This essay describes a theory of emergent music. It is what I have used to create the music shared here.


This essay describes the specific formalism I have adopted in the exploration of emergent musics. Examples of emergence--complex structures arising from the interplay of simple part--have been given in other essays. This essay turns to emergence specifically as it relate to music. Emergence is defined broadly, and when we turn our gaze to music several foundational questions arise: What are the component parts? What interactions can take place? What processes can be brought to exploration of emergent music? How can such explorations be structured is a way so that its results have a chance of being listenable, of being recognizable as music? This essay addresses those questions.

It may be helpful to think of what follows as a theory of music unto itself. Standard music theories are created to describe and comprehend certain kinds of music and take many cultural and aesthetic biases as axiomatic. There is nothing wrong with that, and this theory does not contradict other ways of looking at music. Moreover, it stands slightly orthogonal to other theories, addressing different questions and adopting different axioms. In fact, this theory should probably be viewed as a collection of axioms. They are aesthetic assertions, and their "truth" is bounded by the musics they produce. And so this theory will largely be given as a collection of axioms and aesthetic assertions, with their justification coming from the musics themselves.

Of course, other theories and formalisms and axioms are possible. But while these are my own personal, aesthetic choices, I do assert that they are reasonable on two grounds: they are well-grounded in what emergence is, and they produce interesting musics.

This theory of emergent music hinges on the concepts of sound, space, process, and transformation.

The Core Theory


The most basic component of music is the sound. Sound has many attributes, and the ones addressed in these musics are as follows:

  • A sound has a pitch (frequency).
  • A sound has a volume.
  • A sound has a timbre.
  • The sound has a position in time.
  • The sound has a duration.

In consonance with other descriptions of music, a sound with these five attribute can be referred to as a note.


Sound occupies space. There are three main dimensions of this space:

  • Time.
  • Frequency.
  • Voice.

Note that the attributes of sound act to situate it in this dimensional space. Pitch places the sound in frequency space. Timbre and volume place it along the dimension of voice. The note's position and duration describe the space occupied by the note in the dimension of time.

Spaces can be linear or curved. Linear spaces can be infinite or bounded.

A curved space joins at its ends in a cycle, and is characterized by its extent and size.

Time is a continuum that is measured in beats and fractions thereof; most commonly these are placed at whole numbers or simple fractions (powers or two or three in most extant musics). Musical time is correlated to perceptual time via a specification of tempo.

Frequency is a continuum, but the cycle is used to quantize this continuum for the placement of notes. The defining attribute of the cycle is that it repeats and quantizes frequency, but it may be defined in any relationship to frequency.

The scale is an example of a cycle with certain assumed biases. As an example, in western music a scale is often defined as a cycle that repeats at a doubling of frequency and has quanta defined from a small set of pre-defined patterns. But a cycle need not repeat at frequency doubling, and the quanta can be defined arbitrarily.

Notes can at times be sounded outside of the cycle, but even in these cases a quantization and mapping to frequency is used.

Voice is a quantized dimension, the position upon which defines the instrument or sound source of the note.

In the simplest case there is one space shared by all notes. However, each quanta of voice can have its own spacial limits defined for time and frequency. [This is something of a practical consideration, as the same frequency limits don't necessarily make sense for all sound sources.]


Processes exist in the same space as notes. A process is a collection of actions. These actions either:

  • Sound notes
  • Execute processes

Processes may apply transformations to both notes and processes when they act upon them. All transformations that apply to notes can be applied to processes.

Transformations change a note's (or processes') position in space, and all such changes are relative and local; a process has no knowledge "where it is" and can not specify any spacial quantities in absolute terms. As an example, if a process sounds a note then the note will sound at the position of the process. If it applies the transformation "shift forward in time one beat" the note will sound one beat after the processes' position. But it cannot specify "go to the 20th beat of the composition and sound a note."

A process can call any process, including itself.


All spacial quantities can be shifted in both directions. Time can go forward or backwards; pitch and volume can go up or down; voice can be incremented or decremented.

Time may also be stretched and compressed in one of two ways. It can be changed absolutely, in which case subsequent notes are stretched or compressed in both position and duration. Or position can be stretched, leaving duration unaltered. This stretching and compressing is actually a rescaling of the temporal dimensions, and affects all notes and processes that are children of the process applying the transformation. In actuality this bifurcates space, creating a new space for the descendants of the transformed process.


Processes may be multiply defined, and stochastic probabilities applied to each definition. When a process is named for execution, these probabilities will determine which version is executed.

The Beginning

Music begins with the execution of a process.

The End

Music ends when there are no more notes, either through the end of process or the exceeding of spacial limits.

[In practical terms a limit can be placed on the total number of notes to accept within a musical expression. This is necessary because in curved or infinite spaces recursively defined processes can produce an infinity of notes. Such music is hard to play, and hard to appreciate.]

Musical Expressions

With the above definitions and axioms given, we can now describe the genesis of an emergent musical expression. Such a description may help the reader to understand the above in context by giving it more concrete form. While the above may sound abstract, the process of creating an emergent musical expression is fairly easy and straight-forward.


The beginning process is specified.

This process is placed in space. Cycles (scales) and tempos are defined. The limits of the space are decided upon, as is its structure (linear or cyclical).

The process is defined. This definition can include the sounding of notes or the execution of other processes. Both notes and processes may undergo transformations in their execution. If a process is multiply defined, probabilities of execution are assigned.

Any process referenced by the initial process must be defined. This continues until all references are given definition.

The beginning process is undertaken.



The Extended Theory

The central metaphor of space suggests a set of transformations that have not been included in the above, transformations which follow logically from the mathematics of space. As an example, an axis of time could be reflected as in a mirror, reversing its flow. The dimension of pitch could be scaled in the same way that the time dimension can be scaled. More abstractly, the dimensions given above could be considered orthogonal and subjected to arbitrary rotation (but some serious thought would need to be given in regards to just what this would mean). In its broadest definition, the theory would include any transformation that is defined mathematically for a space.

I have not included these transformations in the core theory for two reasons. First, some of these transformations (such as arbitrary rotation) seem unlikely to me to result in listenable, recognizable music. Of course, this is just an assumption, and until such transformations are empirically explored it remains such.

The second reason is more pragmatic. Even as a subset of possibilities, the core theory provides a vast space of possible musics. In the implementation of the theory I have chosen the core transformations with an eye to keeping the scope in check. I have barely begun to explore the musics that emerge from the core, and at this point in the process I am disinclined to expend the effort at expansion. Perhaps at some later point additional transformations will be brought into the core and implemented.


For information on the implementation of this theory, see Implementation.