CHAPTER 3 The
Twelve-Tone System and Max
 3.1
The history of twelve-tone system
When
there was a Pope who had central governing power to which all others
were subordinate in his whole empire, when a male-dominated
society still existed, and when fathers had too much power
over their
families, the musical pieces at those periods were composed
based on a do-dominant (tonic dominant) harmonic structure. This
still occurs
in contemporary musical pieces. Almost 80 years ago, there
was
a man who dedicated his whole life to making equally importantly
dominated
harmonic structure, which is now called the twelve-tone system.
His name was Arnold Schoenberg (1874-1951).
Among
the new changes that took place during the 1920s, the most dauntless
and doubtlessly innovative change might
have been Schonberg’s
twelve-tone system. However, the twelve-tone system was not created
solely by Schonberg. The advent of this new method of composing
music, based on the twelve-tone system, was inevitable. It followed
as a
natural result of the over-all movement of the writing of many
free and atonal
pieces during the 1910s.
The term “atonality” was derived from the idea of the absence
of a tonic. Due to the fact that all twelve tones have equal standing
in relation to each other, atonality rapidly grew in popularity and
became very important to Western music after 1910. Meanwhile, other
famous composers tried to discover some new ways to work with tonal
centers in terms of not relying on classical, functional harmony. Those
ways included using ostinato, which was seen in Stravinsky’s
The Rite of Spring, modal pitch complexes, seen in Debussy’s
Voiles, polytonality, alternative scale forms, and some new chord
types. These were all musically and historically inevitable, and
they resulted
in the twelve-tone system.
3.2 Kandinsky and the twelve-tone system
Expressing their inner emotions by rupturing traditional conventions
of either musical composition or visual painting is the kinship between
Kandinsk and Schoenberg. This intellectual affinity of these two
artists in terms of spending their whole lives in pursuit of new
innovative ways for the revolutionary changes
in their own provinces resulted from both reciprocal influences
and intelligent exchanges.
Kandinsk’s 7 Compositions are the examples that evoke the affinities
between painting and music and that hint at a metaphor with music.
In his works, it can be seen how much he was captivated by the emotional
power of music. By the same token, the twelve-tone system, which
had stirred Schoenberg to devote his life to abandoning tonal and
harmonic conventions, coincided with the spirit of Kandinsk’s
expressionism.
This sprit
of both Kandinsk’s expressionism and Schoenberg’s
the twelve-tone system could be resurrected by using Max program.
For instance, a multimedia piece that uses both expressionism
painting and Max-based music can be exhibited in a multimedia
museum or
in a concert hall. The example piece that is based on Max will
be introduced
in later sub chapter.
3.3
Basic features of the twelve-tone system
The basic purpose of the twelve-tone system is to have
no tonal center, or to treat all 12 chromatic tones
equally. To avoid the possibility
of emphasis on a particular tone even if it is not intentional,
a compulsory rule was necessary. The compulsory rule is the
twelve-tone system.
To compose music with this twelve-tone system, a composer creates
a row by putting the twelve different pitch classes into a pre-arranged
order. The row that the composer makes determines the pitch texture
of the melodies and chords as well. Like Baroque fugue, one twelve-tone
row has four basic forms, with a possibility of 48 permutations;
prime (forward and right side up), inversion (forward and upside
down), retrograde (backwards and right side up), and retrograde
inversion (backwards and upside down).
The composer may simultaneously use different permutations of
the row. However, there is one rule that the composer must follow.
That
is, once a permutation of the row has been used in a piece, all
twelve pre-arranged pitches must be chosen before repeating the
same permutation
of the row, whether or not it is in melody structure, chord structure,
or in the structure of the combination of both.
In addition,
this idea of the twelve-tone system can be applied to rhythm,
dynamics, and other compositional aspects. In
fact, after
1950, Karlheinz Stockhausen, Pierre Boulez, Milton Babbitt,
and other contemporary composers have extended this idea
to their
own compositions.
By the same token, a Max-based piece can be created by
using this basis of "Integral Serialism."
3.4
Analysis of Arnold Schoenberg’s piano piece op. 33a
(1929)
The row
that Schoenberg used for this piano piece is shown in the
figure 3-1. Although this row contains
twelve different semi
tones, and appears as if the twelve tones were randomly chosen
and arranged, Schoenberg chose these twelve tones very
carefully and
arranged them after listening to them over and over again.
