord to go with the dim7.
"Why should the extra notes come from the middle tetrachord? well, because
it's already right next to that last tetrachord. Wait, what's this?"

A
... ... D F#
Ab Eb
C

"What a pretty little hexachord. Sounds familiar. Oh, yeah,
Stravinsky popularized the same notes (down a whole step) in his
ballet, Petruchka, and so it's called a Petruchka chord. And
Stravinsky made a big deal of the fact that you can regroup it into
two major triads, with their roots a tritone appart. In this case I
have a D major triad and an Ab major triad. If I transpose it by a
tritone, I get the same six notes. Ok, so what do the OTHER six notes
look like?"


Checkmark motif two more notes
\ Bb /
\ E Db / ... ...
B G
F

"Ok, well, that's like an e minor triad on top of a Bb-minor triad, again with
a tritone between e and Bb. So I'll call this a "minor Petruchka chord", and
call the other one a "major Petrucka chord" to distinguish them. I know
the inversion of a major triad is a minor triad, and vice versa, so I bet
some inversion will relate these two hexachords. Let's plot it out on
a circle and see what it is."

Fig.5. Two Hexachords. C#,E,F,G,Bb,B=[1457te]=.X. C,D,Eb,F#,G#,A=[02369t]=.Y.
(.Y. is unmarked)
Rotate .X. 6 stations to
.X. 0 .X. get .X., and same for .Y.
e | 1 Reflect on the shown axis
.X. | to swap .X. and .Y.
t | 2
|
|
|
9 | 3
|
|
|
8 | 4
| .X.
7 | 5
.X. | 6 .X.


Kitty jumps on Composer X's lap, sprawls on the music paper pad, and looks
contented.

"Yes, you're right, kitty, I1 of .X. is .Y., and I1 of .Y. is .X. Let's
see where I've gotten. I started with my Checkmark Motif,

[e54t]

and then I added four more notes up a minor third to make an octatonic:

[e54t] {1278}

"Then I realized that I would eventually want the remaining four pitches, so
I could move beyond that octatonic:

[e54t] {1278} {0369}

"Then I got annoyed at the last four notes, and decided to combine them
with two notes from the middle four to get a Major Petruchka chord; gratis, I
got that the first six notes would be Minor Petruchka chord.

Draft-series: [e54t] {17} {28} {0369}

"So I could move from music based on the Checkmark Motive to Petruchka chord
music just by bringing in two more voices. And I noticed that I1 would
swap the left and right hexachords:

Draft-series: [e54t] {17} {28} {0369}
\________/ \_________/
\ /
X
________ / \ _________
/ \ / \
I1 Draft-series: [2896] {03} {e5} {147t}

"So I could use that relationship to 'modulate' to a new set of notes for my
Checkmark Motive, while maintaining the hexachords. I'd achieve this
'modulation' by first adding two more voices so I'd shift my emphasis to
hexachords, then substitute I1 of my material so I'd get the same hexachords
in the opposite order, then shift my emphasis back to tetrachords.

"And I remember from Figure 4 that T6, I3, and I9 maintain the content
of my Checkmark while permuting its order. What do they do to the
partial order I've got so far for all twelve notes?

Draft-series: [e54t] {17} {28} {0369}
T6 Draft-series: [5et4] {17} {28} {0369}
I3 Draft-series: [4te5] {28} {17} {0369}
I9 Draft-series: [t45e] {28} {17} {0369}

"T6 rearranges the notes within their groups, but leaves the groups in the
same order. I3 and I9 both swap the middle diad. What happens to the
hexachords, then? The first 6 notes now spell a MAJOR Petruchka chord,
Bb major plus E major, while the last six notes now spell a minor Petruchka
chord, C minor plus F# minor. So if I am ambiguous about the order of
the first four notes I play, and, say, play them [e54t] at one time and
[t45e] another time, I can, lessee, if these are horns, I can bring in
a couple bassoons playing either {17} or {28}, so I have my choice of
major or minor sonorities. Right, Kitty? ---Kitty?"

