I’ve recently been somewhat interested in the mechanics/physics of musical harmonics, and in my process of learning more I stumbled upon the claim that it’s impossible to perfectly tune a piano. Several videos I found on YouTube explained that even if you get an octave perfectly in tune, the other intervals (such as a third or a fifth) would be slightly out of tune. Why is this?
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Our musical scale is based on frequencies. If you go up one octave, you double the frequency. However there’s no “rational” way to divide that doubling into equal-sized steps. How a chord sounds is based on the ratio between two notes’ frequencies. A fifth is when one note’s frequency is 3/2 the other’s. A fourth is when the ratio is 4/3. If you start at a note, go up a fifth, and then go up a fourth, your frequency increases by (4/3) x (3/2) = (12/6) = 2. So you go up by exactly one octave. However, most of the intervals don’t consist of ratios that can multiply together perfectly up to 2, so you either can have some of the chords perfect and others (like if you change key) further off of perfect, or you can make everything equally spaced so it loses that perfection but is pretty close at all points.
TL;DR: 2^7 <> 1.5^12.
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Take a starting note (which really means its frequency or pitch)
a0 = 55Hz
If we double the frequency, we get the same note but an octave higher
|a0|55Hz|
|:-|:-|
|a1|110|
|a2|220|
|a3|440|
|a4|880|
|a5|1760|
|a6|3520|
|a7|7040|
So now we have defined the note “a” for seven octaves.
This can get a bit boring, so let’s make another note. The next simplest after 1/2 is 2/3, which is an interval of a fifth. Again we start from a=55 and go up to a fifth above, then a fifth above *that,* and so on:
|A0|55Hz|
|:-|:-|
|E0|82.5|
|B1|123.75|
|F#1|185.63|
|C#2|278.44|
|G#2 / Ab2|417.66|
|Eb3|626.48|
|Bb4|939.73|
|F4|1409.59|
|C5|2114.38|
|G5|3171.58|
|D6|4757.37|
|A7|7136.05|
So we have defined A7 in two ways, and they don’t match.
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