GEMS Guidelines for Beginning Composers
GEMS Guidelines for Beginning Composers
PROLOGUE
Computer networking is one of the amazing feats of engineering of the
late 20th century. Vast, all-encompassing networks now make it possible
for data to be shared among people world-wide. Three basic services
are now commonly available:
1) Electronic mail: person-to-person text communication. A message I type
here in Michigan can be read on a computer screen in New Zealand in minutes.
2) FTP (File Transfer Protocol): general data-sharing. Libraries of
programs and other data have been created in various parts of the world
in such a way that any user can browse or copy these programs to their
own computers at high speeds.
3) Network News. Of the tens of thousands of computers networked worldwide,
some thousands of them form the backbone of a system for mass-distribution of
information in a newsletter format. Any user can read this news, and many
users can post a news article to the closest participating computer, which
in turn mails copies of the article to other computers. Vast amounts of
information is shared this way, and eventually, some of it has to be
retired or deleted.
To make network news more manageable, it is grouped heirarchically into
/Newsgroups/. New newsgroups are created when enough people agitate
for the existence of them. Currently, most sites carry over a thousand
newsgroups.
In the summer of 1992, the newsgroup was proposed
by Joshua Barinstein, as a forum for the discussion of all kinds of
musical composition. The discussion regarding its creation centered
on the issues: were the communication needs of composers being served
by newsgroups devoted to musical performance or synthesizers, and
could these needs be met by these groups without the overhead of a new
newsgroup. Several people argued convincingly that mixing apples with
oranges would force many participating computers to perform the
redundant job of sorting composition articles from the others, and so
the overhead of using existing newsgroups would be substantial. But
the more obvious concern was that composers would not use newsgroups
not devoted to composition. I participated in that discussion as an
advocate of the group.
In July 1992, a vote was taken, in which the idea received
overwhelming support, and in August 1992, came
into being. After a rocky start in which the participants worked
to distinguish the group from existing groups, some heavy discussion
of the complex relationship between compositional craft and intuition
emerged. In that climate, I posted a short message offering to write
a series of educational articles regarding bits of compositional
wisdom that had been passed on to me over the years, with the
following proposed contents:
1) Drama and Climax
2) About Parallel Fifths
3) Shortcuts for Theory Homework
4) Strategies for Canon and Fugue
5) About Serial Materials
In the discussion that followed, the phrase "gems of wisdom" became a
sort of /leitmotif/, so the idea hatched in my mind to use the word
GEMS as the title of the article series, as a way of saying "these are
the articles that I promised." A variety of people wrote news articles
or sent me electronic mail strongly encouraging me to write and post
the series.
The readership of the group ranged from musically-illiterate novices
to top-notch musical scholars, making every kind of music under the
sun, from pop songs to serial music to musical happenings and so
forth. For me, this posed some challenges, because, while my articles
had to be clear and readable to a variety of novices, the slightest
misrepresentation or oversimplification could lead to a flurry of
corrective and explanatory articles, at great expense to the computer
network. On top of this, I wanted to make sure that my articles would
be of interest specifically to composers, but at the same time, be
appealing to a wide variety of composers. My prose style had to be at
once rather precise and quite informal, in keeping with the informal
nature of computer network news.
In writing these articles, I am indebted to the many teachers who have
prodded me towards quality work, especially Richard Hoffmann of
Oberlin College, Ross Bauer, Alfred Lerdahl, and Leslie Bassett of the
University of Michigan.
GEMS 1
==== =
Matthew H. Fields
I mentioned a willingness to post some general and specific observations
regarding music composition, and so far, I've received an enthusiastic
response. Therefore, this is the first such posting.
The topic for today is:
DRAMATIC SHAPE
I have chosen to present this topic first because it seems the most useful
to the greatest number of people, and, of all the topics I've offered to
write about, it is the least tied to a particular style.
Disclaimers: I am presenting the material here mainly as my opinion. If
you try to make use of my suggestions and they don't help you write
fabulous music, I don't accept any liability. Likewise, it is strictly
to your credit and none of mine if you do write fabulous music before or
after reading these posts. Plenty of the ideas I will be discussing
in this series have been mentioned before, and some theorists may even
wish to lay copyright claim or patent claim to some of them. However, I
claim that the core ideas have been known to composers and used by them
long before anybody published any writings on them, and these ideas are
therefore basically in the public domain.
On the other hand, I actually sat down and wrote the text of this posting,
and it took me a bit of time and thought, so if anybody were to exploit
this text as a commodity without consulting me, I might get very mad
(standard disclaimer).
All that having been said, I am interested in getting some feedback
on how interesting or useful you find this article.
ABSTRACT
In this article, I will explore several basic hints for writing pieces
with convincing climaxes.
INTRODUCTION
One way in which I like to classify music is into two kinds: pieces which
move from a beginning towards a climax, and pieces which don't. Really,
the only way which a piece can avoid a sense of climax is to keep a fairly
consistent level of intensity throughout. Many pop/rock songs do this,
and pop/rock composers may feel that this article is irrelevant to their
art. On the other hand, such artists often compose a series of their
songs or performances as an album side, a dance set, or an uninterrupted
portion of a concert, and on this scale they often seek to create a
motion to a climax across many songs. Therefore, perhaps this topic will
be interesting to them and performers in general, too.
DEFINITIONS
Now, by intensity, I refer to a rather abstract psychological variable,
something on the order of "level of frenzy". Typical ways of expressing
increasing intensity are:
a) getting louder (making a more emphatic music);
b) moving towards extremes of pitch, both high and low (again, immitating
spoken expressions of strong feeling);
c) adding additional instruments to those playing (in classical music
we call this "thickening the texture")
d) interspersing melody with more and more irregular, frequent rests
(in emulation of shortness of breath)
and so forth (you may always use your imagination to find more ways
to use in addition to these).
Sometimes, people will refer to the dramatic curve of a composition as
its "form". This is a tricky word to use, at least in English, because
it can also refer to what I call a rhyme scheme for a piece (is it
made of repeating verses with a bridge, is it sonata-allegro form, is
it ABA form, rondo, or what?)... So, if I were called upon to discuss
dramatic shape as a kind of form, I would have to distinguish between
"dramatic form" and "rhetorical form". The two can have clear correlations;
e.g., in a form that goes Refrain-verse-Refrain-verse-Bridge-Refrain,
the climax might typically be towards the end of the Bridge; however,
there's no rule that says the climax has to be in any particular part
of the rhetorical anatomy of a piece. In sonatas, climaxes typically
come at the beginning of the recap, or at the beginning of the coda,
or at the beginning of the recapitulation of the second key area, or
at the very end, or...
FOUR BASIC SUGGESTIONS
OK, so by now, I'm assuming you have a basic idea what I'm talking about.
what are the hints that I can offer on this material?
1) Strongly consider having only ONE main climax. You can have lots of
subsidiary climaxes, but if you make one peak just this much more intense
than all the others, this may give your piece a sense of having really
argued its point, having really expressed its emotion, etc. If you
have two nearly-equal main peaks, you run the risk of the second one
seeming tedious. Consider making the second one bigger/louder or
gentler/softer than the first.
2) DO SOMETHING ASSERTIVE AT THE BEGINNING OF YOUR PIECE. This needn't
be loud or sharp, but if you start too soft or mild in the hopes of
then gradually cranking the intensity up, you run the risk of failing
to grab the listener's attention.
Let me tell a little story. Once, several years ago, I took a composition
lesson with a famous New England composer who will remain nameless. Fresh
off a plane from his backwoods home, still wearing his coonskin cap,
stinking of cheap whiskey and cigarettes, he arrived having had less than
2 hours sleep in the previous 2 days. After listening to a few minutes
of my recent compositions, he said, "Well, I can see that I don't have
to encourage you to get to know any basic mechanical transformations for
your material, Matt."
Then he reached over and yanked me to my feet by my collar. "Your music
has to grab me like this, and NOT LET GO UNTIL THE VERY END!" With this,
he ended the lesson. Perhaps I experienced Zen enlightenment in that
moment; perhaps not.
In any case, the suggestion is to save your softest music for just a
little ways into your piece, or for the ending.
