A sequence of numbers increases or decreases exponentially when you get each previous number 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 that you multiply by obviously changes for different sequences.
Some people claim that mathematicians are usually good musicians, although if you’ve ever heard me play the piano, you might not agree! The use of 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 or subdominant chord, even though we can’t name any of them. So something around us fits a pattern; that pattern is about to be revealed.
When a symphony orchestra tunes, one instrument, usually an oboe, plays a note and the other instruments tune in relation to it. That note is the A-up-middle-C that sounds when the air vibrates at 440 beats per second. A note one eighth (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 A will vibrate at 880 beats per second. The twelve spaces between the thirteen semitones of a scale are divided equally on 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”; At 440 Hertz (Hz).
As the pitch of each note goes up one semitone, the number of beats per second increases 1.0595 times. If you want to check the figures in the list below, you can 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 rounded 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 music fans among you, note that I had to put D# in place of 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’re most likely to stray into from time to time is E, or the dominant key of A, as it’s called. If you want to make a great final bar or two for your next piece of music, you’ll probably end with the E (or E7) chord followed by the final A chord.
Another interesting point here is that the key signature of A has 3 sharps, while the key signature of E has 4 sharps. More on that later.
There is a beautiful word in French: “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 “a foot and a half” (long) and is used sarcastically of people who use long words when shorter words will suffice. Another aside here is that the word sesquipedaliophobia means fear of long words. Sesquipedaliophobes won’t know, 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 by 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 note D in the list above – 294 – and “skew” it, increase it by the ratio 2:3. You will get 294 + 147 = 441 (should be 440, but we are approximating). So? A is the dominant key of D, and the key signature of D 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 one sharp at a time (G has one sharp).
do, sol, re, la, mi, si, fa#, do#. What will he do. Notice that they go up by the musical interval of a fifth. To go “down” from C, a sharp is removed, or in other words, a flat is added.
I don’t have the space to show you how to tune a guitar, but it’s related to this work and it’s much clearer because you can see the relationships of the keys on the fingerboard. Maybe another article later?