I've spent this morning tutoring at a Year 8 Mathematics Masterclass on "Mathematics and Music", for which I have to admit to only having competency in one half thereof, but in spite of that it was very interesting. Towards the end of the lecture, my attention was drawn to the existence of the Fibonacci Waltz, a composition based on the Fibonacci sequence of numbers. How it is composed is by assigning a note to each number, so 1 is C, 2 is D and so on. Sharps are not considered for the purpose of this (although if someone wishes to attempt further composition...), and similarly octaves are disregarded so number 8 corresponds back to C, essentially meaning running the Fibonacci sequence in a form of modular 7 arithmetic.
There are a couple of surprises in this piece, the first of which is that it is somewhat aesthetically pleasing! Mathematically though, I was surprised to see that it has a repeating sequence of 16 notes, and I'm now rather keen to see why. For a start, I would have probably expected the repeat to be some multiple of 7. Further to that, would the sequence repeat if sharps were included (turning it into a form of mod 11 arithmetic)? If it does, what about other modular bases? This is something that requires further investigation, I reckon, and I can already see an A-level class investigation brewing over this one...
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