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A simple waveform

All right, we need to come up with numbers. Repeating patterns of numbers. How on earth do we do that? Well, it's absolutely trivial - any repeating pattern of numbers will create an audible waveform (if you make them span a large enough range, going between 0.0 and 0.00001 is hardly going to catch anyone's attention).

* Any pattern you say? Well, what about 1,0,1,0,1,0,1,0,...?

Yeah, that'll work. But unfortunately you won't hear that one either, even though it's using a large part of the range. Remember that the samples are fed out through the D/A converter at 44100 per second, and those numbers describe an oscillation (back-and-forth, remember?).

...but the pattern repeats itself every two samples, so the resulting speaker oscillation is at 22050 Hz. That's just above 20 kHz which you may recall as the upper limit for human hearing. That's actually the reason for using 44.1 kHz as the sample rate - so we can reproduce sounds over the entire range of the human ear, but not more.

* Ok, what if I hold each of the numbers for a while before changing?

You mean something like 1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0,...?

* Yeah, like that. Or even more of the same before changing.

Whee, you've just created a square wave, which is probably the most classic digitally synthesized waveform. How many times do we repeat a number before changing to the other one? If we increase that count, the waveform will be more stretched out in time, and the oscillations will be slower since it takes more time to complete a full back-and-forth. The repeat count (also called "period" if measured in time) is a direct way of controlling the frequency of the tone that we are generating. A longer period means a lower frequency and vice versa.

A small note of worth here is that you generally want a waveform to oscillate evenly around the zero-point, otherwise there will be an audible pop/click at the start and end of the tone. This is caused by the average signal level being shifted suddenly. It is illustrated rather clearly by how audio systems are constructed. They usually filter out anything that moves slower than a handful of Hz, to always keep the speaker cone oscillating around the physical rest point of the speaker. This is so you'll get the maximum available movement range out of it, even if your audio signal has a DC offset (meaning a rest level other than actual zero).