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Permutations Of The 12 Chromatic Tones

I have a question regarding the permutations of the 12 chromatic notes:  1x2x3x4x5x6x7x8x9x10x11x12 = 479,001,600.  Can anyone give me some advice on established patterns in usage or thoughts on looking for patterns of variation from the vast number of possibilities for arrangements of all 12 notes?.  I'm sure the Serialists have explored this and have reached some conclusions.  I am using the information for non serial compositions, but need some mathematical information.  Thank you.  

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  • To Carson Farley:  You might want to look at D.J Hunter and P.T. von Hippel's paper "How Rare is Symmetry in Musical 12-Tone Rows?" Math Asso. of America Monthly 110, Feb 2003.  This is an excellent paper or parsing the variability of which you may be concerned......

    Just a thought.......   --DT



  • Thank you David.  I will look into it.  

    - Carson

  • It's a lot easier if you zero index:


  • On a humorous note . . . 

    Connor - are you saying that 0 is the same as millions of permutations!  David Thompson's suggested paper (which I read last night) is much more helpful and constructive!

    Apparently the First Viennse School of composers favored symmetry according to the paper.  So everyone jump on the bandwagon.  And remember to add "cyclic shift" to the transformation list of P, I, R, and RI.  

  • Carson-

    If don't know how much you know about combinatorics in general (which is really what you need to understand to dig into such questions).  If you haven't read it before, you might check out Julian Hook's article "Why are There Twenty-Nine Tetrachords?" in Music Theory Online, linked here.  The article goes through a lot of basics about how you enumerate various musical elements and incorporate various symmetries into your calculations, even though it's not focused specifically on 12-note serialism.  The Hunter/von Hippel article is more directly on point to your question, but the Hook article lays out basic musical combinatorial methods in a way that may be more approachable (and might give some hints on how to do your own calculations on more specific questions).

  • Thanks for the information on Tetrachords John (I have certainly explored them in my own work and interested in the ancient Greek formulations). I look forward to exploring your reference.  I am somewhat familiar with combinatorial math having studied level 400 basic atonal theory.  Having decided to not follow that path in composition (serialism) since other math minded composers are so much more adept at it than I am; and other philosophical reasons. However, my interest is related to general logic and pattern and I am more intrigued with Hindemith's direction in composition based on physical relations in frequency and overtone series.  Are you familiar with his book on composition? Other compositional interests center around Synergetic Geometry.  



  • Hi Carson: Also look up Harald Ferptinger who has worked on these questions. You can actually find a lot more symmetries in the permutation group than are recognized as standard 12-tone operations. This paper from the Math and Computation in Music conference for instance addresses this topic:


    --Jason Yust


  • Thanks Jason!  I'm going to download the link immediately.