Why It's Impossible to Tune a Piano

Aah, the sound of shaking animal intestines.. I mean, strings which are traditionally

made out of cat gut but regardless of what it's made out of when a string

vibrates it does so with the ends fixed to the instrument. This means that it can only

vibrate in certain waves, sin waves. Like a jump rope with one bump or two bumps or

three or four or some combination of these bumps. The more bumps the higher

the pitch and the faster the string has to vibrate. In fact, the frequency of a

strings vibration is exactly equal to the number of bumps times the strings

fundamental frequency that is, the frequency of vibrations for a single bump.

And since most melodious instruments use either strings or air vibrating

pipes which has the same sinusoidal behavior it won't surprise you to hear

that musicians have different names for the different ratios between these pitches. In

the traditional Western scale, 1 to 2 bumps is called an octave; 2 to 3 is a perfect fifth;

3 to 4 is a perfect fourth, then a major third, minor third some other things that

aren't on the scale and from 8 to 9 bumps is a major second or whole step. If you play a few of these

notes together you get the nice sound of perfect harmony. Hence the name for this band of

pitches, harmonics. In fact a sound that matches one of the harmonics of a string can cause

that string to start vibrating on its own with their resonant ringing sound. And a bugle

playing taps uses only the notes in a single

series of harmonics which is part of why the melody of taps rings

so purely and why you can play taps with the harmonics of a single guitar string.

Harmonics can also be used to tune string instruments. For example, on a

violin, viola or cello, the third harmonic on one string should be equal to the

second harmonic on the next string up. Bassists and guitarists can compare the fourth

harmonic to the third harmonic on the next string up but then we come to the piano or

historically the harpsichord or clavichord but either way the problem is

this: it has too many strings. There's a string for each of the 12 semi tones of

the Western scale times seven. If you wanted to tune these strings using

harmonics you could for example try using whole steps that is you could

compare the ninth harmonic on one key to the eighth harmonic two keys up which works fine for the

first few keys; but if you do it

six times, you'll get to what's supposed to be the original note an octave up

which should have twice the frequency.

Except that our harmonic tuning method multiplied the frequency by a factor of

nine eighths each time and 9 over 8 to the 6th is not two, its 2.027286529541 etcetera. If you tried

harmonically tuning a piano using major thirds instead, you'd multiply the

frequency by five fourths three times or 1.953125, still not two. Using fourths

you'd get 1.973 not two. Fifths gives 2.027 again. And don't even try

using half steps; you will be off by almost 10 percent and this is the problem. It's

mathematically impossible to tune a piano consistently across all keys using

perfect beautiful harmonics, so we don't. Most pianos these days use what's called

equal tempered tuning where the frequency of each key is the 12th root of two

times the frequency of the key below it. The 12th root of 2 is an irrational number

something you never get using simple ratios of harmonic tuning;

but its benefit is that once you go up 12 keys you end up

with exactly the 12th root of 2 to the 12th or, twice the frequency. Perfect octave!

However, the octave is the only perfect interval on an equally tuned piano. Fifths

are slightly flat; fourths are slightly sharp; major thirds are sharp, minor thirds are

flat and so on. You can hear a kind of "wawawawawa" effect

in this equal tempered chord; which goes away using harmonic tune. But, if you tuned an instrument

using the 12th root of 2 as most pianos, digital tuners and computer instruments are, you can play

any song, in any key, and they will all be equally and just slightly out of tune.

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