This fact is demonstrated in the three main tetrachords that
were derived from
the main row. The main tetrachords are shown in the figure 3-2.
< Figure 3-1 examples of twelve-tone rows>
<Figure
3-2 the three tetrachords from first prime and first retrograde>
These main three tetrachords are used in this piece as if
they, I (do-mi-sol), IV (fa-ra-do), and V (sol-si-re), were
used in many classical pieces. Moreover, although each tetrachord
sounds very much like an emancipated panchromatic dissonance,
the three main tetrachords (see the figure 3-2) are obviously,
from beginning to end, like common classical pieces (see the
figure 3-3). In other words, this emancipated panchromatic
piano piece has symmetrical balance. This is achieved by a
method of beginning with these main tetrachords, which are
derived from the first prime and the first retrograde, and
ending with them as a cadence.
<Figure
3-3 the three tetrachords from first prime and first retrograde
appear from beginning to end>
<Figure
3-3>
3.5
The twelve-tone based piece using Max
I created this piece to demonstrate how easily Max can
help composers create twelve-tone system music, and
also how Max
can possibly create string-based pieces. This first movement
for a string quartet is solely designed for the main concept
of excluding the tonal center.
First,
there is a small difference between the principles of the
pieces based on Schoenberg’s twelve-tone system and
my piece, in that I did not take care of the possibility of
48 permutations from the main row; prime, inversion, retrograde,
and retrograde inversion. In fact, there is no main row in
my piece. Due to the fast calculation ability of today’s
computers1, machines deal with whether or not all twelve-semi
tones are equally treated. Moreover, each string part has its
own row in my piece. Instead, there are a total of 144 permutations
in each string part, which means that there are a total of
576 permutations in the whole first movement. Each semi tone
appears equally, 48 times, in this movement, and shows up 12
times in each part. This results in the whole piece maintaining
a balance in terms of using twelve tones; it is the same as
the result of Schoenberg’s piano piece op. 33a (1929).
1.
The dual PowerPC G4 processors — each
with a sustained performance of over
three billion calculations per second, <http://www.apple.com/powermac> (cited
16 Sep 2000).
To use twelve tones
equally, Max provides a helpful object called an urn. Urn is
the best object for choosing random numbers without repeating
a choice2. In the left inlet, urn receives bang and clear messages.
In the figure 3-4, metro gives bang messages in certain intervals
which two urns generate. Urn 12 means that it generates previously
unchosen random numbers from 0 to 11 according to the bang
messages from its left inlet. After it generates all 12 numbers,
it sends a bang message out to the right outlet and a clear
message refreshes the memory of urn. This circulation will
continue until a stop message is given. This urn circulation
generates 144 permutations for each string part until a stop
message is given.
<Figure 3-4 urn chooses random numbers without
repeating a choice>
For
the rhythmic aspect, there is a formula, {(Y urn + 1) * (X
run +1)} * 20. The right urn 12 object receives bang messages
depending upon the left urn 12’s left outlet. In other
words, during the first row, string I contains the same rhythmic
pattern; however, other 3 string parts have their own rhythmic
patterns, derived from their own urn 12 objects. They create
an overlapping rhythmic sound. It is also possible that the
left urn can predominately generate the rhythm, but then
all 4 urns in 4 string parts will generate little chaotic
rhythmic patterns. The reason why number 1 is added is to
avoid 0 because 0 can only result in 0 when it is multiplied
by other numbers.
2. J. Christopher Dobrian, Max Reference, ed. Jon Drukma (Opcode
Systems, Inc., 1995), 400.
Multiplying
20 means multiplying 20 milliseconds. In the computer, the
following have no meaning: 4/4 beats, 3/4 beats, or 6/8 beats,
etcetera. Finally, the numbers that pass the formula will
be 20, 20, 20, … 20; 40, 40, 40, … 40; and 60,
60, 60, …60, and so on3 (see the figure 3-5). However,
these numbers are also multiplied based on the X value so
the results
will vary slightly. (X * Y) is a variable ranging from 1
to 144. Thus, the final results range from 20 (1 * 20) to
2880
(144 * 20) milliseconds4. Anyway, these numbers go to the
right inlet of metro to generate bangs for certain intervals.
