Lace

Here's some ARNUC for ya'

And since we love our block diagrams, we can make one to generate a square wave by introducing a new device, the comparator, which outputs -1 if its left input is less than its right, 1 if its left output is greater than its right, and 0 if they're equal.

So, to summarize, a square wave is a sawtooth wave, but without the even harmonics (odd overtones).

I should note here that the the harmonic of a waveform with the same frequency as the waveform (in this case, the term "sin(2*pi*f*t)") is called the "fundamental". It is considered a harmonic (the 1st harmonic), but it is not considered an overtone (as the name implies that it is over the fundamental, not equal to it). The next highest harmonic, with twice the frequency of the fundamental, is called the 2nd harmonic, or the 1st overtone. Alternatively, you can think of the fundamental as being the 0th overtone (this is what I like to do).

Made an oopsie on that last post, edited now.

That was graphically, now algebraically:

Start with a sawtooth wave:

sin(2*pi*f*t) + sin(4*pi*f*t)/2 + sin(6*pi*f*t)/3 + sin(8*pi*f*t)/4 + sin(10*pi*f*t)/5 + ...

Double the frequency:

sin(4*pi*f*t) + sin(8*pi*f*t)/2 + sin(12*pi*f*t)/3 + sin(16*pi*f*t)/4 + sin(20*pi*f*t)/5 + ...

Halve the amplitude:

sin(4*pi*f*t)/2 + sin(8*pi*f*t)/4 + sin(12*pi*f*t)/6 + sin(16*pi*f*t)/8 + sin(20*pi*f*t)/10 + ...

Notice we now have every even term of the original sawtooth wave, so when we subtract it from the original, we get every odd term:

sin(2*pi*f*t) + sin(6*pi*f*t)/3 + sin(10*pi*f*t)/5 + sin(14*pi*f*t)/7 + sin(18*pi*f*t)/9 + ...

Notice what happens if you double the frequency, halve the amplitude, and subtract the result from the original, both graphically and algebraically.

In other words, a sawtooth wave with frequency f can be denoted as a function of time (t) as such:

sin(2*pi*f*t) + sin(4*pi*f*t)/2 + sin(6*pi*f*t)/3 + sin(8*pi*f*t)/4 + sin(10*pi*f*t)/5 + ...

For reasons I don't fully understand (maybe you can explain it to me), a sawtooth wave contains all such harmonics, with amplitudes inversely proportional to their frequencies.

In summary, the possible harmonics of a repeating waveform with frequency f are those harmonics with frequencies f, 2f, 3f, 4f, etc.

Conversely, if we know that a harmonic repeats without variation in period t, then potential values for the smallest period of time it repeats without variation in are t, t/2, t/3, t/4, etc.

It is true that, if a harmonic repeats without variation in period t, it does the same in 2t, 3t, 4t, etc.

The selection of possible harmonics that can make up a waveform is limited by the fact that the waveform repeats without variation in a fixed period of time, meaning that all of its harmonics must repeat without variation in the same period of time. Let's call that period of time t (where t=1/f, f is the frequency in Hz).

The answer is that the only waveform that only stimulates one hair in our ears (in theory) is the sine wave. All other waveforms stimulate a number of different hairs in our ears, each of which corresponds to one of the sine waves that make up that waveform (called the "harmonics" or "overtones" of the waveform).

How is it then, that we're able to so easily hear the difference between two waveforms with different shapes, even if they have the same frequency and amplitude?

Anyway, these hairs in our ears send our brains amplitudes, but they send our brains no information about the shape of the waveform (i.e. sawtooth/square/sine/etc).

For now, we're only really concerned with amplitude, not phase (more on this later) so we can safely omit the cosines without loss of accuracy.

In our ears, there are a bunch of little hairs, each of which corresponds to a certain frequency of sound, and resonates with that frequency, sending the loudness of that frequency to our brain.

What I mean by the harmonic content of a waveform is the set of sine waves that, when added together, yield the waveform.