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?
In: 0
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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