3) In works of longer than 30 seconds duration (this figure is chosen
somewhat arbitrarily, but the exact number is irrelevant), the main
climax does not come at the beginning. It does not come at the
middle. It comes anywhere from 60% of the way through the piece to
right at the end. Otherwise you run a terrible risk of having your
listeners get bored with the gradual denoument of your work.
4) Having gotten good at implementing suggestions 1-3, you may still feel
that your climax is somewhat dissappointing. Let's say you now have a piece
which works like this:
i ^
n / \-
t -/ \
e -/ ^ \-
n ---/ | \-
s -------/ | \--
i -\ ---------/ | \---
t --\ /------------/ | \--
y v | \--
time |
|
climax
An easy method that often works to make the climax less disappointing goes
by the name "prolonging the climax". What it often is is a prolongation
of the music just before the climax, and how it works is like this:
1) make sure the music just before the climax strongly suggests that the
climax is coming;
2) write and insert more of it--possibly a lot more of it. In classical
music, this is accomplished by such technicalities as dominant pedals,
deceptive cadences (Fokke: see the passage just after the horn calls
in that piano piece!), etc. My favorite example from pop music is one
almost everybody has heard: Lennon/McCartney's HEY JUDE. It works
up to a frenzy, then spends about half the cut repeating the frenzied
verse over and over. 2 minutes later, the industry-standard fade-out
is applied. When this single was released, the crowd went wild.
Now, this suggestion doesn't guarantee a fix. If you're expecting
a solo flute playing in its lowest octave to sound climactic during
a symphony band piece, you may need to rethink other aspects of the piece.
However, it works so remarkably well so much of the time that it's worth
trying, at least part of the time.
Concerning the dramatic shape we saw above, suggestion no.4 would
revise it to look like this:
i ^
n ---------/ \-
t -/ \
e -/ ^ ^ \-
n ---/ | | \-
s -------/ | | \--
i -\ ---------/ | | \---
t --\ /------------/ | | \--
y v | | \--
time | |
|
prolongation |
climax
Another famous way of carrying out the same procedure is to get almost
to the climax, then suddenly cut back to a very low level of intensity and
build back up to the climax in just a few seconds of music. In fact, there
are so many variations and permutations on variations of these techniques
to be explored that you can have endless fun being creative with them.
As a composer, you might want to listen to a variety of works which you
feel have powerful climaxes, and see how they address the motion to the
climax.
OTHER CONCERNS
Now, I haven't mentioned how words create or don't create climaxes of
their own; a favorite suggestion of mine is to experiment with the possibility
that in the midst of a rising vocal line, the climactic text is suddenly
sung very softly, or whispered, so that the text is understated, and then
the accompaniment may or may not state the climax just afterward. This
can be a particularly spooky, frightening effect.
A lot of people feel that they should compose a piece from the
beginning to the end. Obviously, suggestion no.4 above says that you
needn't feel constrained to do so (in this way, composition differs
from improvisation, in which, once you've played, you can't go
backwards in time and adjust things). This is a general admonishment
of mine: don't feel constrained to work in sequential order! You're
the composer, so you can work in whatever order is best for you. In
particular, when you have a great idea for some part of your piece
which is out of sequence, by all means record it (on tape or in
writing), so you can use it when the time comes. Along with this
admonshment comes another basic one: no note is absolutely sacredly
unchangeable, not one of yours, not one of mine, ... heck, I can even
imagine that some day there might be someone who could improve
compositions that were originally written by Mozart. Finally,
there are two important admonishments: 1) a word to beginning composers:
begin! and 2) sooner or later, you're going to have to be satisfied
with how well you've polished your piece, so you might as well call it
"done" and play it for someone, then start a new piece.
The previous paragraph of admonishments will apply well to techniques
that I describe in detail in future articles, too.
LISTENING ASSIGNMENT
For those who are interested, a work to get to know and study which
demonstrates a lot of what I've been talking about is TREN OFIRAM
HIROSZIMY (Threnody for the Victims of Hiroshima), by the living
Polish composer Krysztof Penderecki (composition date: about 1964).
This piece is scored for 52 violin-family instruments (violins,
violas, cellos, and basses), who play a variety of massed sounds,
screeching noises, scratching noises, etc. The piece has no regular
beat, and no recognizable melodic shapes; really, the main feature of
this work is its undulating, shifting level of sound density,
intensity, and emotional fervor. After a few subsidiary climaxes, the
piece comes to a point about 65% of the way through its length where
the players drop out one by one until, after a brief cello solo, there's
a couple seconds of silence. Then, a renewed build of intensity
leads to several minutes of almost-climax, a brief pause, and a final
climactic ringing chord.
The sounds of this piece are not friendly, but rather fierce. They
are not deeply grounded in the Western Classical Tradition, or in
any folk music either, for that matter. But the dramatic curve of
the piece as a whole is as classical as the motion to a climax in a
Shakespeare play.
The piece can be heard on several recordings, including a current
CD from Warsaw on Conifer Records.
WRITTEN ASSIGNMENT
For those who like to practice principles in little studies, here's
one which I've assigned to beginning students.
Write a composition for 1-4 players.
Limit your duration to about a minute.
Use only "found sounds", that is, noises made by non-musical objects that
you have handy.
Notate your piece with a graphical notation of your own devising, NOT
incorporating any conventional music notation. Preface your piece with
a legend or key so that your players can quickly decode your notation.
Stage a small performance and perhaps even a recording session of
your new work.
Suggestions: don't be overly specific about matters of time or pitch: this
tends to delay your premiere and make you ponder extra considerations other
than dramatic shape.
DO seek out interesting sounds, like attaching a contact mike to a string
from which a wire hanger is dangled. DO seek to express yourself, even
with these (possibly) unfamiliar restrictions on sounds/materials.
DO try to build a convincing climax to your piece. DO try to throw
in something special to mark the end of your piece, if your piece
continues beyond its climax. DO experiment with prolonging the
climax.
If you try this assignment and feel moved to violate some of its rules,
relax! There will be no penalty.
CONCLUSION
I hope some of these ideas are useful to some of you out there. The
only way to learn to use them is to play with them constantly until
they become an automatic part of your musical personality.
For at least a while I will be keeping a copy of this article here
in my disk directory. As long as the volume of "reprint" requests
is reasonably manageable, I will offer to send copies out by e-mail.
I have heard a lot of interest in my hints for canon and fugue, but
as a matter of logical sequence, I intend to delay them until I've
had a chance to post concerning the mystery of parallel perfect intervals
(some of you clearly already have a good idea what I'm talking about
here, and some of you probably have no idea, but I'm most concerned
about the middle third: those who have come across the proscription of
parallel perfections in a theory class, but don't see what it has to
do with the real world) and 4-part harmony.
I can't really tell you when the next article will be ready for
posting, since I haven't written it yet. The feedback I get from
this article may have important consequences concerning how I write the
next one.
11 August 1992 Matthew H. Fields, D.M.A.
GEMS 2
==== =
Matthew H. Fields
I mentioned a willingness to post some general and specific observations
regarding music composition, and so far, I've received an enthusiastic
response. Therefore, this is the second such posting.
In my first GEM article (named after the phrase 'gems of wisdom' that
was passed around a great deal in the discussion that preceeded the
first such posting), I discussed dramatic shape and climax-building,
and passed on several famous hints for building better climaxes to
dramatic musical works.
Today's presentation is a bit more philosophical, and takes a more
round-about route towards being helpful to composers.
The topic for today is:
PARALLEL FIFTHS AND OCTAVES --- WHY I BOTHER ABOUT THEM
I have chosen to present this topic here in my sequence because most
of my later proposed articles will be written assuming you have some
idea of my biases regarding countrapuntal issues. This article will
not contain any hints or suggestions regarding composition, but will
instead talk about some meta-issues of perception.
Disclaimers: I am presenting the material here mainly as my opinion.
If you try to make use of my suggestions and they don't help you write
fabulous music, I don't accept any liability. Likewise, it is
strictly to your credit and none of mine if you do write fabulous
music before or after reading these posts. Plenty of the ideas I will
be discussing in this series have been mentioned before, and some
theorists may even wish to lay copyright claim or patent claim to some
of them. However, I claim that the core ideas have been known to
composers and used by them long before anybody published any writings
on them, and these ideas are therefore basically in the public domain.
