Prologue to this blog..



Self reference invariably leads to contradiction, and Bertrand Russell's eponymous paradox, which he communicated to Gotlieb Frege around the turn of the (20th) century, was no exception, rather, it would become the canonical example that shook the axiomatic foundation of naive set theory and mathematics itself.

It really shouldn't have had such a profound impact, to be honest; It was no more surprising (nor resolvable) than children's diversions such as "This statement is false. Is my statement true or false?". It caused a great deal of consternation amongst logicians because they're a breed that has little tolerance for the indefinitive, unlike Euler for instance, who famously solved the Basel Problem on intuition, but didn't rigorously prove the result, at least by present day standards. The gulf between the two is vast. Russell's Paradox is easily conceived by anyone, whereas Euler's solution was devilishly clever and had evaded the best mathematical minds of Europe for about a century.

To frame the problem for the gentle reader without any knowledge of set theory, imagine the infinite lists of things that could be constructed. Some of those lists themselves contain other lists, eg:


List A

-the list of long haired dogs

-one particular Etruscan knife

-list A


Notice that the last item is 'List A' itself. So, lists can contain any object, includng both other lists, themselves, or other lists that contain themselves. What Russell suggested was The List Of All Lists That Do Not Contain Themselves, and asked, does this list contain itself?


Now, I got to contemplating preludes and fugues last night, and their inherently self referential nature, particularly when, as I often have, the composer of such odious objects constructs the prelude out of motivic material borrowed from the fugue's episodic passages or countersubjects. In fact, early on, I endeavored to construct the prelude exclusively from the elements of the corresponding fugue that were not the fugue's subject(s) proper, in order to lend some thematic continuity and significance to the pairing, instead of merely sticking two pieces together that sounded nice, or complementary, side by side.

Yet this still falls short of an absolute motivic coupling of the prelude and fugue, without the introduction of the subject itself to the prelude. And while we're there, why not just present the exposition, too? Well, you can imagine, this epiphany tore me from my chair, slammed my against the wall, slapped my face and embraced me in a passionate kiss, just as in one of those old, corny war-era movies.

I saw at once the artistic economy and an opprtunity for respite from the arduous and miserable task of composing these verbose, bloated objects of immitative tedium we smugly refer to as 'fugues'. My contempt for fugue welled up inside as I envisioned a musical lanscape liberated from the abject objects of contrapuntal trickery, a brave new world in which preludes are written to themselves.

And these benefits are not constrained to just the prelude and fugue. One can easily imagine the analogous dismissal of books, stories, and essays, as well, where prologues to themselves serve the purpose succintly. The world is cluttered with books and fugues, and I've been guilty of contributing to that littering on my own part, but I'm trying to change my ways.







a draft of Prelude to Itself and Itself in G:





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  • Self-reference is not necessarily a bad thing. Computer programming, for example, often deals with recursion and reification, both of which involve self-reference. Recursion is as old as math itself, and leads to things like the Fibonacci sequence and functional fixed points (cf. orbits and chaotic bifurcation). Reification is the act of representing (some or all aspects of) a program within itself, so that it can inspect itself and direct its computations accordingly. Java's class introspection is an example of an application of this concept.

    Also, there are alternative set theories that actually allow self-referential sets. For example, the so-called New Foundations is a set theory that allows the Quine Atom: a set that contains itself. Interestingly enough, NF is constructed in such a way that Russell's paradox is not constructible within the theory. So the theory does not suffer from such contradictions or paradoxes.

    But enough about math. When it comes to art, there is no obligation to avoid contradiction. In fact, I daresay art thrives on contradiction, on the juxtaposition of unlike or contradictory elements in order to produce a lasting impression on the audience. Like Escher's mind bending geometries, the impossible cube, the tumbling cube illusion, the staircase looped back upon itself upon which you could ascend or descend indefinitely, a river flowing downstream that becomes its own source - such things are impossible in the real world, but art is not confined by such practical considerations. Indeed it thrives on the impossible.

    And so even though I have written 9 odd fugues (and have many more WIPs), I have never written a prelude that goes along with them. Never intended to emulate Bach in that sort of way, and never felt the need to preface my fugues with preludes. Maybe one of these days I could try it out just to get a taste of what it's like, but I feel no obligation binding me to commit to such a form.

    So just as you have a prelude to itself, I have fugues unto themselves. 😄 Maybe I should write a fugue whose subject is the fugue itself... now *that* would be something interestingly self-referentia. 😋

  • HS,

    No matter how you attempt to rally to the defense of necromancers, witches, and warlocks who might practice their dark arts of self referential immitative contrapuntal moral deceipt, and heap its iniquities and corruption upon unsuspecting mankind, I have exposed the dark [motiffs] smile behind their scorcery, under the black cover of night with incantations strange and unholy, and forthwith banish their tarnished souls to eternaal, flaming damnation.

  • But you gotta admit, a fugue on itself as the subject is an awesome self-referential idea that must be explored at some point. 😉🤪

    It would be the musical equivalent of a Grothendieck universe, built up by basic operations (fugal devices) iterated transfinitely to a strongly inaccessible cardinal.

    Grothendieck universe
    In mathematics, a Grothendieck universe is a set U with the following properties: If x is an element of U and if y is an element of x, then y is als…
    • "Fugue on itself"

      Yes, this was the objective of my Fractal Transform Algorithm, if you recall. Those were fun, and challenging, too.


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