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helveticat
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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.

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?

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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helveticat
  • 512
  • 2
  • 11

What transformation corresponds to the z-relation?

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?