In fact, some of these ideas have even been bandied about on
rec.music.compose in recent weeks, often quite well.
On the other hand, I actually sat down and wrote the text of this
posting, and it took me a bit of time and thought, so if anybody were
to exploit this text as a commodity without consulting me, I might get
very mad (standard disclaimer). Furthermore, to the best of my
knowledge, at no time have I herein explicitly quoted anybody's
special article: this prose is all mine.
All that having been said, I am interested in getting some feedback
on how interesting or useful you find this article.
ABSTRACT
In this article, I will describe a perceptual basis for being careful
concerning the use of parallel octaves and fifths. I don't expect to
convince anybody to take on such a concern, and I most especially will
not hand out any rules, generative or proscriptive, on this matter.
On the other hand, it is my intent to argue that this concern is not
obsolete but current, and not a matter of abstract rule-making, nor a
matter of mystical invocation of physics, but rather a matter of
hearing and musical expression.
INTRODUCTION
Parallel octaves and fifths: we hear of a 'proscription' against them
in our music theory classes. Then we find out that Bach's organs had
8-foot, 4-foot, and 3-foot stops, so that every melody he played could
be sounded out in parallel octaves and fifths. Even worse, we
discover that parallel octaves are ubiquitous in ensemble music and
piano music. And then, as we delve into musical history, we discover
early forms of organum in which singers always sang in parallel
fifths.
Why, then, is a big deal made about these things in theory classes?
and why these intervals, only, and not thirds, sixths, and sevenths?
What is the role of dogma and propaganda in this matter?
As I so often do -- perhaps it's a Jewish habit? -- I'll begin my
answer with a story. No, not "we were slaves in the land of
Mitzrayim", but rather: once, I was teaching the rudiments of aural
skills to a total beginner, and he was working on the game of "name
that interval", meaning that given the sound of two pitches played
either sequentially or simultaneously, he was to name the interval
between them. He complained at one point that he was having a bit of
trouble hearing octaves and fifths when the notes were played
simultaneously, and he said it sounded like the upper pitch was
somehow 'hiding' behind the lower pitch. I probed him a bit on this
observation: had he noticed this phenomenon outside of his work with
the aural-skills software? Yes, he had started noticing it in all the
music he heard. Did it apply to other intervals? Yes, especially
strongly to the unison, and quite weakly to the major third.
I was, of course, surprised to hear a beginner mentioning such a
phenomenon. He had never heard of any rule which made a big deal
about parallel octaves and fifths, and was quite surprised by it
when it came up in his theoretical studies---after all, parallel
octaves are ubiquitous in piano music. But he was a dilligent student,
and promptly proposed an abstract theory in which parallel octaves
and fifths were somehow purely timbral events of physics, while other
parallel intervals were events of multiple melodies.
Many authors continue to describe the harmonic series and say, without
further explanation, that it is the cause of the concern with parallel
fifths and octaves. I think that such a description of the physical
world is not sufficient to describe how certain composers have treated
these materials, but coupling that description with some purely
/SUBJECTIVE/ observations (like the ones my student complained of) may
actually bring us closer to an understanding of the matter. Even that
will not be enough to explain the concern with parallelisms, though,
since parallelism is a matter of melodic motion, not of how we
perceive individual intervals.
DEFINITIONS
Before I go any further, let's make sure we're all talking about the
same things.
When I say that two parts are in /unison/, I mean that they are
sounding the same pitch at the same time, i.e. in the same octave.
For the acoustically-minded out there, this means that within
tolerances that our ears define, they are sounding the same
fundamental frequency (where applicable).
When I say that one note is /an octave higher/ than the other, I mean
that it sounds the eighth ascending diatonic step from the other, or
is at an ascending distance of seven diatonic steps, or twelve
half-steps (in 12-tone equal temperment). For the acoustically-
minded, this means tolerably close to a frequency ratio of 2:1, so
A-880 is an octave above A-440, and A-1760 is an octave higher than
that. Naturally, if I say that a note is an octave /lower/ than a
second note, this means just that the second one is an octave higher
than the first. Carrying out the arithmetic, we find that the first
note is seven diatonic steps below the second note, or twelve
half-steps below the second note, or tolerably-close to a frequency
ratio of 1:2 with the second.
When I say that one note is a perfect fifth higher than another, I
mean that there is an ascending distance of 7 half-steps between them.
I don't give this definition in diatonic steps, because while the
fifth diatonic step in the C-major scale over C is G, at a distance of
7 half-steps, the fifth diatonic step over B is F, at a distance of
only 6 half-steps. So, I'm saying that I care about the distance
being 7 half-steps, regardless of where it sits in the scale. For the
acoustically-minded, the frequency ratio this time is 3:2. In 12-tone
equal temperment, this ratio (which can be precisely expressed in
decimal form as 1.5) is approximated by the seven-twelveths power of 2
(~~1.498307077, or a little more than 1% flat).
Finally, by /compound interval/, I mean an interval augmented by the
addition of one or more octave to its distance. In the case of a
perfect fifth, the first few compoundments of it are the perfect twelveth
and the perfect 19th, or distances of 12+7=19 and 24+7=31 half-steps,
or frequency ratios of 3:1 and 6:1 (within tolerances).
The tolerances I mention above have been the topic of quite a lot of
debate over the years, so I'm not going to pin them down, partly
because doing so would not add any vital information to this article.
Mathematicians out there are asked to please refrain from the
temptation to say 'Let epsilon be any positive real number'. If
anybody is tempted to do that, would they please agree that our
tolerances are less than 2% of the lower frequency for the sake of
this article? Ok. I'm not going to talk about quantitative acoustics
much more in this article, because I think it's time to talk about
psychological phenomena.
SO WHAT'S THE BIG DEAL?
All right, we're getting to that. But first, let's talk about melody.
I THOUGHT THIS WAS ABOUT PARALLEL FIFTHS.
Yes, but we're coming to that, and we have to back up and visit melody
and polyphony on the way.
A long time ago, somebody first started coming up with the notion of
'a nice melody' or 'a nice melodic shape' that some of us still use
today (it's the first thing you now study when you learn species
counterpoint). The basics of this concept were things like: it had
one and only one climax point, which was typically its highest note,
or sometimes its lowest note; it started on, ended on, and generally
circled around a main note which was supposed to express a sense of
repose; it moved mainly by step, occasionally by third, and rarely
by fourth or fifth --- any time a string of notes was constructed that
leaped a lot up and down, this was perceived not as a single melody but
rather as a sort of time-sharing between two or more melodies, each of
which moved stepwise (/compound melody/).
Long before people were experimenting with what we now call harmony,
they had gotten pretty good at building interesting and exciting
things that were single melodic lines. After a while, folks tried two
crucial experiments that forever changed the way people made music: 1)
Two folks got together and sang the same melody at the same time; 2)
Two folks got together and sang different melodies at the same time.
Of course, this last sentence is a gross oversimplification of
history, and is not a documented event anywhere in the world. But
let's consider the consequences of the two experiments anyhow. In the
first case, perhaps the people had the same voice range the first time
they tried this, in which case they sang in unison, and the sound
reverberated larger than either of them. Or perhaps, the first time
they tried this, they had such different voice ranges that they sang
in octaves (Perhaps an evolutionary theorist could explain our ability
to recognize melodic content after transposition in terms of our
needing to recognize the same intonation pattern from adults and
children?). Now, the first people to try singing two different
melodies together had a much more complicated result. Certain
combinations of tones came to be called pleasing-sounding, and others,
anxious-sounding; from these basic notions, a variety of complex
systems of consonance and dissonance were developed---which were
different in different eras---and plans were developed for ways in
which various consonances and dissonances could be strung together to
express something vaguely analogous to a sentence-structure. Meanwhile,
folks were listening to, and enjoying, two melodic shapes at once.
At one point, the two shapes crossed through the same note, perhaps.
