Synthesis Waveforms and Harmonics - How do they work?

[NoiseyParker]

Hello,

I am currently studying sound synthesis and I am totally fascinated. However, I don't full understand the harmonics of different oscillators? I understand that a triangle wave has the odd harmonics present and that every other is in reverse polarity, but I don't understand why? It seems relatively close in shape to a saw tooth wave that has all the harmonics? I understand that a square wave has only the odd harmonics present, but not sure why? Can any one help me or point me towards a good source?

Many thanks

 

[amirm]

It can be shown by expanding the fourier series. In that regard, there is infinite number of harmonics. It is just that for a square wave, half the coefficients become zero and those are the even ones.

An intuitive way to think about this is that a sine wave is in phase with a square wave. In other words, it is like a square wave but its corners have been rounded. A cosine term on the other hand (representing the even terms) would be inverse phase of the square wave so can't exist in that expansion.

Here is a good video that shows this but by necessity, is full of math: https://www.khanacademy.org/science...ies/v/ee-fourier-coefficients-for-square-wave

 

[NoiseyParker]

OK so that video is beyond my level of knowledge, but I think I understand that half the coefficients are zero (because the signal is essentially on and off at a regular pulse?) But I don't understand the triangle wave - why are every other at reverse polarity?

 

[RayDunzl]

I understand that a square wave has only the odd harmonics present, but not sure why?

By definition?

It's how the value of the amplitude of the series - where you have a sine fundamental, the odd multiples of that frequency, with the value of 1/number_of_the_harmonic times the fundamental amplitude "adds up".

Audacity (or other software) will allow you to build waveforms by combining tracks.