Simon’s Fibonacci Music Pesano Periods

Simon writes:

I have composed a piece of music based on the Fibonacci sequence, using modular arithmetic (I assigned numbers from 0-6, the remainders after ÷ by 7, to notes C-B, i.e. 1-C, 2-D, 3-E, 4-F, 5-G, 6-A, 0-B. Then I added harmonies to the left hand). I noticed that after 16 notes, the sequence comes back to where it started!

But what really amazed me, is:

> I tried the same with Lucas #s, and Double fibonacci #s, and it always came back to where it started! Not only that, but always with the same length of period as well! It’s amazing!!!!

So, when you see something like this, you try to go over to a whiteboard and prove it, right? This is exactly what I did. In the vid below, you can see my proof of why this happens. I also analyze it a bit more, by seeing what is special of the Fibonacci #s, and also try ÷ by different numbers, instead of 7.

Disclaimer: Numberphile has already done a musical piece based on the Fibonacci numbers and a discussion of Pesano periods. What’s specific to my video:

* Trying different fibonacci-style sequences
* Proof
* What’s then special about the Fibonacci #s
* Making a table of different divisors
* (And, mathematics-aside, doing my composition in a more mathematical way, by being more strict about the melody)

Simon’s original score

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