The listeners became confused, because just after the crossing, it was
hard to tell whether the voices had bounced off each other like this
i ---\v/--- i
*
ii ---/^\--- ii
or crossed through each other like this:
i ---\ /--- ii
X
ii ---/ \--- i
Some folks complained that trying to keep the melodies clear in their
heads detracted from their appreciation of the individual melodies as
well as their appreciation of the consonances and dissonances that
arose between them. So some musicians tried to find pairs of melodies
that eliminated the second possibility altogether, so after a while,
everyone would get used to hearing things the first way anyhow.
Sooner or later, it was bound to happen: the two melodies passed through
two notes in a row exactly the same:
----- i
i ---\__ *<
_>* \____ ii
_/
ii /\/
People had gotten used to keeping the two melodies clear in their
heads for one shared note, but two in a row was just too hard for many
people. It sounded like one of the melodies had momentarily gone
silent while the other had momentarily gotten stronger or louder. At
about the same time, ideas of perspective, shadows, and oclusion were
being developed in the visual arts, and people had analogous ideas
brewing regarding making foreground and background shapes all equally
visible and readily enjoyable. So, some musicians decided that in
their compositions, one was the largest number of consecutive notes in
a row on which two melodic lines would sound in unison, the better to
allow the listeners to follow the shapes of each of the lines up and down.
But the situation in music was more complex. Some folks, like my
talented student, felt a sense of conjunction and aural oclusion at
not just the unison, but the octave as well, and its compoundments.
These folks decided that when two players were supposed to be playing
different musics, they'd never have two consecutive octaves with each
other, again so the melodies wouldn't seem to hide one behind the
other for too long for their enjoyment of each melodic shape by itself
as well as the overall composite. Some folks had the same experience
with the fifth and its compoundments, and foreswore parallel fifths
from their multiple-melody expression (counterpoint). Perhaps some
folks even experienced the same perception with parallel fourths,
thirds, and sixths; if so, those folks probably got disgusted with the
whole thing and went into something like mathematics or geography,
where great new things were being uncovered every day.
Meanwhile, the consequences of experiment number 1 above were still
brewing. Having worked out several melodies to sound simultaneously,
people sometimes had more resources than melodies. They quickly found
that two violins playing the same melody could balance one bass or
cello playing another melody better than one of each (due to the
differences in inherent size and loudness of the instruments).
Furthermore, individual melodies could be played by pairs of players
playing in octaves, often without changing much about the effect of
the music except its perceived loudness and strength. Harpsichord
builders and organ builders made automatic doubling at the upper
octave a feature of their instruments, essentially a simple way of
getting a stronger sound with the same number of perceived melodies.
Orchestrators eventually decided on a rule for groups of players,
which still seems to work pretty well: octave doubling above the
highest melody, and below the lowest melody, but no octave-doubling of
inner melodies, as such doubling was perceived as still confusing to
the ears---except when it was provided by highly-controlled, automated
means, like organ stops, harpsichord stops, or 12-string lutes and
guitars. Organs even came to have extra pipes to produce parallel
12ths (compoundments of fifths) for an even brighter, stronger tone.
So, for a great deal of western polyphonic (multi-melody) music,
parallel octaves and fifths were considered as falling into two
categories: features of a single melody--often highly-desireable
reinforcements of a melody that contributed to its tone color and
perceived loudness; and momentary interactions between two
melodies--usually considered undesirable, because they interfered with
/some/ listeners' ability to enjoy both melodies to the fullest.
Some people continue to hear in these terms, and find ways to treat
these 'sensitive' parallelisms as either constant features of their
music or things that rarely or never occur in their music.
Composers of the classical era worked out some highly elaborate ways
of constructing contrapuntal music so that it avoided parallel octaves
and fifths---yet didn't sound (to them) highly artificial. The study
of the methods and tricks used by these composers (which involved the
resolution of a lot of other preferences and conventions as well as
the avoidance of or isolation and control of these special
parallelisms) eventually blossomed into our modern discipline of
classical counterpoint and harmonic theory. This field and course of
study is now so loaded with interesting tidbits of musical thought
that the concept of parallel octaves and fifths is often dismissed
with the shorthand comment "they're forbidden"---occasionally with a
brief mention of the harmonic series, or of the vague idea that they
interfere with independent motion. But, of course, the truth of the
matter is a bit more subtle.
LISTENING ASSIGNMENT
Once again, the assignments are purely optional.
Give serious consideration to playing around with parallel fifths and
octaves. Do your ears tell you anything about them? Do you have an
attitude about them? How do you perceive music that avoids them?
(try the first or second fugue from Book One of the Well-Tempered
Clavier of JS Bach) music that uses them constantly? (try the
sarabande from Pour le piano by Claude DeBussy) music that uses them
indifferently? (supply your own example) music that uses them
constantly for long stretches, then not at all, but never
indifferently (try the Tenth fugue in e minor from book one of the
Well Tempered Clavier of JS Bach) ? See if you can find sources and
recordings documenting the effect of different tuning systems on the
sound of the music. Do your discoveries suggest anything for your own
compositional preferences?
WRITTEN ASSIGNMENT
No written assignment this time. Go compose.
CONCLUSION
I hope this article was interesting. In writing it, I've tried to
condense an enormous amount of information and ideas into a small
space. While the resulting article is still rather long, some of the
topics treated--especially the musical history--are quite eliptical,
abbreviated, and abstract. However, I hope that for those readers who
find the article too hurried in its descriptions, the subject matter
may at least be intriguing, and those readers may wish to look into it
further, starting perhaps with the New Grove Dictionary of Music and
Musicians s.v. /counterpoint/.
For at least a while I will be keeping a copy of this article here
in my disk directory. As long as the volume of "reprint" requests
is reasonably manageable, I will offer to send copies out by e-mail.
I can't really tell you when the next article in this series will be
ready for posting, since I haven't written it yet. The next article
will be aimed at the student enrolled in the typical undergraduate
theory course, who has been asked to demonstrate proficiency at 18th-
century harmonic counterpoint. It will consist of a very short list
of things to try as shortcuts, so that the reader might finish their
theory homework earlier and have more time available for composing.
4 September 1992 Matthew H. Fields, D.M.A.
GEMS 3
==== =
Matthew H. Fields
I mentioned a willingness to post some general and specific
observations regarding music composition, and so far, I continue to
receive an enthusiastic response. GEMS 1 was a discussion of drama
and climax, while GEMS 2 was about parallel octaves and fifths.
Therefore, this is the third such posting.
The topic for today is:
SHORTCUTS FOR UNDERGRADUATE THEORY HOMEWORK
We're composers, and we want more time to compose. But those of us in
conventional conservatory programs spend a lot of time mastering
4-part harmony and choral counterpoint. This takes time.
Can these shortcuts help? Maybe.
Disclaimers: I am presenting the material here mainly as my opinion. If
you try to make use of my suggestions and they don't help you write
fabulous music, I don't accept any liability. Likewise, it is strictly
to your credit and none of mine if you do write fabulous music before or
after reading these posts. Plenty of the ideas I will be discussing
in this series have been mentioned before, and some theorists may even
wish to lay copyright claim or patent claim to some of them. However, I
claim that the core ideas have been known to composers and used by them
long before anybody published any writings on them, and these ideas are
therefore basically in the public domain. But here let me acknowledge
the inspiration of the great late Russel Dannenburg, a fine composer and
the first person to teach me music theory.
On the other hand, I actually sat down and wrote the text of this posting,
and it took me a bit of time and thought, so if anybody were to exploit
this text as a commodity without consulting me, I might get very mad
(standard disclaimer).
All that having been said, I am interested in getting some feedback
on how interesting or useful you find this article.
ABSTRACT
In this article, I will list a couple hints for (possibly) breezing
through theory homework.
INTRODUCTION
These hints worked for me, although I no longer conciously think of
them or any other methodology when doing that sort of problem.
DEFINITIONS
I'm not defining anything this week. If you're in Freshperson Theory,
you're probably already bogged down in definitions.
THREE BASIC SUGGESTIONS
Try these on one or two examples. If they help, they help, if not,
discard them and stick to the instructions in your theory book.
The bass line can either move by step/7th, third/sixth, or 4th/5th.
1) If the bass moves by step, try moving the other 3 voices as little
as possible, in contrary motion to the bass. If it moves by 7th, treat
it as if it were moving by step and the second note just got moved to
a different octave---then apply this rule.
