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Forte's list of set classes includes some pairs that are "z-related" -- you can't get one from another through transposition and inversion, but they have the same interval vectors. Every interval vector is shared by at most two set classes.

Given a set class, such as 6-z3 (012356), is there a known transformation that produces its z-partner, in this case 6-z36 (012347)?

By "transformation" I mean a mathematical procedure to change from one PC set to the other. For example, multiplication by 11 (mod 12) transforms each PC set to its inverse -- which operation transforms each PC set to its z-partner?

In the specific case of 6-note sets we have the Hexachordal Theorem, so we can define the required transformation as taking the complement. I'm looking for a generalized transformation that works for any cardinality.

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The problem identified here is the subject of deep research in set theory, category theory, crystallography, and music theory — to name a few. There is no known generalized transformation that works across cardinalities.

Emmanuel Amiot's textbook Music Through Fourier Space: Discrete Fourier Transform in Music Theory (Springer, 2016) might be of interest. Also research articles by David Lewin, Allen Forte, Robert Morris, Emmanuel Amiot, Joseph Straus, Ian Quinn, Franck Jedrzejewski, and Cliff Calendar, among others.

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    Is the fact that every interval vector is shared by at most two set classes specific to 12-tones, or would it also be true for scales with different numbers of (equal-tempered) notes? I would strongly suspect there is provably no possible generalized transformation - I don't know what the precise mathematical statement might be, but maybe that finding the z-partner(s) is an NP-hard problem (if you consider arbitrary sized sets in scales with arbitrarily many equally-tempered notes).
    – Alexander Woo
    Commented Oct 2, 2023 at 4:37
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    Note - for contrast, that inverse is always multiplication by -1, no matter the number of notes in the scale.
    – Alexander Woo
    Commented Oct 2, 2023 at 4:38
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    @AlexanderWoo I believe that in 16-TET there are Z-triplets and that there are other groupings that pop up as the number of tones increases, but I can't immediately find a reference.
    – Aaron
    Commented Oct 2, 2023 at 4:43
  • Thank you -- that makes me feel better about not being able to figure it out on my own :-). I'll check out Amiot's book, it looks extremely interesting! I'd be interested to know about the links with crystallography (via group theory, I assume?) but can make another question for that if it's appropriate.
    – helveticat
    Commented Oct 2, 2023 at 8:30
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    @helveticat In crystallography, it has to do with non-congruent structures that produce identical diffraction patterns, because they contain the same sets of distances between atoms. Search for "homometry".
    – Aaron
    Commented Oct 2, 2023 at 8:43

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