Sine, square, triangle, and sawtooth waveforms
Before we dig deeper, let’s remind ourselves of some basics:
Typically we humans hear down to 20 Hertz (Hz) or vibrations per second) and up to 20,000 Hz (also said 20 KHz).
If we were to hear a note at 440 Hz, that note would be the A above middle C, also known as “Concert A” which is the note an orchetra tunes to.
The question we pose is, “How would we be able to tell if a 440 Hz ‘Concert A’ sound came from a violin or a clarinet?” The answer is, we can tell by the harmonics, or the mathematically related sine waves above 440 hz that give each instrument their characteristic sound or timbre.
Let’s understand this better by looking at a mathematically ideal square wave and sawtooth wave. For reference, a square wave sounds somewhat string-like -any early string emulation was built on these square waves. However, a sawtooth wave sounds somewhat reedy, like a clarinet.
Animation of the additive synthesis of a square wave with an increasing number of harmonics
So mathematically, a square wave contains the odd harmonics (1, 3, 5, 7, 9, etc), each one half as quiet as the previous while a sawtooth wave contains all harmonics (1, 2, 3, 4, 5, etc).
What we see as we add harmonics, is that the waveform gets less wobbly, more mathematically precise, and eventually (with the harmonics going out to a theoretical infinity requiring an infinite frequency response) we have a perfectly sharp corner.
Thinking about sound as sine waves lets us make sense of a lot of things which we will talk about soon.