This post provides a working java program for hearing different combinations of strange circles; the program is described below and is available for free as a Java jar file at this location: http://www.bcp.psych.ualberta.ca/~mike/BlogStuff/Circles/StrangeCircles.zip.
Also, this post is a revised version of this previous post on this blog. The text is a bit different; the big difference is that at the bottom of this blog I provide a java program that lets you compose with a variety of strange circles and hear the results. Feel free to download the program and play with it. If you have any difficulties please leave a comment; this is my first attempt at distributing code in this fashion and I would be surprised if I don't make some mistakes. To run the program, download the zip file, unpack it, and double-click on the .jar file's icon. You need to have Java installed on your program for this code to function.
Figure I-8. The first four bars of an atonal piece composed
with some strange circles found within musical networks. See text for details.
Atonal music has no discernible musical key or tonal center
because all twelve pitch-classes from Western music occur equally often. Arnold Schoenberg invented a method, called
the twelve tone technique or dodecaphony,
for composing atonal music. His 1923
piece Fünf Klavierstücke, Opus 23 was
the first to be composed using this technique.
In dodecaphony one begins a new composition by arranging all
twelve pitch-classes in some desired order; this arrangement is called the tone row. The first note from the tone row is then used
to begin the new piece. The duration of
this note, and whether or not it is repeated, is under the composer’s control. However, once the use of this note is
complete, dodecaphony takes control: the twelve tone method prevents the
composer from using again until all of the other eleven notes in the tone row
have first been used. Their use,
naturally, follows the same procedure used for the first note: the composer
decides upon duration and repetition, uses the note, and then moves on to the
next note in the tone row. The final
movement of Schoenberg’s Fünf Klavierstücke,
Opus 23 was the first to be composed using a complete (twelve note) tone
row in this fashion.
In the preceding chapter we saw that musical pitch-classes
could be arranged in a number of different strange circles: for instance, four
different circles of major thirds ([C, E, G#], [C#, F, A], [D, F#, A#], and
[D#, E, G]) or two different circles of major seconds ([C, D, E, F#, G#, A#]
and [C#, D#, F, G, B]).
We also saw that when artificial neural networks are trained
to solve problems in harmony, they often use these strange circles to organize
pitch-classes into different equivalence classes. For instance, all of the pitch-classes that
belong to one circle of major seconds may all be assigned the same connection
weight (e.g. to the connection from a pitch-class input unit to a hidden unit).
In a musical network, the connection weight from an input
unit to a hidden unit is essentially the ‘name’ that identifies the
pitch-class. If all of the pitch-classes
belonging to a strange circle are assigned the same connection weight, then
they are all being assigned the same ‘name’.
This means that the hidden unit is deaf to any differences between
members of this subset of pitch-classes.
For a hidden unit that uses equivalence classes based on circles of major
seconds, there are only two pitch-classes: some ‘name’ x (the weight assigned to C, D, E, F#, G#, and A#) and some other
‘name’ y (the weight assigned to C#, D#, F, G, and B).
Why do networks use strange circle equivalence classes to
represent musical structure? One reason
is that networks discover that notes that belong to the same strange circle are not typically used together to solve
musical problems, such as classifying a musical chord. Instead, the network discovers that combining
notes from different strange circles
is more successful.
This use of equivalence classes -- combining pitch-classes
from different circles, but not from the same circle – suggests an alternative
approach to composing atonal music.
Imagine a musical composition constructed from a set of
different musical voices. Each of these
voices could be derived from a strange circle.
The notes sung by this voice are selected by randomly choosing from the
set of pitch-classes that belong to the strange circle. For instance, if one voice was associated
with a particular circle of major thirds, then one could write its notes by
randomly choosing one note at a time from the set [C, E, G#]. To make the voice more musically interesting,
one could add a randomly selected rest to the mix by selecting from the set [C,
E, G#, R] where R indicates a rest (i.e. no note is to be sung).
