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The song has been described as a "self-referential and infinitely iterative children's song". I would like to use a word that encapsulates this description.

I have considered recursive, but does that necessarily imply self-referentiality? I don't think it would imply an infinite process either.

Eternal or never-ending only describes one facet.

A quine may be apt, but does the analogy really hold? Again, a quine doesn't imply infinite reproduction, or does it? They output themselves only once in examples I've seen.

Is there any appropriate term (via linguistics, computer science, physics, etc.) that completely captures the essence of the song (and similar works)?

Edit: I'm also considering ouroboros. My goal is to find something that is descriptive or directly analogous to the underlying pattern. I'm not sure if there's some jargon in other fields, biology, music, art, etc. that could apply. The fact that it's a children's song is irrelevant, so I have emphasised the crux of the quoted phrase.

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wouldn't that be the adjective 'ouroboric'? –  Jim Aug 30 '12 at 23:36
    
@Jim That'd work great, but I only see a definition on Wiktionary. Is this word found in other sources? –  Zairja Aug 30 '12 at 23:39
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Good question, Wordnik has it also but I don't know if OED does. But if you are looking for a word to fit in the blank: "The Song That Never Ends is _______." Then 'ouroboros' does not fit because it is a noun. A description of the structure should be an adjective. Maybe your could say, "It is like an ouroboros." "Recursive" fits, but recursion implies nesting which I don't the song has- it simply repeats. –  Jim Aug 30 '12 at 23:44
    
Cyclical definitely fits. It cycles and repeats. –  Rob Aug 31 '12 at 4:20
    
Maybe Fractal? –  Alexander Gladysh Aug 31 '12 at 6:08

5 Answers 5

Recursive is the correct answer, in my opinion.

I have considered recursive, but does that necessarily imply self-referentiality? I don't think it would imply an infinite process either.

By definition, recursion is self-referential, isn't it?

The number one definition you linked on wiktionary:

  1. drawing upon itself, referring back.

Recursion can be infinite:

Infinite Recursion

If you wanted to be more specific, you could say that The Song That Never Ends is an infinitely recursive children's song.

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I wouldn't be too obsessive about finding a single word. Do you need a single word to convey the fact that it is also a children's song? If english were akin to german we could say it is an infinitelyrecursivekindersong. –  ghoppe Aug 30 '12 at 21:00
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Also, I can't think of any definition of recursive that is not self-referential — that's the essence of the word's meaning, that is, it is implicit in the meaning of the word. I'm not sure what you're getting at when you say "language supports recursion but I don't see how it's self referential…" –  ghoppe Aug 30 '12 at 21:02
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But The Song That Never Ends is self-referential, right? (Another possible word for this kind of song would be headache-inducing.) –  J.R. Aug 30 '12 at 21:05
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The song is not infinitely recursive. As with recursive subroutines, someone will eventually blow their stack and interrupt you. –  MετάEd Aug 31 '12 at 4:52
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Recursion is not always self-referential. Consider routines which call one another recursively. And the self-referential character of the song is not just expressed in its recursive nature: it also speaks of itself. A word for this is autological. –  MετάEd Aug 31 '12 at 4:58

I think you’re looking for a sonic barber pole, or Shepard tone. From Wikipedia:

A Shepard tone, named after Roger Shepard, is a sound consisting of a superposition of sine waves separated by octaves. When played with the base pitch of the tone moving upward or downward, it is referred to as the Shepard scale. This creates the auditory illusion of a tone that continually ascends or descends in pitch, yet which ultimately seems to get no higher or lower.

It also notes that:

In his book Gödel, Escher, Bach: An Eternal Golden Braid, Douglas Hofstadter explains how Shepard scales can be used on the Canon a 2, per tonos in Bach's Musical Offering (called the Endlessly Rising Canon by Hofstadter) for making the modulation end in the same pitch instead of an octave higher.

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The Hofstadter reference is interesting. I initially thought of using "strange loop", as well; however, I felt that it was an overly complex description of the key attributes of the song. I'm still thinking and searching, though. . . –  Zairja Aug 30 '12 at 21:47
up vote 3 down vote accepted

I think that ouroboros may be the best choice for a single word.

If I had to pick two words, I would definitely choose infinite recursion, but I am limited to one word. I am probably going to accept my own answer, but I've given credit to the other answers. They were very useful and spurred me on to find the best fit for what I was looking for. If a better answer comes along (I don't care about the obscurity of the term), then I may change it.

If I may (mis?)appropriate some mathematical terms, the song could also be considered impredicative and idempotent -- unless a mathematician cares to interject.

idempotent - adj.

  1. (mathematics) (computing) Describing an action which, when performed multiple times, has no further effect on its subject after the first time it is performed.
  2. (mathematics) Said of an element of an algebraic structure (such as a group or semigroup) with a binary operation: that when the element operates on itself, the result is equal to itself.

impredicative -adj.

  1. . . .impredicativity is the property of a self-referencing definition. More precisely, a definition is said to be impredicative if it invokes (mentions or quantifies over) the set being defined, or (more commonly) another set which contains the thing being defined.
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NB: “idempotent” ≠ “impotent”. You may wish to fix your typo. –  tchrist Aug 30 '12 at 22:29
    
@tchrist Good catch. Typo fixed. I'm making all kinds of errors today. If the terms are too much of a stretch, then I may retract them. –  Zairja Aug 30 '12 at 22:32

My first thought was cyclical, then recursive.

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You might call the song tail-swallowing instead of ouroboric.

Also note that the specific kind of recursion used here is tail recursion, which (per link) “can be implemented without adding a new stack frame to the call stack”. That is, tail recursion, like the song itself, need not entail nesting.

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