> So all overtones are harmonics.
Not at all! It's the other way round. Harmonics are those overtones that are integer multiples of the fundamental. You can have other, non-harmonic, overtones.
Not at all! It's the other way round. Harmonics are those overtones that are integer multiples of the fundamental. You can have other, non-harmonic, overtones.
Now what happens when I play a chord and get overtones out of the interaction between two strings or a choir? What's happening there?
Nothing. Sound superposition is linear. There's no interference between different frequencies. By playing several strings together you obtain the sum of the sounds played by each of them separately. No new frequencies can appear.
The beats have a frequency and are audible, but their frequency is not a pitch.
You can deconstruct any given sound wave into its partial sine waves. This is typically done using a Fourier Transform algorithm (FT).
Of course, we can also do it the other way around, create a signal out of many sine waves. In practice, this is called additive synthesis.
Instruments not only sound different because of the timbre. Another thing to look at is the loudness over time, e.g. when plucking a string. We usually do this with an ADSR representation (Attack, Decay, Sustain, Release).
Knowing these chracteristics is enough to recreate an instrument with a synthesizer. Of course, it gets more complex when there are inharmonics and if they are irregular (different depending on each tone). That's why synthesizing instruments realistically is a pretty time consuming science.
Since you mention "octave" here I want to point out that this is a common misconception. Harmonics are not just octaves of the fundamental. The octave scale is logarithmic but harmonics are linear and include all integer multiples.
If your fundamental is 100, the octaves are 200, 400, 800, 1600, .... But the harmonics are 200, 300, 400, 500, 600, ... There are many extra harmonics that aren't overtones.
This is important for many reasons, but a fun one is that you can use this in sound design by relying on a clever thing our brains can do. We are so good at doing frequency analysis in our heads that we can figure out what fundamental must be present even when it isn't. If you play sine waves at 200, 300, 400, 500, 600, etc. your brain can figure out that those would all be multiples of a 100-Hz fundamental, even though that fundamental isn't present [1].
This lets you do a neat trick where you hi-pass a sound to remove some of the lowest frequencies in order to make room in the mix for other bass sounds. Even though the sound loses its fundamental, listeners will still hear it as "functionally" having a bass register. (This is also why when you listen to music on a crappy tiny speaker, you still hear the bass as bass even though it's actually quite tinny and high-frequency.)