One of my favorite songs of all times is Jimmy Buffett’s Come Monday. Here is visual proof that I like it.

Denny's shoes

As I suspect you have, I too have been curious about the mathematics of the songs I like. A few weeks ago I looked into some of the mathematics of Come Monday.

On February 22, 2009, I posted an article on the note vibration frequency in Jimmy Buffett’s Come Monday. The notes of the song are played on a guitar and each note rings out with a particular vibration frequency when a string is plucked.

The article came from a discussion I had with a colleague about Chebyshev’s Theorem in statistics. The theorem is named after the Russian mathematician Pafnuty Chebyshev and deals with the relationship between the values of data in a collection and the mean (average) of those values. In summary,

The Mean: The mean of a collection of values can be thought of as the “expected value” of those values. If, from the collection, we randomly choose one the values, the mean value would be a good guess as to what that value would be. In a song, if we were to randomly select a note, the mean of the note vibration frequencies would be a good guess for the frequency of that selected note.

The Standard Deviation: A standard deviation might be thought of as a typical distance between a data value and the mean of the data values. For Come Monday, we might think of the typical distance between the frequency of any note and the mean of 319.7 Hz as 67.25 Hz.

Chebyshev’s Theorem says that

At least 75% of all the values in any set of data lie within 2 standard deviations of the mean of the data set. So, at least 75% of all the note frequencies in Come Monday should lie within 2 standard deviations of the 319.7 Hz. That is, at least 75% of all the note frequencies should like between 185.2 Hz and 454.2 Hz.

At least 89% of all the values in any set of data lie within 3 standard deviations of the mean of the data set.

At least 94% of all the values in any set of data lie within 4 standard deviations of the mean of the data set.

The Process: From the sheet music I bought for Come Monday, I noted the names and frequencies of the notes in the first 19 measures. I stopped at 19 measures only because I got tired of looking up all the names of the notes. But 19 measures is a lot and close to all the measures in the song. Here is an example.

Note names and frequencies

think sampling just 19 measures is fine because the Central Limit Theorem guarantees us that the mean of all the sample means is exactly the mean of the populations. I’ve got 19 sample means, and the mean of those means, so I know I am close to the population mean frequency of Come Monday.

I then computed, in an Excel spreadsheet, the mean frequency of the notes in each of the 19 measures. Here is the result.

The means of the 19 measures

You can see in the table that the mean of the frequencies is 319.7 Hz and that the standard deviation is 67.25 Hz. We can take this to mean that if we randomly chose a note from Come Monday, we could reasonably expect its frequency to be about 319.7 Hz. Also, we could reasonably expect most notes to have frequencies about 67.25 Hz from 319.7.

I don’t know the name of the note having frequency 319.7 Hz, but it is between a D# in the 4th octave at 311.13 Hz and an E in the 4th octave at 329.63 Hz. How do I know that? I that because I wrote down all the note frequencies for 8 octaves!

The 319.7 is between the D# and E of the 4th octave

Here is the frequency list. I keep mine in my pocket.

Notes and their frequencies

I tried to tune my Stratocaster to 319.7 Hz so I could hear what that note sounded like and then play it over Come Monday to see if that note frequency was dissonant or harmonious with the other frequencies. I could not tune any string to 319.7 Hz because guitar tuners apparently tune only to standard preset frequencies. Here is a picture of my guitar tuner with the setting on the high E string.

Almost a perfect E note

I visited a few music stores and none of them sold tuners that tuned to a specific frequency. In fact, when I asked the sales person at Guitar Tuner’s Universe, he called security and they tossed me out into the street and then threw my statistics book and slide rule at me. Punks!

I have the great fortune at teaching at a college that employees many bright and talented people. I asked Mel, Doug, LungYee and Anne, of the physics department about frequency tuners and they said “Hey man, use a frequency generator.” They got me one from the stockroom, and showed me how to use it. I like this thing! Here is what it looks like. I have it set at 319.7 Hz, the mean frequency for Come Monday.

A frequency generator set at 319.7 Hz

The Experiment: The following QuickTime videos demonstrated how Come Monday sounds with the 319.7 Hz frequency mean played over it as well as a couple of frequencies 1 and 2 standard deviations away from the mean frequency.

Each video starts with Come Monday playing. Then I start the frequency generator and turn it towards the desired frequency. Its interesting to start with a lightly different frequency and then tune it to the desired frequency. See if you can hear the notes move toward the Come Monday notes.

Video One is with the 319.7 Hz mean played over it.

Video One

Video Two is with the 319.7 Hz mean plus one standard deviation played over it.
319.7 + 67.25 = 387.2

Video Two

Video Three is with the 319.7 Hz mean plus two standard deviations played over it.
319.7 + 2*67.25 = 454.2

Video Three

Video Four is with the 319.7 Hz mean minus one standard deviations played over it.
319.7 – 67.25 = 252.4

Video Four

You will have to decide for yourself if the tones are dissonant or harmonious.