2) If the bass moves by 3rd or 6th, try holding on to as many common
tones as possible.
3) If the bass moves by 4th or 5th, try modeling the progression on
the tail of a familiar cadence formula.
LISTENING ASSIGNMENT
Play and listen to your solutions. Don't just work them out on paper.
This is not a chess game: it's a craft that may be useful to you some
day. Train your ear to tell you when you've made a mistake.
WRITTEN ASSIGNMENT
You've already got enough written assignments.
CONCLUSION
I hope some of these ideas are useful to some of you out there. The
only way to learn to use them is to play with them constantly until
they become an automatic part of your musical personality.
For at least a while I will be keeping a copy of this article here
in my disk directory. As long as the volume of "reprint" requests
is reasonably manageable, I will offer to send copies out by e-mail.
Next time: hints for canon and fugue (from a composers' point of view).
I can't really tell you when the next article will be ready for
posting, since I haven't written it yet. The feedback I get from
this article may have important consequences concerning how I write the
next one.
14 September 1992 Matthew H. Fields, D.M.A.
GEMS 4
==== =
Matthew H. Fields
Folks have continued to express enthusiasm for my educational postings.
The story so far:
Shortly after the opening of rec.music.compose, 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 dealt with dramatic
shape and the expression of climaxes; GEMS 2 dealt with the concept of
parallel perfect intervals, and their implications for melodic
perception; GEMS 3 was a quick list of heuristics for solving tonal
harmonization homework exercises.
As posted elsewhere, Nathan Torkington has arranged an anonymous FTP site
in New Zealand where these articles are warehoused. For the time being,
they are also available from me via e-mail.
I have downloaded GEMS 1-3 to my mac, checked their spelling, and
cleaned up some details of grammar, so they are now available from me
in hardcopy (about 15 pages, so far) for a SASE (when figuring
postage, include weight charges for 15 sheets of standard 8.5 inch x
11 inch 20-lb-test paper, and if necessary, figure airmail charges
between your location and the USA). I don't believe I'm violating any
netiquette here, since I'm not actually selling anything.
The topic for today is:
HINTS FOR COMPOSERS OF CANON, FUGUE, AND OTHER INTELLECTUAL MATERIALS
If you are rigidly opposed to the application of intellect to creative
processes, you may wish to skip this article. If you expect this article
to take the place of a theory text on contrapuntal devices and conventions,
you will be disappointed, since I will be mainly addressing issues of
interest to composers, and will assume that you can find definitions and
rough descriptions of various intellectual musical procedures in plenty
of existing textbooks.
Disclaimers: I am presenting the material here mainly as my opinion. If
you try to make use of my suggestions and they don't help you write
fabulous music, I don't accept any liability. Likewise, it is strictly
to your credit and none of mine if you do write fabulous music before or
after reading these posts. Plenty of the ideas I will be discussing
in this series have been mentioned before, and some theorists may even
wish to lay copyright claim or patent claim to some of them. However, I
claim that the core ideas have been known to composers and used by them
long before anybody published any writings on them, and these ideas are
therefore basically in the public domain.
On the other hand, I actually sat down and wrote the text of this posting,
and it took me a bit of time and thought, so if anybody were to exploit
this text as a commodity without consulting me, I might get very mad
(standard disclaimer).
All that having been said, I am interested in getting some feedback
on how interesting or useful you find this article.
ABSTRACT
In this article, I will attempt to address the role of intellectual
techniques in the composer's mental toolkit. Along the way, I will
make some general suggestions that I think may be helpful whether
you're working in serialism, Fibonnacci numbers, canon at the twelveth
at 3-3/4 beat delay, or whatever. I expect to also explain some
details of specific techniques, mainly to use them as examples of the
general principles I will be describing.
INTRODUCTION
In musician's core theory classes, beyond 4-part harmony, all further
exploration tends to be analytical. By that dense sentence, I mean that
most people are taught how to analyze music, but not how to go about trying
to construct it themselves.
Occasionally, a theorist will be talented at fugue or serialism, and will
inspire the composers among her/his students to figure out how to use such
materials themselves. On rare occasions, composition classes are taught
which combine rigorous planning of material with general compositional
strategy.
For those of you who are thirsting for such guidance but have not
found it yet, may this article serve as a primer on the matter.
DEFINITIONS
I'm going to use without definitions a lot of terms from the standard
jargon of classical music theory. The definitions are omitted mainly to
save space and time. If you want, you can look up these words in e.g.
the New Grove Dictionary of Music and Musicians:
Canon
Fugue
Sequence
Parallel Motion
Intervals
Serial Music
Fibonacci Numbers
Algorithmic Music
Stochastic Music
Inversion
Stretto
etc.
A MODEL OF A COMPOSITIONAL PROCESS
Many composers have spoken of a search for the El Dorado of algorithms,
the rigorous method which, if applied, always yeilds great music. The way
I compose certainly feels methodical to me, but when I consider the
oft-voiced goal of automating my work on a computer, I realize that what
I do is very hard to quantify or describe in clear categories---that is,
clear enough categories to form the basis of a computer program.
Why might this be so?
It helps to remember that two of the most widely accepted measures of
"good" music are that it is A) entertaining to people who don't already
know all about it, and B) memorable. Entertainment value, by which I
mean entertainment in the widest sense--not just light entertainment
but also tragic, spiritual, and other kinds of entertainment---is
essential if the listeners are going to pay much attention.
Memorableness is essential if you hope the listener will ever seek to
hear your piece a second time. As it turns out, both entertainment
value and memorableness are tricky quantities which psychologists are
still just starting to classify, so I don't expect anybody to turn up
a mathematical formula mapping patterns of notes or sounds to specific
values in either domain. You're just going to have to try to judge
for yourself how well your music meets these goals. So any serious
methodology which I propose for composition is going to have to incorporate
these unexplained, unquantified judgements.
So, let's suppose that I'm doodling with a melody, and I decide that it
might be neat to try to make a convincing piece in which this melody is
a subject of a four-part fugue. If I'm not really all that knowledgeable
about fugue, I
1) STUDY THE CATEGORY OF MATERIALS that I'm thinking of using, to see whether
it stimulates my imagination. At about the same time, I may begin to
2) PLAY WITH THE MATERIALS that I might use, and try to learn as much as I
can about their qualities.
In the case of my fugue subject, I may decide that I'm looking for 3 counter-
subjects, so when all four parts are playing, they are all playing material
that can be heard elsewhere in my piece. Soon, I realize that these subjects
not only have to form decent 4-part harmony with each other, but also, to
be interesting and to be easily recycled into other parts of the piece, they
have to form decent harmony when parts are exchanged (counterpoint is inverted)
so lower parts become upper parts, outer parts become inner parts, etc.
(this means that music is transposed by, e.g., an octave or two when it is
assigned to a different player or melodic strand).
Now, depending on my mood and the nature of the materials, I might
either want to 2a) refamiliarize myself with (or learn) the principles
resulting from my decision, or 2b) begin immediately trying to
construct music based on the ideas in my head, then fix it as I go
along.
If, in the example of the fugue, I choose No.2a, I'll go back to look
at invertable counterpoint at, e.g., the octave, and find out its
consequences for parallel fifths: parallel (or direct) fourths become
parallel (or direct) fifths after revoicing, so if I'm trying to avoid
parallel fifths, I better not allow any parallel fourths either.
Knowing that ahead of time can save me some time on process No.2b,
which I'll eventually have to confront anyhow.
So now I'll try to build the first such countersubject, by laying out
my main subject on a stave, with a blank staff above and another one below.
I can now go into the following cycle:
2b1) Have I written enough music for this subject?
If so, I can quit this cycle, else I continue.
2b2) Add a note to the subject. Put it in the upper
staff, and at the same time, transpose it down an
octave and put it in the lower stave.
2b3) Consider, in isolation, my countersubject so far.
Does it groove? Does it sound ok? If not, erase this
note from both staves, and go back to 2b2.
2b4) Consider the music in the upper two staves. Does it
violate any of the rules I've discovered in step 2a?
