# 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 Pi Composition

Simon has finished working on his Pi Composition, a piece of music entirely based on Pi (in the treble clef). Simon used only the digits of Pi he knew from memory (the first 19 digits), assigning each digit to a specific frequency on the grand piano (the 4th octave and three notes in the 5th octave). The accompanying tune for the left hand (bass clef) is not based on Pi and is purely added for the purpose of harmony (with the help of Simon’s piano tutor.

Simon has already made two other music videos about number sequences: one about the Recaman Sequence and one about the Fibonacci sequence. The Pi Composition, however, is the first time he has tried turning a sequence into an artistic peace, with rhythm and harmony.

# Home Concerts: Pachelbel

Just a typical evening scene from our home. Simon practicing Canon in D by Johann Pachelbel, Neva improvising a dance.

Below is another recording of Simon playing the whole piece in a more serious manner:

# Map of the piano chords

During his piano lessons, Simon has been working on a diagram that would map all the possible chords on the piano. I gave him a huge roll of paper to draw on that he spread on the floor of his piano teacher’s studio. He said he wanted to create a network of chords and how you go to other chords. “Music is basically like finding a path through the network that you like, messing with but preserving the chords in the network. What do I mean by preserving? There’re 4 things that you can do to a chord to preserve it:

• move some of the notes in the chord by multiples of an octave (like 1 octave, 2 octaves, 3 octaves, etc);
• split up or mix some of the notes in the chord (taking one note and splitting it into two copies of the same note you start with but in different octaves or mixing the note from two octaves into one);
• get rid of some notes in the chord;
• there are some notes you can add to the chord and preserve it.

So far, Simon has been able to map C Major and A Minor tonalities. He got a little bit stuck, but is determined to continue.