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Exponential Growth and Music

A sequence of numbers increases or decreases exponentially when each number from the previous one in the sequence is obtained by multiplying it by a given number. For example, the sequence 1, 2, 4, 8, 16 is generated by multiplying by 2. This number that is multiplied by obviously changes for different sequences.

Some people say that mathematicians tend to be good musicians, although if you ever hear me play the piano, you might disagree! Using music to introduce exponential growth follows a valuable theme: there is a mathematical pattern to something we can recognize around us. Most of us can hear an octave in music, or a dominant chord, or a subdominant, although we may not be able to put a name to any of them. Thus, something around us fits a pattern; this pattern is about to be revealed.

When a symphony orchestra tunes, one instrument, usually an oboe, plays a note and the other instruments tune it in relation. This note is the Aa above middle C that sounds when the air vibrates at 440 beats per second. A note an octave (8 notes or 13 semitones) below this is heard when the air vibrates at 220 beats per second (220 is half of 440). An octave above A will vibrate at 880 beats per second. The twelve spaces between the thirteen semitones of a scale are divided equally these days. This division is called “equal temperament” and is what JS Bach meant when he used the title “The Well-Tempered Clavier” for one of his major works.

The technical term for “beats per second” is “Hertz”; A has 440 Hertz (Hz).

As each note increases in pitch by a semitone, the number of beats per second increases by a factor of 1.0595. If you want to check the numbers in the list below, you might like to take this increase as 1.0594631.

Here is a list of the beats per second for each of the notes (semitones) in a scale starting with A. The numbers are to the nearest whole number. Note that the sequence of numbers is an exponential sequence with a common ratio of 1.0594631. Who would have guessed?

A is 220 Hz, A# is 233, B is 247, C is 262, C# is 277, D is 294, D# is 311, E is 330, F is 349, F# is 370, G is 392, G # is 415, A is 440.

For those of you out there, please note that I had to put D# instead of E flat because there is a symbol for “sharp” (the hash sign) on the keyboard, but not one for “flat”.

When playing music in the key of A, the other key you’re most likely to encounter from time to time is E, or the dominant key of A, which is what it’s called. If you want to make a big final bar or two in your next piece of music, you’ll probably end up with the E chord (or E7) followed by the final A chord.

Another interesting point here is that A’s armor is 3 sustained, while E’s armor is 4 sustained. More on that later.

There is a lovely word in English: “sesquipedalian”. “Sesqui” is a Latin prefix meaning “one and a half”, while “pedal” gives us “feet”. Notice the word “pedal” here. Hence, the word means “a foot and a half” (in length) and is used sarcastically for people who use long words when shorter words will do. Another aside here is that the word sesquipedaliophobia means fear of long words. Sesquipedaliophobes won’t know, of course!

Now back to the music: sesqui, or the 2:3 ratio takes us from the beats per second of a key, to the hertz of its dominant key. A has 220 Hz. Increase it by 2:3 and you get 330, the Hz of E, the dominant key of A.

The fun continues! Look at the Hertz of the D note in the previous list – 294 – and “sesqui”, increase it by the ratio 2:3. You will get 294 + 147 = 441 (it should be 440, but we are approximating). So? A is the dominant key of D, and the key signature of D is 2 sharps to A’s 3.

In short: here are the keys in “different” order, starting with C which has no sharps in its key signature, and going up one sharp at a time (G has a sharp).

C, G, D, A, E, B, F#, C#. This will do the trick. Notice that they raise the musical interval by a 5th. To go “down” from C, remove a sharp, or in other words, add a flat.

I don’t have space to teach you how to tune a guitar, but it is related to this work and is much clearer because you can see the relationships of the keys on the fingerboard. Maybe another article later?

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