Does it violate any aesthetic rules (like the
avoidance of parallel fifths and octaves) that I've
decided to stick to? Is it in any way less than optimal
in grooviness or some similar nameless quantity that I
want to preserve? If the answer to any of these
questions is yes, erase the note and go back to 2b2.
2b5) Apply the same rules as from 2b4 to the music in the
lower two staves (remember, we have 3 staves here:
countersubject above, main subject, and countersubject
below).
2b6) If we get to here, this note is one we're going to
try to stick with, for now. Go back to step 2b1 to
continue building more of the countersubject.
This procedure looks like a backtracking algorithm, but notice that none of
the aesthetic judgements are quantified. With a little struggle, it can
probably be adapted to quite a wide variety of musical structures. I've used
a sort of variant on it as a way of working out serial music, and as far
as I can tell, it hasn't failed me there.
The important test: DOES IT SOUND GOOD? DOES IT GROOVE? is the glue that
holds this method together.
3) SOLVE THE MOST INTELLECTUALLY DIFFICULT OR COMPLICATED PART FIRST.
In the case of my four-part fugue, I'm going to try to construct music
in which the subject and all three countersubjects sound together, and verify
that every pair of them fits in good invertable counterpoint together. In
so doing, I'm going to use all the tricks I have from steps 1 and 2.
4) CONSIDER EXPLOITING YOUR SOLUTION FROM STEP 3 when constructing other
parts of your piece.
Now, by simply copying this music to a fresh sheet of paper (or a
fresh range of measures in my notation program, or a fresh notelist
file in my sequencing program, etc.), maybe transposing it, maybe
revoicing it, maybe erasing one or more voices, I can come up with an
incredible number of musics with slightly different feelings to them,
different textures and densities, but all with a sense of relatedness
to each other. I don't have to be at all careful about (e.g.)
parallel fifths because I already arranged that they wouldn't arise
back at step 3. So, out of my quartet texture, I can pull 23 other
quartets by merely rearranging the voices; 24 trio textures,
constructed by erasing one voice and permuting the others; and 12 duet
textures, obtained by erasing two of the voices, and optionally
inverting the counterpoint of the two voices I have left. Plus, I can
try putting the music in a different mode (e.g. major or minor),
transposing it, etc. In all these cases, I'm taking advantage of all
the work that went into step 3, so I don't have to work hard to get
any of these materials.
Since the materials are so closely related, if I construct a piece mainly
out of these materials, it will have a sort of redundancy that may help drive
the melody into the listener's memory in a more effective manner than mere
repetition.
4) Sooner or later, I've got ENOUGH MATERIAL, and it will be time for me
to STOP PLAYING WITH MATERIAL AND START ORGANIZING A COMPOSITION.
Under this heading I include all the ideas from GEMS 1, including top-down
planning of a dramatic push towards a climax, choosing an assertive gesture
to start the piece, finding a convincing and special-sounding ending to the
piece, prolonging the climax, etc. I already have a sketchbook loaded
with both explicit materials and ways I've found to make materials that I
want to use, so now it's merely a matter of choosing among these materials,
linking them together, and taking a step back to look at the big picture.
DOES IT GROOVE? If not, it doesn't matter how perfectly I've applied my
intellectual technique: I'm going to have to go back and adjust the
eigenvalues or coefficients, prolong the climax a bit more or less, and
maybe throw in a contrabass clarinet solo for good measure---whatever it
takes to make it finally sound good.
5) REMEMBER, BY MECHANICAL MEANS IT'S EASY TO PRODUCE MORE VARIANTS
THAN YOU'LL NEED. If I systematically presented all the duets, trios,
and quartets that I could extract from my 4-part fugue exposition, my
piece would probably get to be quite long and boring. I may only want
3, or 5, or 7 or 8 of the variants that I've found. That may be
enough to build a fairly long, dramatic piece. It's all right for me
to know that all the other variants exist and all work perfectly well
as counterpoint or whatever, but I don't get any brownie points for
cataloguing them all to the audience. So I'm going to have to just
learn to let go of, cross out, and ignore most of the variants which I
have generated, once I've used all the ones that fit my piece. Or, I
might separately create another piece out of some of the leftover scraps
of fabric. But unless I'm incredibly clever, I should never have both
of these pieces played on the same concert.
MATT RANTS ON
While I'm on the soapbox, let me remind folks that our listeners hope
to be entertained. We can't count on them reading program notes, so
they're going to have to get something out of the music without any of
the knowledge that could be imparted there, whether it amounts to an
explanation of poetic allusions, an intriguing essay on the
intellectual techniques underlying your piece, a story that the music
is supposed to tell programmatically, or whatever. If the listeners
are interested enough in the music, they may be a bit more likely to
read the program after hearing it, and if they're really interested in
the music, they may pour over the program looking for information on
how to buy a recording of the music. At that time, they may learn
something about the music which may, after the fact, enhance their
appreciation of it. But you just can't count on them gleaning the
important fact from the program notes which turns their listening
experience from a mystified sitting through a wash of sound into an
enlightened experience of a scientific principle. The music has got
to draw them in and get them interested all on its own.
On the other hand, even the worst concert-goers (with the exception of
a few psychotics) go to concerts to have a good time, and will try to
have a good time with your piece. So, your relationship with your
audience is not necessarily adversarial...although some listeners will
bring a healthy skepticism (or a pathological fear of anything new) to
their listening. While it's certainly reasonable to simply not worry
about the few who have already prejudged your music on the basis of
the fact that it's new to them, and to not worry about reaching the
few who will groove simply on the fact that any sound is being made at
all, it would not be a good idea to ignore the middle of the audience,
the folks who don't yet know whether they can dig your music. If you
can guide them into your way of hearing things, it doesn't matter
whether they can describe your piece in theoretical terms: at some
human level, they're following along with the course of your musical
argument, and they stand a chance of getting something out of it.
LISTENING ASSIGNMENT
Here I list some of my favorite examples of beautifully passionate but
rigorously intellectually-structured music. Most emphatically, let me
repeat that you can gain a great deal by looking at the score while
listening.
I recommend two compositional publications by J.S.Bach as informative
sources on musical intellection (and sources of delight and
wonderment, as well): The Well-Tempered Clavier, which is a set of 48
preludes and 48 fugues arranged in two sets of 24 each, where each set
cycles through all 12 major and all 12 minor keys; and Art of Fugue, a
collection of some 16-odd fugues and 8-odd canons for unspecified
instruments (plus an arrangement of two of the fugues for keyboard
duet), all based on variants of a single melody (and a fairly small
set of counterpoints to that melody). If you really cannot read
music, for about twice the price of the scores to these works, you can
acquire CD's of performances of them (on WTC, I recommend the
harpsichord performances of Gustav Leonhard or Kenneth Gilbert; on Art
of Fugue, I recommend the performance by Musica Antigua Ko"ln, who have
also produced a superb recording of another recommended Bach piece,
The Musical Offering--but unfortunately, this recording is out of print
now).
Other great examples that come to mind are Bela Bartok's Music for
Strings, Percussion, and Celeste, and Arnold Schoenberg's Variations
op.31.
Part of what I hope listeners to these works will come to realize is that
for composers who use such intellectual material all the time, the
intellectual structure eventually becomes so basic to their art that the
focus of the art is on how they improvise expressive shapes in, with,
and around these materials, rather than how they assemble these materials
themselves.
WRITTEN ASSIGNMENT
For those inclined to think harder when there's a written assignment,
here is a short one. But don't forget to work on your current opus!
Compose a piece of 8-40 bars in two melodic lines, using one of the
following: invertable counterpoint of the octave, twelveth, or tenth.
Decide whether to work in tonality or not. Decide what rules to apply.
After some number of bars, swap parts betwen the melodic lines.
Add whatever is needed to the beginning, the middle, or the end to
make a convincing piece. Consider only using additional material that is
relatively simply derived from the other material of your piece.
Sculpt this music to provide a convincing climax, where applicable.
Choose a keyboard, or, preferrably, two voices and/or instruments to
play your piece; if necessary, either adjust your choice of
instruments to meet the demands of the piece and the availability of
players, or adjust your music to meet the demands of writing for these
players, or both.
Stage a performance of the piece. Perhaps record it.
