Algorithmic composition is art of composing music by writing and running algorithms. These algorithms usually are implemented as computer programs, although there are notable examples of algorithmic scores computed entirely by hand, e.g. Leonid Hrabovsky’s * Concerto Misterioso* (many works of Cage could be considered to fall into this category as well).

An algorithm is computationally irreducible if, and only if, its output cannot be predicted in advance of actually running the algorithm, even by a close reading and analysis of its code. Only irreducible algorithms are of any *intrinsic *artistic interest, because reducible algorithms are essentially alternative forms of music notation — more convenient, perhaps, than neumes on ledger lines, but not fundamentally different. Thus, only irreducible algorithms enable the computer to bring anything really *new *to the capabilities of the human imagination. Irreducibility is a consequence of the undecidability of formal systems. It follows that irreducibility is a truly fundamental feature of logic and, indeed, because logic is embodied in computing devices, of Nature itself.

One possible approach is parametric composition. This is the art of composing music by interactively exploring a parameter space, each point of which represents a different input to the same computationally irreducible score generator. If the points are colored by some musically relevant feature of the generated scores, examination and exploration of the parameter map is far more informative than merely picking points at random and generating the corresponding scores. Note that such a map cannot make any algorithm computationally reducible — but it *does *present the outputs of an irreducible algorithm in organized, intelligible form. Of course, for each point in a map, the corresponding score still has to be computed in advance.

I have elsewhere proved, using attractors of iterated function systems interpreted as scores, that there exists a parameter map for all music. However, this IFS map covers far too much ‘noise’ in comparison with ‘music.’ This is because (a) the map covers infinitely more infinite than finite attractors, and the infinite ones can only be interpreted as scores by discretization, and even more importantly, (b) the operators and attractors contain no specifically musical structure.

I have somewhat refined my approach to parametric composition into the following list of requirements for implementing it:

- The score generating algorithm must be irreducible, so that it amplifies the musical imagination and does not serve as a mere shorthand form of music notation. That is the musical motivation for the approach, and is therefore the most basic requirement. This requirement is settled.
- The score generating algorithm must compute each piece of music as a fixed point, so that the algorithm possesses mathematical integrity and is not subject to arbitrary musical interpretation. This requirement is settled.
- The fixed points must be finite or periodic attractors, so that each point in the attractor can be interpreted as a definite element of music such as a grain of sound, a note, or a chord. This requirement did not used to be sufficiently clear. Now that I have clarified this requirement, I may be able to figure out how to implement it. The most important open question is whether every finite piece of music has in fact a corresponding periodic attractor. What passes for my mathematical intuition suggests that it does.
- The space in which the attractors live must have some inherent musical structure, again so that the attractors are not subject to arbitrary musical interpretations. This requirement is sufficiently clear, but has not completely been implemented. I need to be able to represent either a rest, or a note, or a chord as a single point in time. I am confident this is possible but not completely confident that I can easily do it, or do it at all. Implementing this requirement would probably also mean that each operator in the algorithm would have some specifically musical meaning.
- The parameters for all fixed points must be transformed into 2 or 3 dimensional Hilbert indexes so that the parameter space for the algorithm, no matter how many independent numerical parameters there actually are, can be visually mapped, thus enabling parametric composition by exploring the map. This requirement is settled. Although I have not worked out the details, I am completely confident that it can be done.

I have recently been developing insight by examining the 3-dimensional voice-leading space of Tymozcko. The 1-voice chords are the edges of the prism that are parallel to its orthogonal axis. The 2-voice chords are the faces of the prism that connect the 1-voice edges. The 3-voice chords are the interior and caps of the prism. I think that the dimensionality of the prism can be increased (e.g. to 12) to obtain every arity of chord by translating to some subspace of that prism.

By also extending octave equivalence to range equivalence, bundles of prisms arise in which these same relations are extended across octaves. It then becomes possible to move up and down a scale, to create the P, L, R, and D transformations, etc., just by using different translations.

Arpeggiation (and playing a scale is a special case of arpeggiation) can be done by iteratively translating a chord such that it reduces to the single tone on each axis of the chord space. This can be considered a projection. It can be done by multiplying the chord times a unit vector that becomes in turn each element of the basis.

In short, the use of independent dimensions to represent translation, pitch-class set, voicing, arpeggiation, etc., is a non-starter, it is not even a coherent idea. All the required effects can be accomplished by projecting chords to various subspaces of chord space under range equivalence.

Although it is possible to use one real point for any chord whatsoever, it is not possible to have ‘memory’ of some chord or scale, for the purposes of arpeggiation or iteration, without multiplying by an additional group.

My best take on this now is chord space of the desired maximum arity under range equivalence, times the group of permuting the dimensionality.

The irreducible representations of the product group are what we seek