All
twelve tones, consequently, will have an equal duration if
all individual tones are summed up from the tone in the first
row to the same tone in the 12th row. For example, the total
duration from C1 in the 1st row to C12 in the 12th row is
240*(1~144); (20+20+20+20+20+20+20+20+20+20+20+20)* (1~144)=240(1~144)
(see the figure 3-6).
<Figure
3-5 the rhythmic patterns of 4 parts based on their own urn
12 objects>
3. A* object multiplies only when it receives a number in the
left inlet. The right urn 12 object in figure 3-4 sends next
number only when it receives a bang message from the right outlet
of the left urn 12 object. Therefore, the * object does not send
the next number to metro and metro keeps same rhythmic pattern
until it receives another value from the * object.
4. One minute is 60 seconds or 60000 milliseconds. Thus, metronome
60 means 60000 milliseconds and a quarter note is 1000 milliseconds.
If 2880 is divided by 4 to make whole note, its quarter note
will be 720 milliseconds, so 60000/720 is equal to 83(83.33).
Thus, the metronome speed is 83. In this piece, the overall speed
could be 83, and 20 could be converted to a sixty-fourth note.
Therefore, in this whole movement, the note values vary from
a sixty-fourth note to a whole note in the metronome speed of
83.
<Figure
3-6 the total sum from C1 in the 1st row to C12 in the 12th
row>
Furthermore,
in terms of fairness, there is something else to be considered.
When using twelve tones, within the whole 4 string parts,
for example, there is no overlapping register among them.
In other words, the violin I has its own register from C5
to B5, and the cello’s melodies are chosen from C2
to B2 (figure 3-7). These works can very easily be done by
using the first Max’s patcher object and copying and
pasting it three times. The last step is to change numbers
of the + objects; + 60, + 48, and + 36.
<Figure 3-7 fair
registers among 4 parts>
The
first step is to create a patcher for violin I (figure 3-8).
I called it “row1,” it may be called
anything. The main purpose of using a patcher object is for
encapsulation. Todd Winkler notes the importance that encapsulation
embeds and masks complexity, and will help keep the overall
program structure simple and greatly improve readability5.
Moreover, a patcher object works very well when representing
a specific instrument when a main patch is designed for composition.
For example, if an orchestra piece is composed using Max, patcher
objects can help to neatly organize all instruments by giving
each patcher object a name.
<Figure
3-8 the patcher of violin I>
5. Todd Winkler, Composing Interactive Music: Techniques and
Ideas using Max, (Cambridge, Massachusetts: The MIT Press, 1998),
77.
To
finish the first movement after generating all possible 576
permutations, combining a histo object and
an if object works appropriately (figure 3-9). The object histo
counts how many times it has received the same number, sends
out the number of times to the right outlet, and carries the
number itself out the left outlet6. The object if works like
C programming language so several relational operators can
be typed in an if object7. The if statement: [if $i1 == 0 && $i2
== 12 then bang] means that if the value of the first inlet
is equal to 0, and if the value of the second inlet is equal
to 12, then send bang.
<Figure 3-9 the patcher” itstops” finishes
the first movement>
6.
J. Christopher Dobrian, Max Reference, ed. Jon Drukma (Opcode
Systems, Inc., 1995), 205.
7. For more examples see Todd Winkler’s Composing Interactive
Music: Techniques and Ideas using Max, (Cambridge, Massachusetts:
The MIT Press, 1998), 104-105.
The
next step is to put musical value into the whole piece. Adjusting
crescendo and diminuendo is easily done
by drawing graphics in a table object8. Table is the best for
editing the stored numbers in a graphic mode. Table keeps a
pair of X and Y values; X is an address and Y is possible values
from 0 to 127. For the entire dynamics, including extreme pianissimo
and fortissimo, of this piece, the Y values were set from 0
to 95 (figure 3-10). Table receives its X addresses from counter.
Counter counts the number of bangs from metro. In a subpatch
named “row1”, there is a table object called “cre”,
which is connected to the second outlet of “row1”,
and its values pass to all 4-string parts as dynamics for 4
makenote objects (figure 3-11 and 3-12).
< Figure 3-10 Table object used for crescendo and diminuendo>
8. J. For
more specific examples see Christopher Dobrian’s,
Max Getting Started, ed. Jon Drukma (Opcode Systems, Inc., 1995),
Tutorial 32, 140-146.