Kitty comes running from beyond a doorway and looks up expectantly. Composer
X strokes Kitty a bit while. "Ah. There you are. Did you get all that?"

"Mrrrrrrrrrrrrrr."

"Let me play that last bit for you at the piano, just to make sure it sounds
right.

Minor Petruchka Chord Major Petruchka Chord

B-F-E-Bb G E-A#-B-F G#
C# D

"Ok. Now, what part of my material has the least order already
imposed on it? Well, the last four notes make a chunk of length 4
with no particular order, {0369}. I'm beginning to really look at order
a lot here, so it'll help if I have names for the POSITION of notes in
this order. Just for fun, I'll number THEM zero through eleven too.


POSITION NUMBERS 0123 4-5 6-7 8-e
Draft-series: [e54t] {17} {28} {0369}

"Ok, the tetrachords at positions 0-3 and 4-7 are both members of (0167).
If I take the {28} that lives at positions 6-7 and combine it with {39},
which I'll take from the notes in positions 8-e, I get {2389}, which is
ANOTHER member of (0167). Ok, so that means I'll put {39} in positions
8-9, and in positions t-e I'll have {06} left over. That's also neat,
because if I repeated the series immediately, {69} would be followed by
[e5] and those FOUR notes would again add up to a member of (0167). So now
I have a more specific ordering of pitch classes, that has my Checkmark
Motive for the first four notes, a variant on it for the next four notes,
an octatonic for the first 8 notes, a Minor Petruchka chord for the
first six notes, a Major Petruchka chord for the last six notes, a variant
of the Checkmark Motive in positions 6-9, and another variant of it in
positions t-e plus 0-1."

POSITION NUMBERS 0123 4-5 6-7 8-9 t-e
Second-Draft-series: [e54t] {17} {28} {39} {06}

"Now, here's an interesting effect of my choices. Each of the pairs, in
POSITIONS 0-1, 2-3, 4-5, 6-7, 8-9, and t-e...each of those diads is a
tritone, i.e. a member of Interval Class 6. So if I wrote a music out of that
for two players, where each diad between the players was one of THESE diads,
they could be in parallel motion.

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

Let's play that at the piano:


B Bb G G# A F# F E C# D D#
F E C# D D# C B Bb G G# A


"If somebody plays that really really fast, that'll make nice swirling
motions...which I could programmatically associate with winds and turbulent
rivers, to continue my metaphor of forces of nature.

"Speaking of tritones...well, I was speaking of tritones, wasn't I?...
I noticed that each hexachord, positions 0-5 and 6-e, was made of a pair
of similar triads. Let's see if I can do something to bring those triads
to the fore. Lessee, the first six notes contains an e-minor triad,
E-G-B, or {4-7-e}. Where are those tones? Well, PC e is in location 0,
and PC 4 is in location 2. I can't wedge PC 7 between them because I'm
already committed to delaying PC 7 to positions 4 or 5, so I can have my
Checkmark Motive in positions 0 to 3. But I could have PC7 in position
4, and now I've got a pattern: every second note is a member of this
triad.

Hmmm... suppose I were to write the notes alternating between
short clipped high notes for the piccolo and percussion, and long low notes
for some other group, and I did it so that the major and minor triads
came out.

piccolo [e 4 7 2 9 6]

others [5 t 1 8 3 0]

"Let me try it at the piano. I'll play the piccolo line in the top octave,
and spread the other notes out down low, and I'll sustain them with the pedal
so I can hear those triads more clearly.

piccolo B E G D A F#


C-------
Ab---------------------
Eb--------------
Db-----------------------------
Others
F---------------------------------------------

Bb-------------------------------------

"Sounds kinda pretty to me. It looks like I have another hexachord to
work with, a diatonic hexachord, one note short of a diatonic scale.
Let's look at it on the clock face.