CONCLUSION
I hope some of these ideas are useful to some of you out there. The
only way to learn to use them is to play with them constantly until
they become an automatic part of your musical personality.
For at least a while I will be keeping a copy of this article here
in my disk directory. As long as the volume of "reprint" requests
is reasonably manageable, I will offer to send copies out by e-mail.
I can't really tell you when the next article will be ready for
posting, since I haven't written it yet. The next and final article
in the GEMS series will address the concept of serial materials: what
they are, where they come from, how they're used, and how they relate
to the art of composition.
17 November 1992 Matthew H. Fields, D.M.A.
GEMS 4
==== =
Matthew H. Fields
Folks have continued to express enthusiasm for my educational postings.
The story so far:
Shortly after the opening of rec.music.compose, 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 dealt with dramatic
shape and the expression of climaxes; GEMS 2 dealt with the concept of
parallel perfect intervals, and their implications for melodic
perception; GEMS 3 was a quick list of heuristics for solving tonal
harmonization homework exercises.
As posted elsewhere, Nathan Torkington has arranged an anonymous FTP site
in New Zealand where these articles are warehoused. For the time being,
they are also available from me via e-mail.
I have downloaded GEMS 1-3 to my mac, checked their spelling, and
cleaned up some details of grammar, so they are now available from me
in hardcopy (about 15 pages, so far) for a SASE (when figuring
postage, include weight charges for 15 sheets of standard 8.5 inch x
11 inch 20-lb-test paper, and if necessary, figure airmail charges
between your location and the USA). I don't believe I'm violating any
netiquette here, since I'm not actually selling anything.
The topic for today is:
HINTS FOR COMPOSERS OF CANON, FUGUE, AND OTHER INTELLECTUAL MATERIALS
If you are rigidly opposed to the application of intellect to creative
processes, you may wish to skip this article. If you expect this article
to take the place of a theory text on contrapuntal devices and conventions,
you will be disappointed, since I will be mainly addressing issues of
interest to composers, and will assume that you can find definitions and
rough descriptions of various intellectual musical procedures in plenty
of existing textbooks.
Disclaimers: I am presenting the material here mainly as my opinion. If
you try to make use of my suggestions and they don't help you write
fabulous music, I don't accept any liability. Likewise, it is strictly
to your credit and none of mine if you do write fabulous music before or
after reading these posts. Plenty of the ideas I will be discussing
in this series have been mentioned before, and some theorists may even
wish to lay copyright claim or patent claim to some of them. However, I
claim that the core ideas have been known to composers and used by them
long before anybody published any writings on them, and these ideas are
therefore basically in the public domain.
On the other hand, I actually sat down and wrote the text of this posting,
and it took me a bit of time and thought, so if anybody were to exploit
this text as a commodity without consulting me, I might get very mad
(standard disclaimer).
All that having been said, I am interested in getting some feedback
on how interesting or useful you find this article.
ABSTRACT
In this article, I will attempt to address the role of intellectual
techniques in the composer's mental toolkit. Along the way, I will
make some general suggestions that I think may be helpful whether
you're working in serialism, Fibonnacci numbers, canon at the twelveth
at 3-3/4 beat delay, or whatever. I expect to also explain some
details of specific techniques, mainly to use them as examples of the
general principles I will be describing.
INTRODUCTION
In musician's core theory classes, beyond 4-part harmony, all further
exploration tends to be analytical. By that dense sentence, I mean that
most people are taught how to analyze music, but not how to go about trying
to construct it themselves.
Occasionally, a theorist will be talented at fugue or serialism, and will
inspire the composers among her/his students to figure out how to use such
materials themselves. On rare occasions, composition classes are taught
which combine rigorous planning of material with general compositional
strategy.
For those of you who are thirsting for such guidance but have not
found it yet, may this article serve as a primer on the matter.
DEFINITIONS
I'm going to use without definitions a lot of terms from the standard
jargon of classical music theory. The definitions are omitted mainly to
save space and time. If you want, you can look up these words in e.g.
the New Grove Dictionary of Music and Musicians:
Canon
Fugue
Sequence
Parallel Motion
Intervals
Serial Music
Fibonacci Numbers
Algorithmic Music
Stochastic Music
Inversion
Stretto
etc.
A MODEL OF A COMPOSITIONAL PROCESS
Many composers have spoken of a search for the El Dorado of algorithms,
the rigorous method which, if applied, always yeilds great music. The way
I compose certainly feels methodical to me, but when I consider the
oft-voiced goal of automating my work on a computer, I realize that what
I do is very hard to quantify or describe in clear categories---that is,
clear enough categories to form the basis of a computer program.
Why might this be so?
It helps to remember that two of the most widely accepted measures of
"good" music are that it is A) entertaining to people who don't already
know all about it, and B) memorable. Entertainment value, by which I
mean entertainment in the widest sense--not just light entertainment
but also tragic, spiritual, and other kinds of entertainment---is
essential if the listeners are going to pay much attention.
Memorableness is essential if you hope the listener will ever seek to
hear your piece a second time. As it turns out, both entertainment
value and memorableness are tricky quantities which psychologists are
still just starting to classify, so I don't expect anybody to turn up
a mathematical formula mapping patterns of notes or sounds to specific
values in either domain. You're just going to have to try to judge
for yourself how well your music meets these goals. So any serious
methodology which I propose for composition is going to have to incorporate
these unexplained, unquantified judgements.
So, let's suppose that I'm doodling with a melody, and I decide that it
might be neat to try to make a convincing piece in which this melody is
a subject of a four-part fugue. If I'm not really all that knowledgeable
about fugue, I
1) STUDY THE CATEGORY OF MATERIALS that I'm thinking of using, to see whether
it stimulates my imagination. At about the same time, I may begin to
2) PLAY WITH THE MATERIALS that I might use, and try to learn as much as I
can about their qualities.
In the case of my fugue subject, I may decide that I'm looking for 3 counter-
subjects, so when all four parts are playing, they are all playing material
that can be heard elsewhere in my piece. Soon, I realize that these subjects
not only have to form decent 4-part harmony with each other, but also, to
be interesting and to be easily recycled into other parts of the piece, they
have to form decent harmony when parts are exchanged (counterpoint is inverted)
so lower parts become upper parts, outer parts become inner parts, etc.
(this means that music is transposed by, e.g., an octave or two when it is
assigned to a different player or melodic strand).
Now, depending on my mood and the nature of the materials, I might
either want to 2a) refamiliarize myself with (or learn) the principles
resulting from my decision, or 2b) begin immediately trying to
construct music based on the ideas in my head, then fix it as I go
along.
If, in the example of the fugue, I choose No.2a, I'll go back to look
at invertable counterpoint at, e.g., the octave, and find out its
consequences for parallel fifths: parallel (or direct) fourths become
parallel (or direct) fifths after revoicing, so if I'm trying to avoid
parallel fifths, I better not allow any parallel fourths either.
Knowing that ahead of time can save me some time on process No.2b,
which I'll eventually have to confront anyhow.
So now I'll try to build the first such countersubject, by laying out
my main subject on a stave, with a blank staff above and another one below.
I can now go into the following cycle:
2b1) Have I written enough music for this subject?
If so, I can quit this cycle, else I continue.
2b2) Add a note to the subject. Put it in the upper
staff, and at the same time, transpose it down an
octave and put it in the lower stave.
2b3) Consider, in isolation, my countersubject so far.
Does it groove? Does it sound ok? If not, erase this
note from both staves, and go back to 2b2.
2b4) Consider the music in the upper two staves. Does it
violate any of the rules I've discovered in step 2a?
Does it violate any aesthetic rules (like the
avoidance of parallel fifths and octaves) that I've
decided to stick to? Is it in any way less than optimal
in grooviness or some similar nameless quantity that I
want to preserve? If the answer to any of these
questions is yes, erase the note and go back to 2b2.
2b5) Apply the same rules as from 2b4 to the music in the
lower two staves (remember, we have 3 staves here:
countersubject above, main subject, and countersubject
below).
2b6) If we get to here, this note is one we're going to
try to stick with, for now. Go back to step 2b1 to
continue building more of the countersubject.
