A sequence of numbers increases or decreases exponentially when you get each number from the previous one in the sequence by multiplying it by a particular number. For example, the sequence 1, 2, 4, 8, 16 is generated by multiplying by 2. This number you multiply by obviously changes for different sequences.
Some people claim that mathematicians are often good musicians, although if you ever heard me play the piano, you might not agree! Using music to introduce exponential growth follows a valuable theme: there is a mathematical pattern in something we can recognize around us. Most of us can hear an octave in music, or a dominant chord or a subdominant, even though we may not be able to put a name to any of them. So, something around us fits a pattern; that pattern is about to be revealed.
When a symphony orchestra tunes up, one instrument, usually an oboe, plays a note and the other instruments tune relative to it. That note is the A-above-middle-C which sounds when air vibrates at 440 beats per second. A note one octave (8 notes or 13 semitones) below this is heard when air vibrates at 220 beats per second (220 is half of 440). An octave above the A will vibrate at 880 beats per second. The twelve spaces between the thirteen semitones of a scale are equally divided these days. This division is called “equal temperament” and is what J. S. 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 rises in pitch by one semitone, the number of beats per second increases by 1.0595 times. If you want to check the figures 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 at A. The figures 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 the musical fuss-pots among you, note that I had to put D# rather than E flat because there is a symbol for “sharp” – the hash sign – on a keyboard, but not one for “flat.”
When playing music in the key of A, the other key you are most likely to drift into from time to time is E, or the Dominant key of A which is what it is called. If you want to make a grand final bar or two to your next piece of music, you will probably finish with the chord of E (or E7) followed by the final chord of A.
Another interesting point here is that the key signature of A is 3 sharps, while the key signature of E is 4 sharps. More of that later.
There is a lovely word in English: “sesquipedalian.” “Sesqui” is a Latin prefix meaning “one and a half,” while “pedalian” gives us “feet.” Notice the word “pedal” here. So the word means “one and a half feet” (in length) and is used sarcastically of people who use long words when shorter words will do. Yet another aside here is that the word sesquipedaliophobia means a fear of long words. Sesquipedaliophobics will not know that, of course!
Now back to music: sesqui, or the ratio of 2:3 takes us from the beats per second of a key, to the hertz of its dominant key. A has 220 Hz. Increase it in the ratio 2:3 and you get 330, the Hz of E, the dominant key of A.
The fun continues! Look at the Hertz of the note D in the list above – 294 – and “sesqui” it, increase it in 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 D’s key signature has 2 sharps to A’s 3.
To summarize: here are the keys in “sharp” order, starting with C which has no sharps in its key signature, and increasing by one sharp at a time (G has one sharp).
C, G, D, A, E, B, F#, C#. That will do. Notice they go up by the musical interval of a 5th. To go “downwards” from C, you take away a sharp, or in other words, you add a flat.
I do not have space to show you how to tune a guitar, but it is related to this work and is a lot clearer because you can see the relationships of the keys on the fret-board. Perhaps another article later?