Andrew M. Cavallo

Multiplicity in Unity: A Mark of Great Music

J. S. Bach, of course, excels at deriving musical variety from highly economical means, i.e., at unfolding, as it were, a simple musical idea into a complete work often filled with surprising turns and great intricacy.

However, no matter how surprising or intricate Bach's music becomes, each passage emerges intelligibly from what precedes it: even the most striking contrast does not amount to musical discontinuity. There is thus in Bach's work a principle of continuity, for continuity is a property of true unity, allowing changes and transformations to produce multiplicity without destroying unity.

But what determines musical continuity? This is indeed a fascinating question to which I propose the following rather rough answer: musical continuity is only possible when the composer stays within the possibility space determined by a given musical idea. Musical ideas that reside on the boundary of this space are of special interest: a point closely related to my 2012 paper on the topological problem of system boundary, wherein I describe how a system often shares a boundary with its environment that permits the local entanglement of elements of the system with elements of the environment without thereby collapsing the environment and the system into one unit.

In other words, every composer begins with certain ideas or material. In order to be well-formed, this starting material should be constructed according to established musical rules that are, in turn, rooted in the diatonic scale and the objective mathematical relationships that give rise to it. For purposes of unity and identity, the starting material should be as economical as possible given the desired effect. Once in place, this starting material implies a musical possibility space, i.e., a space of musical ideas that are compossible¹ with the starting material. Musical ideas that fall outside this space create, when joined to the starting material, some sort of musical absurdity or contradiction.

Returning to Bach, his music unfolds in such a way that new material is not discontinuous with what came before; it may be surprising or even jarring, but it remains genetically related to the material from which it emerged.

And this suggests a notion of musical creativity which, as far as I am aware, has not been formulated in precisely this way before. Namely: musical creativity amounts to unexpected ways of navigating the possibility space implied by a starting motive or other given musical idea.

This notion of musical creativity thus differs from the oft-assumed idea that creativity is something like mere novelty. On this view, musical creativity does not stand in opposition to well-formed musical rules but rather assumes them. The creative act is not about merely making never-before-heard noises, which could, after all, be a rather unpleasant experience, but rather about gaining new perspectives and interpretations of objectively rooted relationships.

Bach is extraordinarily creative because he can travel enormous distances, arriving by most ingenious paths, even while remaining within the possibility space determined by his starting material. Indeed, the farther he travels while preserving continuity, the more remarkable the achievement becomes.

In other words, the music of Bach, like the music of all great composers, is rooted in continuity within the musical space implied by a piece's starting material, thereby creating a true and coherent identity which is itself a mode of multiplicity in unity.

Multiplicity in Unity & Optimization Problem

There is an important consequence of the foregoing discussion, namely: multiplicity in unity imposes competing demands upon a composition. An uncompromising pursuit of unity tends toward repetition and musical poverty, whereas an uncompromising pursuit of multiplicity tends toward discontinuity and chaos. Great composition must therefore generate sufficient variety on the basis of a highly economical, unified musical idea.

What is accordingly at issue is an analogy with mathematical optimization. Suppose an engineer is designing a car and cares about two things: speed and safety. One can maximize speed by minimizing safety. Similarly, one could maximize safety by minimizing speed. Both approaches are rooted in straightforward maximization, but neither produces an optimal car if the goal is to make it as fast as possible while simultaneously as safe as possible. In other words, speed and safety are competing value factors.

Something analogous occurs in great music. Economy of means and musical variety are likewise competing value factors. Purely maximize economy and one arrives at musical poverty; purely maximize variety and one approaches chaos. The compositional problem is therefore not one of pure maximization but of optimization: to derive the greatest musical variety from the most economical adequate means.

Bach's genius is exemplary precisely in this respect. The extraordinary variety of his music does not require an equally extraordinary proliferation of independent musical ideas. Instead, astonishing multiplicity unfolds from remarkably economical material while remaining unified with the material from which it emerged.

Musical Creativity Is Not Algorithmic

My emphasis on structure and, in particular, the objective mathematical relations that give rise to the diatonic scale may give the reader the false impression that I view musical creation as a mechanical process. Nothing could be further from the truth.

The manifold of possible scores implied by an initial musical idea is typically infinite, leaving plenty of room for discovery even if the starting material bears a family resemblance to other known ideas. Moreover, musical discovery—the process of discovering unknown paths through a possibility space—is, by definition, non-algorithmic. The fact that one can later study this act or process and approximate it algorithmically implies neither that the original act was algorithmic nor that all processes are algorithmic.

As my basic metaphysical view is monadological, I follow Leibniz in folding subjectivity and objectivity together. Each rational spirit reflects not only the entire universe but, being made in the image of God, certain divine truths as well. Each rational spirit thus reflects infinitely many truths, but always from its own finite and particular perspective. A few things follow:

1. As certain truths come into sharper focus, others become less distinct or even recede beyond the horizon of conscious apprehension.
2. Although rational spirits reflect the same objective order, they remain capable of error because that order is apprehended from a limited and potentially distorted perspective.
3. Perspective therefore helps account for differences of opinion, judgment, and conclusion despite the reflection of the same objective truths in every rational spirit.

The creation of music, and the interpretation of existing music, are very much bound up with the finite perspective of a rational spirit, which involves not only the rational apprehension of mathematical relations and metaphysical principles but also private experience, values, emotions, and more.

I have said much here that I must necessarily defend elsewhere. For those interested in these underlying axiological and metaphysical ideas, see my Journey into Monadology and my 2020 book Gödel's God Theorem.

Private Online Guitar and Composition Lessons

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Students may pursue a Performer Track, a Composer Track, or a combined Performer–Composer Track. Depending on the student's goals, lessons may include guitar technique, improvisation, composition, arrangement, recording, mixing, mastering, and the development of complete musical works.

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