If one associated different voices with different strange
circles, and composed via random selection as described above, then one would
be following the general principle discovered by the network: pitch-classes
from different strange circles can occur together, but pitch-classes from the
same strange circle cannot.
Furthermore, one could use this method to compose atonal
music by wisely choosing which strange circles to use to create different
voices. For instance, imagine creating a
piece that included four voices, each associated with a different circle of
major thirds. This composition would be
atonal, in Schoenberg’s sense, because the four circles combine to include all
twelve possible pitch-classes. Randomly
selecting pitches from each of these circles would produce a composition that
did not have a tonal center because each of the twelve pitch-classes would
occur equally often when the composition was considered as a whole.
Figure I-8 provides a short score created by using the
approach described above. This score
includes six staves, one for each voice.
Each voice is generated by randomly selecting from one strange circle (and
including rests in this sampling procedure).
The top two staves, written in quarter notes, are each drawn from a
different circle of major seconds. The
bottom four staves, written in half notes, are each drawn from a different
circle of major thirds.
The score illustrated in Figure I-8 is created by applying
two additional musical assumptions.
First, while each wheel generated a pitch-class name, I decided how high
or low (in terms of octave) each note was positioned. Second, in order to ensure that all notes tended
to occur equally often in the score, I sampled the two circles of major seconds
twice as frequently relative to the other four strange circles. That is why the upper two staves use notes
that are half the duration of those in the bottom four staves.
Figure I-8 provides the first four bars of a longer
composition that can be found at this website: http://cognitionandreality.blogspot.ca/2013/03/composing-atonal-music-using-strange.html. At the bottom of this web page one can find
links that play some of the voices individually, some combinations of a small
number of the voices, and all of the voices played together.
On listening to these samples, one discovers that individual
the strange circles are musical, but are not really musically interesting. Music that is more interesting emerges from
combining the random outputs of different circles. For instance, I enjoyed the results of
pairing the two circles of major seconds together. I was also surprised at the musicality of the
full composition. My impression of this
piece was that it is a modern, atonal composition. I am no Schoenberg, but I humbly submit that
composing music by combining strange circles provides an interesting and
alternative method to dodecaphony.
Of course, there are other strange circles that could be
incorporated into this approach to composing, such as the three circles of
minor thirds or the six circles of tritones.
What kinds of atonal pieces can be created when many different strange
circles are available?
To answer this question, I created a Java program that uses
David Koelle’s music package jFugue (Koelle, 2008). This package lets the programmer define
strings of musical notes, and then takes care of playing them. The program that I wrote lets the user choose
a composition’s tempo and length with a mouse, and then make a checkmark beside
every strange circle to be used in a piece.
All fifteen circles in Figure I-9 can be used at once! The user can decide whether or not to include
rests, and set the duration and the octave (2 is lowest, 5 is highest) for each
set of circles.
A press of the compose button leads to a pause while the
various voices are constructed, and then the piece is played through the
computer’s speakers. One can easily
explore the possibilities of strange circle composing with this program and listening
to the sounds that it creates.
This program is available for free as a Java jar file at
this location: http://www.bcp.psych.ualberta.ca/~mike/BlogStuff/Circles/StrangeCircles.zip. Save the zip file to your computer, move it to a desired location, and unpack it. You will see a program called StrangeCircles.jar and a lib directory; these two items have to be in the same location on your computer. To run the program from a command line, when you are in the proper location type:
java –jar StrangCircles.jar. On a
windows machine, the program can also be run simply by double-clicking on the
program’s icon after it has been downloaded. The program requires that Java be installed on your program.
Figure I-9. A screenshot of a Java program that randomly selects from various strange circles to compose atonal music. In the figure, one circle of major thirds, one circle of minor thirds, one circle of major seconds, and two circles of tritones have been selected to be used in a four bar composition that includes rests. See text for details.
Koelle, D. (2008). The Complete Guide to JFugue: Programming Music in Java:
www.jfugue.org.
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