Non-euclidean geometry

yeah the title is click-bait-y, what I really mean is: complements of Euclidean patterns. For example:

euclid 3 8 = [1,0,0,1,0,0,1,0]  
euclid 5 8 = [1,0,1,1,0,1,1,0]

Since 3 + 5 = 8, we have eight 1 events in total (over both patterns).

Now: when we shift the second pattern to the left (one step)

euclid      3 8 = [1,0,0,1,0,0,1,0] 
euclidOff 5 8 7 = [0,1,1,0,1,1,0,1]

then events are disjoint (and their union is the full set). Can you prove that this always works? E.g., formalized like this:

For each 0 <= k <= n, there exists an offset s such that events in
euclid k n and euclidOff (n-k) n s are disjoint?

I think one can construct an answer from the facts that 1. Euclidean (that is, evenly spaced) sequences are essentially unique (that is, up to rotation); and 2. that Bjorklund's algorithm does produce the lexicographically smallest such sequence (I assume this is well-known - but I can't access some paywalled papers).

[EDIT] "lex. smallest" - of all rotations that start with 1.

NB - current implementation computes the complementary pattern directly. The above exercise (if true) would give an alternate implementation.


Yes, this is known, e.g., Emmanuel Amiot: David Lewin and maximally even sets, 2007, , cited in Toussaint: Geometry of Musical Rhythm.

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