Consider $n$ evenly spaced points on a circle representing $\mathbb{Z}^n$.
Two sets of points
with the same multiset of distances between them (measured by the shortest distance around
the circle) are said to be *homometric*.
In the music literature,
homometric point sets correspond
to pitch-class sets with the same intervalic content, and this
theorem is known as the "hexachordal theorem":

**Hexachordal Theorem**: Complementary sets with $k=n/2$ (and $n$ even) are
homometric.

In particular, Schoenberg realized that two complementary chords of six
notes each in a twelve-tone scale have identical intervalic content,
and so have analogous "aural effects."

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Figure from: Ballinger, B., Benbernou, N., Gomez, F., O’Rourke, J., & Toussaint, G. (2009, June). The Continuous Hexachordal Theorem. In International Conference on Mathematics and Computation in Music (pp. 11-21). Springer Berlin Heidelberg.
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The following history was uncovered by Godfried Toussaint in the early 2000's.

The hexachordal theorem was originally proved in the music literature by Lewin in 1960 (1), and
subsequently followed by many different proofs in the music-theory and
mathematics literature, including a proof by Blau in 1999 (2):

(1) Lewin, David. "Re: The intervallic content of a collection of notes, intervallic relations between a collection of notes and its complement: an application to Schoenberg's hexachordal pieces." *Journal of Music Theory* 4.1 (1960): 98-101.

(2) Blau, Steven K. "The hexachordal theorem: A mathematical look at interval relations in twelve-tone composition." *Mathematics Magazine* 72.4 (1999): 310-313.

However, the theorem was known to crystallographers about thirty years earlier,
who were interested because X-rays depend on inter-atom distances, and so
homometric sets have ambiguous X-rays. The theorem was first proved by Patterson (3),
and again followed by many different proofs in the separate
crystallography literature, including most recently
a proof by Senechal (4):

(3) Patterson, A. Lindo. "Ambiguities in the X-ray analysis of crystal structures." *Physical Review* 65.5-6 (1944): 195.

(4) Senechal, Marjorie. "A point set puzzle revisited." *European Journal of Combinatorics* 29.8 (2008): 1933-1944.

The separate literature threads were united by Toussaint, as mentioned above.

propositional logic(synonym:sentential logic). Yes, on the level of propositional logic, Fermat's last theorem is equivalent to 1+1=2, but these two theorems are not equivalent in most other logical frameworks. Another concept useful to answer such objections analytically and precisely isconstructive logic, where, roughly speaking, equivalences have a material existence, and the way from the product rule to Fermat's last theorem is rather long indeed. Maybe experts on constructivist logic could follow up on this? $\endgroup$namefor theconstant$\textsf{T}$. In a sense, there is a forgetful functor fromanyusual logic to sentential logic, treating any well-formed formula (in its entirety) as a variable-name, and any well-formed sentence as a name for either $\textsf{T}$ or $\perp$. Sentential logic is a very coarse framework. Note also that the comment does not mean to say that Christian Remling's comment is meaningless---to the contrary, it raises an important point. $\endgroup$5more comments