This procedure looks like a backtracking algorithm, but notice that none of
the aesthetic judgements are quantified. With a little struggle, it can
probably be adapted to quite a wide variety of musical structures. I've used
a sort of variant on it as a way of working out serial music, and as far
as I can tell, it hasn't failed me there.
The important test: DOES IT SOUND GOOD? DOES IT GROOVE? is the glue that
holds this method together.
3) SOLVE THE MOST INTELLECTUALLY DIFFICULT OR COMPLICATED PART FIRST.
In the case of my four-part fugue, I'm going to try to construct music
in which the subject and all three countersubjects sound together, and verify
that every pair of them fits in good invertable counterpoint together. In
so doing, I'm going to use all the tricks I have from steps 1 and 2.
4) CONSIDER EXPLOITING YOUR SOLUTION FROM STEP 3 when constructing other
parts of your piece.
Now, by simply copying this music to a fresh sheet of paper (or a
fresh range of measures in my notation program, or a fresh notelist
file in my sequencing program, etc.), maybe transposing it, maybe
revoicing it, maybe erasing one or more voices, I can come up with an
incredible number of musics with slightly different feelings to them,
different textures and densities, but all with a sense of relatedness
to each other. I don't have to be at all careful about (e.g.)
parallel fifths because I already arranged that they wouldn't arise
back at step 3. So, out of my quartet texture, I can pull 23 other
quartets by merely rearranging the voices; 24 trio textures,
constructed by erasing one voice and permuting the others; and 12 duet
textures, obtained by erasing two of the voices, and optionally
inverting the counterpoint of the two voices I have left. Plus, I can
try putting the music in a different mode (e.g. major or minor),
transposing it, etc. In all these cases, I'm taking advantage of all
the work that went into step 3, so I don't have to work hard to get
any of these materials.
Since the materials are so closely related, if I construct a piece mainly
out of these materials, it will have a sort of redundancy that may help drive
the melody into the listener's memory in a more effective manner than mere
repetition.
4) Sooner or later, I've got ENOUGH MATERIAL, and it will be time for me
to STOP PLAYING WITH MATERIAL AND START ORGANIZING A COMPOSITION.
Under this heading I include all the ideas from GEMS 1, including top-down
planning of a dramatic push towards a climax, choosing an assertive gesture
to start the piece, finding a convincing and special-sounding ending to the
piece, prolonging the climax, etc. I already have a sketchbook loaded
with both explicit materials and ways I've found to make materials that I
want to use, so now it's merely a matter of choosing among these materials,
linking them together, and taking a step back to look at the big picture.
DOES IT GROOVE? If not, it doesn't matter how perfectly I've applied my
intellectual technique: I'm going to have to go back and adjust the
eigenvalues or coefficients, prolong the climax a bit more or less, and
maybe throw in a contrabass clarinet solo for good measure---whatever it
takes to make it finally sound good.
5) REMEMBER, BY MECHANICAL MEANS IT'S EASY TO PRODUCE MORE VARIANTS
THAN YOU'LL NEED. If I systematically presented all the duets, trios,
and quartets that I could extract from my 4-part fugue exposition, my
piece would probably get to be quite long and boring. I may only want
3, or 5, or 7 or 8 of the variants that I've found. That may be
enough to build a fairly long, dramatic piece. It's all right for me
to know that all the other variants exist and all work perfectly well
as counterpoint or whatever, but I don't get any brownie points for
cataloguing them all to the audience. So I'm going to have to just
learn to let go of, cross out, and ignore most of the variants which I
have generated, once I've used all the ones that fit my piece. Or, I
might separately create another piece out of some of the leftover scraps
of fabric. But unless I'm incredibly clever, I should never have both
of these pieces played on the same concert.
MATT RANTS ON
While I'm on the soapbox, let me remind folks that our listeners hope
to be entertained. We can't count on them reading program notes, so
they're going to have to get something out of the music without any of
the knowledge that could be imparted there, whether it amounts to an
explanation of poetic allusions, an intriguing essay on the
intellectual techniques underlying your piece, a story that the music
is supposed to tell programmatically, or whatever. If the listeners
are interested enough in the music, they may be a bit more likely to
read the program after hearing it, and if they're really interested in
the music, they may pour over the program looking for information on
how to buy a recording of the music. At that time, they may learn
something about the music which may, after the fact, enhance their
appreciation of it. But you just can't count on them gleaning the
important fact from the program notes which turns their listening
experience from a mystified sitting through a wash of sound into an
enlightened experience of a scientific principle. The music has got
to draw them in and get them interested all on its own.
On the other hand, even the worst concert-goers (with the exception of
a few psychotics) go to concerts to have a good time, and will try to
have a good time with your piece. So, your relationship with your
audience is not necessarily adversarial...although some listeners will
bring a healthy skepticism (or a pathological fear of anything new) to
their listening. While it's certainly reasonable to simply not worry
about the few who have already prejudged your music on the basis of
the fact that it's new to them, and to not worry about reaching the
few who will groove simply on the fact that any sound is being made at
all, it would not be a good idea to ignore the middle of the audience,
the folks who don't yet know whether they can dig your music. If you
can guide them into your way of hearing things, it doesn't matter
whether they can describe your piece in theoretical terms: at some
human level, they're following along with the course of your musical
argument, and they stand a chance of getting something out of it.
LISTENING ASSIGNMENT
Here I list some of my favorite examples of beautifully passionate but
rigorously intellectually-structured music. Most emphatically, let me
repeat that you can gain a great deal by looking at the score while
listening.
I recommend two compositional publications by J.S.Bach as informative
sources on musical intellection (and sources of delight and
wonderment, as well): The Well-Tempered Clavier, which is a set of 48
preludes and 48 fugues arranged in two sets of 24 each, where each set
cycles through all 12 major and all 12 minor keys; and Art of Fugue, a
collection of some 16-odd fugues and 8-odd canons for unspecified
instruments (plus an arrangement of two of the fugues for keyboard
duet), all based on variants of a single melody (and a fairly small
set of counterpoints to that melody). If you really cannot read
music, for about twice the price of the scores to these works, you can
acquire CD's of performances of them (on WTC, I recommend the
harpsichord performances of Gustav Leonhard or Kenneth Gilbert; on Art
of Fugue, I recommend the performance by Musica Antigua Ko"ln, who have
also produced a superb recording of another recommended Bach piece,
The Musical Offering--but unfortunately, this recording is out of print
now).
Other great examples that come to mind are Bela Bartok's Music for
Strings, Percussion, and Celeste, and Arnold Schoenberg's Variations
op.31.
Part of what I hope listeners to these works will come to realize is that
for composers who use such intellectual material all the time, the
intellectual structure eventually becomes so basic to their art that the
focus of the art is on how they improvise expressive shapes in, with,
and around these materials, rather than how they assemble these materials
themselves.
WRITTEN ASSIGNMENT
For those inclined to think harder when there's a written assignment,
here is a short one. But don't forget to work on your current opus!
Compose a piece of 8-40 bars in two melodic lines, using one of the
following: invertable counterpoint of the octave, twelveth, or tenth.
Decide whether to work in tonality or not. Decide what rules to apply.
After some number of bars, swap parts betwen the melodic lines.
Add whatever is needed to the beginning, the middle, or the end to
make a convincing piece. Consider only using additional material that is
relatively simply derived from the other material of your piece.
Sculpt this music to provide a convincing climax, where applicable.
Choose a keyboard, or, preferrably, two voices and/or instruments to
play your piece; if necessary, either adjust your choice of
instruments to meet the demands of the piece and the availability of
players, or adjust your music to meet the demands of writing for these
players, or both.
Stage a performance of the piece. Perhaps record it.
CONCLUSION
I hope some of these ideas are useful to some of you out there. The
only way to learn to use them is to play with them constantly until
they become an automatic part of your musical personality.
For at least a while I will be keeping a copy of this article here
in my disk directory. As long as the volume of "reprint" requests
is reasonably manageable, I will offer to send copies out by e-mail.
I can't really tell you when the next article will be ready for
posting, since I haven't written it yet. The next and final article
in the GEMS series will address the concept of serial materials: what
they are, where they come from, how they're used, and how they relate
to the art of composition.
17 November 1992 Matthew H. Fields, D.M.A.
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.
O
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 m