Fig.6. Two other hexachords. D,E,F#,G,A,B=[24679e]=.Z.
C,Db,Eb,F,Ab,Bb=[01358t]=.W. (not marked)
Rotate 6 stations
.Z. 0 to swap .Z. and .W.
e | 1 Reflection on double
| .Z. axis swaps .Z. & .W.
t | 2 Reflection on single
| axis fixes them.
===== |
===== |
.Z.9 ===*=== 3
| =====
| =====
|
8 | 4
| .Z.
7 | 5
.Z. | 6
.Z.

"Well, that's cute. So what have I got? Each of the unordered diads that
I used to have just acquired a definite order. So my pitch classes now go
like this:

Prime Series (P): [e54t71289360]

"or, B, F, E, Bb, G, C#, D, G#, A, Eb, F#, C. I can't impose any
further order on these pitch classes. So, that means, what? It means
that so long as I build material from chunks of P, or transpositions
of P, or inversions of P, or...lessee, yeah, if I play P backwards I
get pretty much the same stuff, so that goes for retrogrades of
transpositions of P and retrogrades of inversions of P...If I do that,
I can have my Checkmark motif followed by a variation of it, which
adds up to an octatonic, with another variation of it made by the last
two notes of the first variation and the next two notes, and another
made by the remaining two notes and the first two notes that I
originally started with...and I get an easy transition to major and
minor Petruchka chords, my whirling tritones, and finally, an easy
gateway to the diatonic hexachord music...all with chunks of music
with all 12 pitch classes in them, so the chunk-breaking event of an
old pitch class returning happens as infrequently as possible, and the
music can give the impression of larger smooth phrases... and I just
noticed that when I shift to the diatonic hexachord, the notes come up
in a shape that closely resembles my Checkmark Motif:

F#
D
A
G
B
E

"Well, I think that about makes enough material to build a pretty
fancy piece, don't you, Kitty? What? No, I haven't cleaned your
litter box yet today. Of course not: I've been busy playing with
compositional materials, and I've just reached step 3 from GEMS 4, the
point at which I've solved the most difficult part and have an idea
how to use my solution to make all the different parts of my piece.
No, don't climb on my lap right now, I'm busy composing. It's time for
me to string my ideas together into a...oh, all right, you can get on
my lap, but---no, wait, don't squat on me like that---why you
#@$)(*)(* cat!"


SERIALISM AS EXPERIMENTAL MODALITY
As the somewhat degenerate story above suggests, a serial composer
doesn't use a tone row as JUST an ordering of pitch classes, but rather
as an ordering PLUS a bunch of connotations, groupings, purposes, and
meanings that they have put INTO the tone row so those things will be
there, handy, when they are called for.

The situation is somewhat like that of a mode or tonality, which, we
will recall, consists of more than just a set of 6, 7, or 8 pitch
classes. A mode is a comprehensive package, that comes with the idea
of one of it's tones being a point of rest; the standard church modes
originally came bundled with a small repertoire of standard melodic
fragments, each connoting a particular kind of grammatical phrase or
rhetorical device, and these phrases later got augmented into standard
chord sequences when polyphony became popular. For a brief while, in
the 18th century, the set of standard phrases of Western music was
reduced to a minimum, called Tonality...almost immediately, composers
became interested in bits of earlier modal practices that had survived
in folk musics and liturgical musics, so musical "romanticism" was
born. In the early 20th century, individuals like Bela Bartok added
more "exotic", non-Western-sounding modal practices to the set of
available ideas, and experimented with choosing new groupings of
pitches and establishing new patterns within them that would serve as
"artificial modes". Since the phrases and patterns such composers
created were usually NOT standard phrases that everybody used, such
composers took on the job of making such phrases SOUND standard---or
sound right in the context of a given piece. This, of course, meant
repetition and variation, for these are the main devices for driving
something into a listener's memory (No, I haven't just quantified
memorability, because the recognizability of a *variation* is still
unquantified here).

Seen in this light, what Arnold Schoenberg sought to do with his
already- extremely chromatic and chunky music begins to make a lot of
sense. He organized it into phrases and chords which may not have
been part of a standard practice before-hand, but by the time you got
done listening to a piece of his, the phrases began to really SOUND
standard---to the particular piece.

