What are Harmonics

This post goes a bit off-course in places to tie in other synthesis and sound knowledge and is more stream of consciousness than proper science

Short answer: the spikes you see on a spectral analyzer after the tallest, lowest pitched one are harmonics.

Long answer: every object has some amount of resonance, and depending on factors like material, size, and frequency, there’s different nodes at which it vibrates, or specifically is silent between areas of vibration. For our purposes we’ll assume a solid metal rod is the object in question. Those nodes are typically equidistant along the length of an object, and in perfect integers (if the fundamental is no nodes and a vibration along the entire rod, the first harmonic has 1 node in the middle, the second harmonic has 2 nodes dividing the rod into 3, the 3rd harmonic has 4 nodes dividing it into quarters, and so on). If we were to tighten this rod as if it were a guitar string (or just use a guitar string), we could place another object, like a finger, over these node points to sound a harmonic when the rod is plucked or struck, typically playing octaves above the fundamental at each harmonic node.

Quick interjection, when I say resonance I don’t mean resonance like in a filter, which is more feedback; although, resonance within a physical object is in some ways akin to feedback in that it’s a finite space and cannot typically dissipate all vibrational energy along the length of the object once, so the vibrations will feed back through the object, clashing at those nodes. It’s also worth noting that modal synthesis, a popular form of physical modelling, doesn’t use basically any methods physical objects employ to create sound.

When it comes to electronic signals such as that in an oscillator, the harmonics come not from resonance, but from the inherent properties of the waveshapes themselves. Sharp transients will naturally create high end content purely because the sharper the transient, the higher pitch the sound and the more harmonics will be present. Look at a saw wave compared to a triangle wave. They are, in many respects, the same waveform, except the saw you can see as either a highly skewed triangle or a cut off triangle (and in fact many synthesizers will employ either of those tactics or the reverse of those tactics to create such waveforms). The difference between a slow transient, such as in a triangle wave, and a fast transient, such as in a saw wave, creates the harmonic difference you hear. It’s also worth noting that a popular way to make triangle waves is to simply integrate a square wave. Integration is basically a special form of lowpass that maintains the sharpness of a signal, but slews the rising and falling edges. Slewing a square wave enough results in a triangle wave.