Way back in time on March 8, 2009, I posted the blog article Note Frequencies in Jimmy Buffet’s Come Monday. In that article I computed and determined the average note frequency in Jimmy’s song Come Monday. By average note frequency I mean the average frequency in string vibrations measured in Hertz (Hz). One string vibration per second equals 1 Hz. Higher pitched notes have high Hertz values and lower pitched notes have low Hertz values. I noted in that article that the average (mean) note frequency of Come Monday was 319.78 Hz and used a frequency generator to produce the sound associated with that frequency. If you are reading this article you probably don’t have much to do, so check out the Come Monday article at http://www.dennymath.com/?p=351.
Since I posted that article, I have been thinking about connections between mathematics and music. I started wondering if songs might have equations associated with them and if so, what the graphs of those equations would look like. The thought didn’t seem that strange to me since sheet music looks somewhat graphical. As you look at a song’s sheet music, dots rise and fall, just like one sees when looking at the graph of discrete data points.
Because my niece Randi Jean loves Beatles music and because she can exert great influence on cosmic events, I decided to try to graph a Beatles song. I bought a book of Beatles sheet music and decided to see if I could create a graph of the song I’ll Follow the Sun.
Randi Jean listening intensely to a Beatles tune
You can see a youtube version of I’ll Follow the Sun as performed by Paul McCartney at
http://www.youtube.com/watch?v=LfO1nbCX0g8&feature=related
The Process
1. I copied the music for the song on my printer.
2. The song is in 4/4 time and the shortest note in the song is an 1/8 note. So, I decided to count 8 beats to each measure.
3. From wikipedia I copied a nice picture of notes so that I would have an idea of each note’s frequency value. See Figure 1.
http://en.wikipedia.org/wiki/Musical_notes
Figure 1 - Note frequency vs note name
4. Using Excel, I made a list of the notes and their corresponding frequencies. See Figure 2. Click on the picture to see it larger.
Figure 2 - Notes and their frequencies
5. Next to each note, I wrote in red ink the note’s frequency in Hertz (Hz).
6. Then next to each note I wrote in blue ink the number of beats the note required to be played.
Figures 3 and 4 show the a cropped version of the sheet music and my notation. Click on the picture to see it larger.
Figure 3 - Verse 1
Figure 4 - Refrain
The following steps show how I created the lists of data, the scatter diagram, the coordinate system, the piecewise function, and the graph of the data using the T-Nspire. They are just a how-to-do-this type of thing. If you want to skip it, scroll down to the The Graph.
7. I accessed my copy of the new TI-Nspire software that emulates the Texas Instrument’s TI–Nspire calculator.
8. Using Nspire, I first created a list of values. The first column in my list is the beat number, and the second column, the note frequency. See Video 1.
9. Then, still using Nspire, I created a scatter diagram of my data. See Video 2.
10. Still using the Nspire, I created a piecewise linear function of the data in the beat vs Hz lists. See Video03.
11. Video04 shows a summary of steps 8-10.
12. Video 5 shows a minor change in the horizontal axis, one that makes the graph look a little better.
13. The following four figures shows the four piecewise functions I constructed to create the graph.
Function, 
1 produces verse 1.
Function 2 produces verse 2.
Function 3 produces the first part of the refrain, and
Function 4 produces the last part of the refrain.
The Graph
Figure 5 shows a black and white version of the graph of I’ll Follow the Sun when plotted as a piecewise function.
Figure 5
Figure 6 shows a colorized view of the graph to make it easier to see where the verses and refrain start and end. Click on the picture to make it larger.
Figure 6
I don’t know yet how to label axes on the Nspire. I will work on that.
I think what I will try next is to graph another Beatles song, or even a few more Beatles songs of approximately the same size, (number of measures) and compare them to each other. I am not sure just yet how I will compare them, but my colleague Joe suggested I use a subtraction method to see how various differences compare with each other.
Mathematical Marvels
