I am taking a long time to check whether or not pitch class set inversions are right, but there may be a shortcut. That's why I am asking for your help instead of procrastination and erroneousness.

If, e.g., there is a pitch class set like:

40 4-11 0135 121110 Phrygian Tetrachord

  • The PRIME column is 0135.
  • The first inversion from the second pitch (0{1}35) of the 0135 prime goes like 024E;
  • The second inversion from the third pitch (01{3}5) of the 0135 prime goes like 029T;
  • The third inversion from the fourth pitch (013{5}) of the 0135 prime goes like 078T.

So the number of inversions depends on the type of chord for example: trichord, tetrachord, pentachord, hexachord etc.

Two questions, basically:

  1. Is there a website floating around the internet which has pitch class set prime inversions?

  2. If there aren't any pages you could share for pitch class set prime inversions, could you help with a simple software or a mathematical method for inputting pitch class set prime inversions?

  • 2
    Are you trying to create a new meaning of the word "inversion" as it applies to pitch sets? It sounds like you're trying to apply a basic version of triadic chord inversion to sets, but changes in ordering like that aren't what we usually mean when we talk about inverting a set. In fact, the rotations you're doing don't produce inversions, they are just transpositions. Commented Apr 9, 2016 at 10:45
  • These rotations aren't even transpositions. They are simple rotations which generate new pitch class sets that share the same prime form. A transposition, when normalized by modulo 12 should claim the same pitch classes as the original set, which is not the case when you simply rotate a non-symmetric set. Rotation, nonetheless, is also an important operation for composition and analysis based upon post-tonal theory.
    – SeuMenezes
    Commented Apr 9, 2016 at 22:43
  • 3
    @SeuMenezes No, they're transpositions. OP started with normal form [0135]. The first rotation was 02411, which is [E024], T-11 of the starting set. Next was 02910, which is [9T02], T-9 of the original. I'm not sure I follow your definition of transposition, a set class is by definition the set of all possible transpositions and inversions of the same basic collection. If two sets have the same prime form, they are by definition either inversions or transpositions (or both) of each other. OP seems to just be moving the first number to the end, then transposing so the new ordering starts on 0. Commented Apr 10, 2016 at 0:47
  • But anyway, the problem is that it's not clear what transformation the OP is trying to do. Set Theory is explicitly not interested in specific ordering when classifying sets. I suspect there's some basic confusion about the inversion operation in the set-theoretical sense vs. chord inversion in the common-practice sense. Commented Apr 10, 2016 at 0:50
  • @PatMuchmore that's nice! I did not realize that the resulting sets were rotated transpositions. Regarding the question, to me it is clear now that the OP is taking the definition of "inversion" as applied to tertian chords and trying to do this to pitch class sets. I am editing my answer to reflect this fact.
    – SeuMenezes
    Commented Apr 10, 2016 at 2:09

1 Answer 1


Edit: As noticed by Pat Muchmore in his comment, it looks like you are trying to apply inversion to a Pitch Class Set as you would do to a common chord (by simple rotation, which in fact results in transposition rather than inversion). The more straightforward way to invert a Pitch Class Set is to take each pitch class and subtract it from 12. In your example, 0 1 3 5 becomes 12 11 9 7, and then 0 11 9 7 via Mod 12 for pitch classes greater than 11.

Anyway, regarding the second item in your question, there is a piece of software called PCN (Processador de Classes de Notas) written by Brazilian professor Jamary Oliveira, which automates pretty much every operation involving pitch class sets. I think you would benefit from giving it a try. You can find the software at http://www.angelfire.com/music2/bahia/pcn/pcn-en.html.

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