Monday, March 18, 2013

Composing Atonal Music Using Strange Circles

In the 1920s, Arnold Schoenberg invented a technique, called dodecaphony, for composing atonal music.  Atonal music has no discernible musical key, because all twelve notes from Western music occur equally often.  In dodecaphony one begins by arranging all twelve notes in some desired order; this arrangement is the tone row.  One then takes the first note from the tone row to use in composing a new piece.  The composer chooses the duration of this note, and decides whether to repeat it.  However, once the use of this note is complete, the composer cannot use it again until all of the other eleven notes in the tone row have also been included.

In this blog, I explore an alternative approach to composing atonal music.  This technique uses ‘strange circles’ as sources of notes for a musical composition.  Strange circles are representations of musical notes that we have discovered in many networks that we have trained on musical classification problems.

First, I will briefly state what strange circles are.  Then, I will explain how I used them to generate the notes of a musical score.  I am going to provide that musical score, and also provide some sound files that illustrate the completed piece, as well as the components from which it was constructed.  If you are not interested in the details of my method, then feel free to skip to the end to hear its results! (Links are in the text, but are not evident in some browsers.  So, I have also set aside the links at the end of this post.)

Strange Circles

Most students of music encounter a device called the circle of fifths.  It arranges all twelve notes from Western music in a single circle; adjacent notes in this circle are a musical interval of a perfect fifth (or seven semitones) apart.  This circle helps determine the number of sharps or flats in a key signature, or guides chord progressions in jazz.

The Circle of Fifths

Other circular arrangements of notes are possible that separate adjacent notes using musical intervals different from the perfect fifth.  These are ‘strange circles’ because they are not typically taught to music students.  When my students and I train artificial neural networks to classify musical chords, and then examine the internal structure of the trained network, we often find that the network assigns notes to a variety of strange circles.

For example, we often find that networks employ two circles of major seconds, both shown below.  In each of these circles, adjacent notes are a major second (or two semitones) apart.  With this separation between notes, one circle captures half the notes in Western music, and the other captures the other six notes.

Two Circles of Major Seconds

We also frequently discover the four circles of major thirds in our networks; all four appear below.  In each of these circles, adjacent notes are a major third (or four semitones) apart.  With this separation between notes, one circle only captures a quarter of the notes in Western music.  As a result, four difference circles are required to represent all of the possible notes.
Four Circles of Major Thirds
Composing With Strange Circles

Why do networks use strange circles 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.

I thought that a network’s approach to strange circles could create new music.  Networks combine notes from different strange circles, but do not expect notes from the same strange circle to co-occur.  What if the notes for a voice or a staff of a musical composition were provided by a strange circle?  If one created this voice by choosing only one note at a time from a strange circle, then its notes would not co-occur.  However, if one used different strange circles in this way to create different voices in the same composition, then this might be musically interesting.

My hypothesis led to action.  I created a six-voice piece of music, in which a strange circle provided the notes of each voice.  I used each of the six strange circles illustrated above.  To choose a note for a staff, I randomly selected a note from its strange circle.  (This move is also consistent with our network interpretations: networks treat each note from the same strange circle as being exactly the same note!)

I made three additional musical assumptions.  First, while each wheel generated a note name, I decided how high or low (in terms of octave) each note was positioned.  Second, I added a rest to each circle, so that at a randomly selected moment any strange circle could be silent.  This was to provide some musical variety.  Third, in order to ensure that all notes occurred 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, I used the circles of major seconds to generate four quarter notes per bar, while the other four circles were used to generate two half notes per bar. The score generated from the ‘strange circles’ in this fashion consisted of 14 bars.  Its first four bars are below; a pdf of the entire score is available here.
Individually, the strange circles are not musically interesting, though they do seem musical.  For instance, listen here to a single circle of major thirds.  There is not much you can do using only three notes and the occasional rest!  You can also listen here to a single circle of major seconds.  It is a bit more interesting, because it uses six different notes.

Music that is more interesting emerges from combining the random outputs of different circles.  I enjoyed the results of pairing the two circles of major seconds together, which you can hear here.  Combining two circles of major thirds also shows promise, as you can hear; even more interesting results result from combining all four circles of major thirds.

Of course, the full composition involvescombining the notes of all six circles simultaneously.  I was surprised at its musicality.  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.



  1. How do you define "musicality? | And if a work is not atonal, is it not modern - even if it has been recently composed?
    I can understand exploring sounds and how they fit together. But is this kind of music really aurally interesting? Or is it that it is simply aurally incomprehensible, and we are drawn to it just because we so desperately need to discern some kind of pattern that we can latch on to? Why do composers really compose atonal music?
    If I take a chair apart, and rebuild it according to some new original design so that in the end it neither looks like a chair nor can hold weight, is it still a chair?

    1. I would imagine that judging the musicality of a piece requires listening to it, and then evaluating it according to one's own criteria. You can do so too -- that is why I provided the audio files with this post.

  2. An interesting piece; the notes appearance on the page are intriguing. But if I close my eyes, I hear something else. A fascinating subject.

  3. Interesting method. Although it seems to me that combining multiple sets of circles in your composition provides a much less controlled result than only using circles of a single interval between voices which would allow you to 'filter out' certain intervals (M2/m7 if you use the two circles of M2) and focus more on specific sounds during the aleatoric process. Which may or may not be what you are after. You could also potentially enhance the method by adding more restricting parameters to the selection process (like the rest) and/or alternating circle types, using non-symmetrical circles etc..

    In any case, great stuff :)