<Figure
3-11 Table “cre” object connected to the second
outlet>
<Figure 3-12 Second outlet in subpatch “row1” is
connected to all 4-string parts to control velocity>
In
addition to dynamics, there is another musical consideration:
making vibrato. While a human violinist relies
on a rapid wrist-and-finger movement on the string for a slight
alteration of the pitch, most modern synthesizers offer a pitch
wheel to alter the pitch. To control this pitch wheel, Max
provides a bendout object, which sends a MIDI pitch bend message
to its connected synthesizer. To make MIDI vibrato, the combination
of random, split, and bendout is used in this piece. Random
20 generates random numbers from 0 to 19 and split 62 66 controls
the random numbers which are added 50 passing + 50 object in
the range from 62 to 66 (figure 3-13). With these values, bendout
a 1 sends MIDI pitch messages to the channel 1 of the synthesizer
specified as “a”. Bendout a 2, bendout a 3, and
bendout a 4 send MIDI pitch messages to the channel 2, 3, and
4 (figure 3-14).
< Figure 3-13 split splits the numbers in the range from 62 to
66>
<Figure
3-14 split numbers go to 4 channels to make vibrato>
3.6
Conclusions
The
first movement is designed solely for the possibility of making
musical pieces based on the twelve-tone
system and showing Max’s potential ability for composition.
The result is quite successful even though it took only a few
days. The total duration of this movement ranges from 2 minutes
11 seconds to 3 minutes, but the result will really vary depending
on the computer. There is, however, a small limitation to this
piece, which is based on Max and a synthesizer, in that it is
not easy to imitate the conventional string instruments’ special
playing techniques such as harmonics. This piece will be more
sophisticated if I consider all possible string techniques and
playing styles. Furthermore, there is another limitation in the
hardware itself. I use Kurzwel K-2500, which has optional sound
ROMs installed, but the sound itself is not yet the same as the
sound of real strings.
References
J. Christopher Dobrian, Max 3.5 manual Addendum, (Opcode Systems, Inc., 1996),
8.
Joel Chadabe, Electric Sound: The Past and Promise of Electronic
Music, (Prentice-Hall, Inc., 1997), 61.
J. Christopher
Dobrian’s, Max Getting Started, ed. Jon
Drukma (Opcode Systems, Inc., 1995), Tutorial 32, 140-146.
J. Christopher Dobrian, Max Reference, ed. Jon Drukma (Opcode
Systems, Inc., 1995), 205.
J. Christopher Dobrian, Max Reference, ed. Jon Drukma (Opcode
Systems, Inc., 1995), 215.
J. Christopher Dobrian, Max Reference, ed. Jon Drukma (Opcode
Systems, Inc., 1995), 400.
Joseph Machlis and Kristine Forney, The Enjoyment of Music,
Eight edition, Standard version. (Norton, 1999), 649.
Joseph Machlis and Kristine Forney, The Enjoyment of Music,
Eight edition, Standard version. (Norton, 1999), p. 672.
Joseph Machlis and Kristine Forney, The Enjoyment of Music,
Eight edition, Standard version. (Norton, 1999), 684.
Stanley R. Alten, Audio in Media, Fifth edition. (Wadsworth
Publishing Co., 1999), 3.
Stanley R. Alten, Audio in Media, Fifth edition. (Wadsworth
Publishing Co., 1999), 278.
Schwarts,
Elliott, and Barney Childs, eds. Contemporary Composers on
Contemporary Music, Expanded ed. (New York” Da Capo,
1998), 201.
Slonimsky,
Nicolas. Lexicon of musical invective: critical assaults on
composers since Beethoven’s time, 2nd ed., Seattle:
University of Washington Press, 1969.
Todd Winkler, Composing Interactive Music: Techniques and Ideas
using Max, (Cambridge, Massachusetts: The MIT Press, 1998), 77.
Todd Winkler’s
Composing Interactive Music: Techniques and Ideas using Max,
(Cambridge, Massachusetts: The MIT Press,
1998), 104-105.
Underwood. The Western Concert Tradition Since 1950, (New York
University, 2000), 14.
Underwood. The Western Concert Tradition Since 1950, (New York
University, 2000), 24. |
