Work and World

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. This Minute Physics video is brought to
you in part by audible.com, the leading provider of audio books across all types
of literature including fiction non-fiction and periodicals. If you go to
audible.com/minute physics you can try audible out by downloading a free
audiobook of your choice. I just read ‘The Name of the Wind’ by Patrick Rothfuss.
It’s a fantasy novel with a very music and scientifically oriented protagonist and I thoroughly enjoyed it. You can download this audiobook or a free audiobook
of your choice at audible.com/minutephysics and I’d like to thank audible
for helping me continue to make these videos
Video source: https://www.youtube.com/watch?v=1Hqm0dYKUx4

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