What, if any, is the relation between Cantor's function and Ligeti studio: Devil's Staircase?

  • $\begingroup$ This is not an appropriate forum for your question. I've voted to close. $\endgroup$ – Ryan Budney Jul 10 '13 at 0:07
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    $\begingroup$ This is a legitimate question since Ligeti's etude seems to be specifically inspired by the mathematical function (see monashcomposers.files.wordpress.com/2009/09/ligeti-13-page1.png for the start, and hear youtube.com/watch?v=1ZTaiDHqs5s for the whole thing). It might even belong on mathoverflow, at least as a reference request for a published exploration of this connection. $\endgroup$ – Noam D. Elkies Jul 10 '13 at 5:38
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    $\begingroup$ A few minutes on Google turn up "On Musical Self-Similarity" by Di Gabriel Pareyón, which reports on p.359 that the piece was written "under the advice of mathmeatician Heinz-Otto Peitgen" and "exhibits recursive qualities, whilst its hemiola rhythmic cells of 2 to 3 recall the binary-ternary geometry of the devil's staircase." That's a start, and maybe there's a more thorough exploration to be found. $\endgroup$ – Noam D. Elkies Jul 10 '13 at 5:42
  • $\begingroup$ [Oops, I didn't notice that this had already been answered $-$ I might have noticed that the scroll bar indicated an extension below the "put on hold as off-topic" block] $\endgroup$ – Noam D. Elkies Jul 10 '13 at 5:58
  • $\begingroup$ math.stackexchange.com/questions/141082/… $\endgroup$ – Henry Zorrilla Jul 10 '13 at 6:08

There is a Masters thesis by Lauren Halsey entitled, "An examination of rhythmic practices and influences in the keyboard works of György Ligeti" (UNCG link), which addresses your question:

"...the idea for this etude emerged: “an endless climbing, a wild apocalyptic vortex, a staircase it was almost impossible to ascend.”28 This etude shares the name and characteristics of the mathematical concept of a “devil’s staircase.” This phenomenon, based on Cantor Sets, involves the relationship of disproportional segments combining to create a self-similar group.29 This concept is also used in the “mode locking” features of clocks and pendulums.30 Ligeti expresses this concept with the inclusion of groups of two and three eighth notes that, when combined, create a self-similar rhythmic set. This grouping structure creates pulse streams and defines the formal boundaries of this piece. The structures seem to spiral infinitely up the piano, suddenly falling down to the lowest octaves.31

     Figure 4.4
And here is a graph from Wikipedia's Cantor function article of the Devil's Stairase:
 Cantor Graph

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    $\begingroup$ I followed your link @Joseph and found Halsey thesis. Nice work, in pages 51-54 we can see Ligeti's concern about chaos and fractals and how he developed this etude. Thanks for your reference. $\endgroup$ – tatojo Jul 10 '13 at 0:49

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