[anamexis]

In a nutshell:

An octave (for example from a C to the next C) is a doubling in frequency. In the Western diatonic system, there are 12 notes per octave. (C, C#, D, D#, E, F, F#, G, G#, A, A#, B). Notes are "evenly spaced" within the octave - every note has the same ratio between its frequency and the frequency of the next note. Hence, that ratio is ¹²√2

[n4r9]

I always knew this but used to wonder why 12?, as I'm sure a lot of people did. It turns out of course to be a human choice, but the convenience of that choice can be codified mathematically.

On a piano you move up an octave by going up by 8 white keys, or 12 semitones (white and black keys). Going up by 4 semitones is called a "major third", which multiplies the frequency of the note by 5/4. If you do three major thirds you get an octave. However, notice that (5/4) multiplied by itself three times is 125/64 which is actually slightly less than 2.

In fact there is no way to tune a piano perfectly - there has to be a compromise in the intervals somewhere. The reason for this is exactly that no rational number (fraction) equals 2 when raised to an integer power.

[soulofmischief]

This is referred to as equal temperament, but one can also use just intonation. Each approach comes with tradeoffs and Western music mostly decided that equal temperament was worth it because instruments can play in any key without retuning.

Just intonation also suffers from harmonic issues when building certain chords, but the tradeoff is that there isn't "beating", or resonant pulsing due to frequency mismatches, since in equal temperament, the notes are slightly detuned in order to fit into the scale, as you've mentioned. Another benefit of just intonation is that it's been observed to be the instinctive intonation used by humans.

https://en.wikipedia.org/wiki/Just_intonation

[someonebaggy]

This is the equal-temperament tuning, which is used a lot now because it's simple and consistent. Other tunings were common in the past, where they'd set notes to simple fractions of other notes like 3/2 of the note 5 below, idk the actual numbers they used. 3/2 isn't any power of the 12th root of 2 so nothing in this new system is actually as harmonic as it should be, but nothing is particularly dissonant either and the system is simple.

[stavros]

Oh, that's the twelfth root of 2? That does make sense when you explain it like that, thank you. That's 12-TET, then?

[anamexis]

Yes, exactly. 12-TET stands for 12 tone equal temperament, where "temperament" is a tuning system.

There are other tuning systems, which I intentionally avoided discussing because it starts involving LOTS more music theory very quickly. But to quickly describe it: pleasant sounding combinations of notes generally occur at simple ratios. If you look at a simple major chord, like C-E-G, the E would be at 5/4 the frequency of C, and the G would be at 3/2 the frequency of C. However if you tuned a piano like this, it would be specifically anchored to the root note of C, as that's what we're referencing those ratios from. (This would be "the key of C major.") It just so happens that in 12-TET tuning, we get ratios that are very close to these simple fractions, and since the tuning is "equal", it works for any key/root note.