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Comment: Re:cure but... (Score 1) 234

by gnufrog (#37957476) Attached to: Mathematically Pattern-Free Music

Yes! That was, perhaps, the main point of the whole talk! The music is highly-structure in that it contains no repetition, at many levels.

1. No note is repeated. (It's a permutation)
2. The interval between all pairs of notes which are separated, in time, by k notes is different. (The Costas Property)
3. The difference between the the starting time between any two notes is different. (Using the Golomb Ruler for starting times)

Doing 1 & 2 requires the usage of a Costas Array, which we only know how to form thanks to a problem in SONAR - and requires Galois Field Theory to construct.

Comment: Re:Mathematics of Ramsey (Score 4, Interesting) 234

by gnufrog (#37953838) Attached to: Mathematically Pattern-Free Music

True. Apologies. What I was trying to say was that it's really hard to, via brute force search, find large Costas arrays. In fact, we've only just been able to enumerate all 29-by-29 sized Costas arrays (took nearly 400 years of CPU time). To find all 30-by-30's will take 5 times longer; Each time we increase the size of the array by one, it takes about 5x longer to enumerate the space (don't know why that's the case). So, needless to say, we're going to have to wait a while to find even a single array of size 88-by-88 by brute force search. But, thanks to Galois+Golomb+Costas, we can just multiple by 3, 87 times, and find one. So we can construct what is very difficult to find via brute force search. To use 'computation' to mean 'brute force search' was a poor choice. My bad...

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