What physically happens when notes in singing or an instrument harmonise? Why do those certain notes sound good together and others sound awful?
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Pluck a string, it will vibrate. There will be a main vibration, say, 110 times a second, but most of the times there will also be other quieter vibrations happening at multiples of that main vibration. The higher the number of vibrations per second, the higher in pitch the sound. That’s called an [harmonic series](https://en.wikipedia.org/wiki/Harmonic_series_%28music%29).
So, let’s start with this set of vibrations:
* 110 times a second.
* 220 times a second.
* 330 times a second.
* 440 times a second.
* 550 times a second.
* …
When talking about music, we seldom use the number of vibrations per second; instead we have note names:
* 110 times a second: A₃
* 220 times a second: A₄
* 330 times a second: E₄
* 440 times a second: A₅
* 550 times a second: C♯₅
* …
It just happens our ear/brain *really* likes combinations of those first vibrations. A₃ to A₄ is an octave, which is the most consonant pair. A₄ to E₄ is a fifth, which is the second most consonant pair. E₄ to A₅ is a fourth; again, very consonant, but less. A₅ to C♯₅ is a major third, sounds good too, but less than all the previous. The farther you go in the vibration set, the more dissonant it will sound.
Note that frequency ratios that are very simple sound consonant; A₃ to A₄ is 1:2; A₄ to E₄ is 2:3; E₄ to A₅ is 3:4; A₅ to C♯₅ is 4:5.
tl;dr: A choir where singers sing the notes A, C♯ and E sounds good, because the notes they sing are in simple ratios to each other.
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Music geeks: this explanation obviously draws on just intonation; equal temperament breaks the elegance of the idea but our ears have been accustomed to the margin of error introduced by non-pure intervals.
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