Are the discrete hexachords of a tone row always either the same set class or z-related?

Question

In a traditional 12-note tone row, it seems to me that the first six notes will always either be in the same set class as the last six notes, or will be z-related, but I'm having trouble confirming this. For example:

Schoenberg's Fourth String Quartet: 2 1 9 T 5 3 | 4 0 8 7 6 E

Both discrete hexachords are (014568)

Berg's Violin Concerto: 7 T 2 6 9 0 | 4 8 E 1 3 5

The first hexachord is (013468) and the last is (012469), both have interval vector <233331> and are thus z-related.

Are there exceptions to this? Alternatively, is there a proof of this fact without having to try all 12!=479,001,600 possibilities?

Answer 1

For the sake of future readers, I'd like to synthesize a full answer to the question so that it isn't only buried in a link inside a comment.

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1.

As Robert Fink's answer suggests, the TL;DR answer to the question is yes, the final hexachord of any tone row will either be the same as the first hexachord, or will be its z-partner.

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2.

A slightly longer answer suggested by a non-SE friend is as follows: The complement of any set-class of any size (that is, the set class of the notes remaining out of all 12 possibilities in the traditional Western system once you've removed the notes of the original set) will always be the same regardless of the specific notes. For example, the notes D, D#, F#, G#—a member of the all-interval tetrachord (0146)—has the complement C, C#, E, F, G, A, A#, B, a member of (01234689). A completely different member of (0146) would be G, A, C, C#; but its complement—D, D#, E, F, F#, G#, A#, B—is still of member of (01234689).

Obviously, this will still hold true for hexachords. Once you have the first hexachord of a traditional 12-tone series, the remaining hexachord is, by definition, the complement of the first. All that remains to prove the original hypothesis is the fact that the complement of any hexachord is either itself or its z-partner, as can be verified by looking at any hexachord list like the one in the Appendix of Straus' Introduction to Post-Tonal Theory.

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3.

A somewhat more formal proof by Steven K. Blau was provided in a link by Micah, the full paper is here: http://www.maa.org/sites/default/files/269122711809.pdf

I'll provide a brief summary. If we pick an arbitrary hexachord, let's call it A, then it implies a complementary hexachord (B) that marks its completion in a tone row. We can envision these two hexachords in a clockface diagram like so:

     _B   ^A   _{A
  B              B
 A                A
  B              A
      B       B}
          ^A

Now we imagine switching one of the A notes with one of the B notes, I'll do a switch of the 2 o'clock and 3 o'clock positions.

     _B   ^A   _{A
  B              A
 A}              ^⤹_B^⤣
  _{B              A
      B       B}
          ^A

The only intervallic changes in A will be matched by identical intervallic changes in B. For example, the A note that used to be at 3 o'clock was 3 semitones away from the A note at 12 o'clock, but it is now only 2 away. This changes the A hexachord's constituent interval content but we've simultaneously changed the B hexachord in that the B note that used to be at 2 o'clock was also 3 semitones away from the B note at 5 o'clock and is now only 2 away. In other words, at the same time that we changed one of the intervals in A from IC3 to IC2, we also changed one of the intervals in B in precisely the same way. The article at that link covers all possibilities (although I think the author accidentally reverses n+1 and n at one point)

Therefore, the interval content of the two hexachords will always be the same no matter what, which is just another way of saying that the discrete hexachords of any row will either be the same set class or z-related. QED

Answer 2

Yes, this is true. We refer to it, in the business, as The Hexachordal Combinatoriality Theorem. It can be mathematically proven, but I can't (without reliving some traumatic graduate school experiences) do it for you here. :)

Answer 3

As the author of some horribly complicated proofs of the hexachordal theorem, I am happy to provide the simplest (I think it was first published around 2004, not sure who found it first). It uses two mathematical tools, but fairly simple ones.

1. the characteristic function of a pc-set X is simply a list of 0's and 1's, depending whether the index is or not an element of X. Thze characteristic function of, say, C E G B or 0 4 7 11 is thus 1_X = (1 0 0 0 1 0 0 1 0 0 0 1).
2. the interval vector, complete, or rather IFunc(X, X) for a pc-set X can be expressed as something called a convolution product of characteristic functions. The convolution product of two functions f and g is defined by f*g (x) = sum f(t) g(x-t) —the sum runs for all t's, if a value exits the range 0-11 then add or remove 12 as needed. The iv satisfies iv(x) = sum 1_X(t).1_X(x+t) since the running term is 1 only when both characteristic functions give a 1, id est when both t and x+t are elements of X. Summing up, iv = 1_X * 1_(-X) where -X is the inverse of X.

Now the hexachordal thm is a one-liner: just compute the iv for the complement of X. Let us call it Y for clarity: 1_Y = 1 - 1_X (exchanges 0 for 1 and conversely !), hence

iv(Y) = (1 - 1_X)*(1 - 1_(-X))= 1*1 - 1*1_X - 1*1_(-X)+ 1_X * 1_(-X)= 12-6-6 + 1_X * 1_(-X) = iv(X)

(take care, 1 denotes here not a constant but the vector of values all equal to 1). Qed.