I was trying to test music with math, so why not Fibonacci? I'd use it lowering every number of the sequence to 7 or minor, I mean, if a certain number n is greater than 7, I'd do n-1, until getting something lower than 7, then assimilate it to some musical note (A=1;B=2;...;G=7). As example, 10 would be 3, a C, and 22 would be 1, an A. I didn't make it number by number, simply programmed ten lines of code and they did the job.
The thing is, after 16 numbers, they repeat again and again. I called the pattern ƒ(7). Tried the same for 2, 3, 4, 5, 6, 8 and 12. Such that (let e[ƒ(n)] be the number of numbers in the pattern, so that e[ƒ(7)] = 16):
- e[ƒ(2)] = 3;
- e[ƒ(3)] = 8;
- e[ƒ(4)] = 6;
- e[ƒ(5)] = 20;
- e[ƒ(6)] = 24;
- e[ƒ(7)] = 16,
- e[ƒ(8)] = 12; and
- e[ƒ(12)] = 24
My question is: is there always a pattern for any positive integer? And if so, is it possible to know how many numbers would it have?