For those who are curious, yes, Composer X's tone row has in fact been
used in a piece, with exactly the set of groupings listed above. The
piece is copyrighted, but the row and groupings aren't and can't be.

NATURAL EXTENSIONS

Everything I've said so far could certainly be used in sets of pitches
other than twelve notes equally tempered in an octave. What makes
material serial is that we have some operations of transposition, etc.
that take a motive and yeild a recognizable variant---and that the
motives we start with can be overlapped in such a way as to grow
smoothly out of each other, with the consequences that we become
interested in organizing and ordering chunks of music larger than the
motives themselves. These larger chunks then serve as a helpful
midpoint between organizing the notes of the motives and organizing a
piece as a whole.

So, for instance, if you were working in some other temperment than equal
temperment, you could group things all the ways I've described above, PLUS
get the added connotations that the NON-equivalence of the intervals might
supply. Suppose, e.g., that I limit myself to a pentatonic set, e.g.
only the black keys of the piano. Well, I still get five transposition
levels---each with a different quality because of the inequality of
the intervals---and five inversions, again each with a different quality.

Or suppose you divided the octave into more than 12 parts. So you could,
perhaps, sound more than 12 pitch classes without returning to one---but
if you divided the octave too finely, and tried to use all the divisions,
it might SOUND like you had returned to an old pitch class when actually
you'd gone to one of its neighbors, so you might not want to compose with
chunks any longer than, say, nineteen tones, even if you divided an octave
into, say, 120 equal (or unequal) parts.

I could say a lot more about neat, easy ways of using serial materials
to build expressive musical statements---using chains of rows with
one or more notes overlapped or shared to build segments larger than
a single row, using "proximal" forms of a prime row to help present
e.g. a now-familiar tune in a new harmonic context, etc. But I think
I've said enough, and some would argue that I've said far too much on
this topic. For those who want to pursue this topic further, John
Rahn's "Basic Atonal Theory" would be a good place to start (his
notation differs from mine somewhat). For everybody else, I hope
this exploration has proven thought-provoking, if not immediately
imagination-stirring.

---And I hope to interest some of you in some of the standard literature
of serial music.



LISTENING ASSIGNMENT
As usual, these assignments are provided strictly for those who want them.
You get no special Internet Brownie Points for doing them, and no special
Internet Karma Points for not doing them.

Here are a couple of different serial works that I think merit listening.
As you listen, a couple of questions to ask yourself might be:
1) Is this music expressive of a mood?
2) Does it feel like it expresses a tension between variety and unity?
3) Can I pick out the main motifs out of which it is built?

Alban Berg: Violin Concerto
Arnold Schoenberg: Variations for Orchestra Op.31
Anton von Webern: Second Cantata
Luigi Dallapiccola: Quaderno di Annalibera


WRITTEN ASSIGNMENT
For those who really want one, here's one.

Find a short tune of your own. Experiment with the consequences of moving
some of its pitches into different octaves. Experiment with transposing
and inverting it. Do the variants you get from this process stimulate your
imagination? Can pairs of them be fit together contrapuntally? Do any
of them suggest another motive that you might cause to emerge from them?

CONCLUSION
Some of you may have joined USENET (Network News) or rec.music.compose
relatively recently, and might be curious about the contents of the
preceding 4 articles. As I said before, I don't plan on reposting
them to rec.music.compose. If interest is great enough, I'll collect
addresses for a distribution list.

At this time I don't plan any further articles of this sort. I think I'm
about gemmed-out, and besides, I did this series for free.

Whew. The GEMS series was quite challenging to write, and this final
installation was the most challenging of the bunch. The first of the
GEMS series came out on 11 August 1992, so the entire series has stretched
over nine months...an extended gestation. I hope you have found this
series interesting, maybe a bit informative, and maybe stimulating.


13 May 1993 Matthew H. Fields, D.M.A.