49: The Overtone Series

The overtone series is a key musico-mathematical concept. The way I’ll explain it will largely be in terms of string lengths because I’ve been around string instruments for 20 years, but the math translates exactly to columns of air and their lengths, for wind players.

Imagine you have a string that vibrates at 196 Hz—the G below middle C, the lowest violin string. If you barely touch your finger to the string at exactly the point that cuts its length in half, and then you play that note, you’ll get a note that sounds exactly 394 Hz—exactly double, or exactly an octave higher. This is the “first harmonic” of that string. (The “zeroeth” harmonic is the fundamental frequency of the string when played completely open, in other words, the original 196.)

Following a series of completely-reduced (i.e., 1/2, not 2/4 or 3/6) ratios, it is possible to build the next several harmonics—the second, third, fourth, etc.

·       The fundamental

·       An octave above the fundamental (2:1 with the fundamental)

·       A perfect fifth above the previous (3:1 with the fundamental, 3:2 with the previous)

·       A perfect fourth above that (4:1 with the fundamental, 4:3 with the previous)

·       A (slightly flat) major third above that (5:1 with the fundamental, 5:4 with the previous)

·       A (nearly-perfectly-in-tune) minor third above that (6:1 with the fundamental, 6:5 with the previous)

·       And so on

You are guaranteed to be perfectly in tune at every ratio/harmonic which is an exact power of 2—1:1 with the fundamental is the fundamental itself, 2:1 is an octave up, 4:1 is 2 octaves, 8:1 is 3 octaves, and so on. You are also guaranteed to be—sometimes slightly, sometimes very noticeably so—some amount of out-of-tune everywhere else, since the way we actually build other intervals isn’t so nice and neat.

Keeping one string and only changing the ratios involved with where your finger “cuts” the string, it is possible to play up and down that string’s overtone series.

There are a few key takeaways that should be absolutely clear before the next article:

1.       The series of n:1 for every integer n, against the fundamental, produces the harmonic series of that fundamental

2.       Powers of 2 are always perfect, and everyone else is off by some amount

3.       The sequence of intervals of the various inputs of n:1 does not change if the fundamental does

4.       Natural horns, trumpets, etc., can only play the notes of one series; valved instruments have access to every series, so they can play every note perfectly in tune

 

 



 

 

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