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Sine wave

While the square wave has the nice property of being very simple to describe with a digital computer, it is in fact very complex from an auditory or musical view. It contains a lot of resonant frequencies above the one we create by selecting the period length, which can make it sound "full" and characteristic. Actually some may describe it more as "hollow" or "airy", but it's still got character.

From a physical standpoint, the simplest waveform is the sine wave. It is quite simply a description of ideal oscillation similar to the kind you can get from a pendulum or a weight suspended by a spring, for instance. Objects tend to prefer this shape of movement when they're swinging back and forth across a midpoint.

Subjectively, a sine wave is rather dull as it can only be described as a very pure tone and nothing else. For making the cleanest possible sound it has some merit, and it's also useful as a component in building more complex sounds. Most of all it's an important theoretical and mathematical construct.

A sine wave is somewhat complex to generate computationally, but in most cases there will be a sin() function providing it at a reasonable speed (and in an easy-to-use manner). A full oscillation period of sin(t) is attained by cycling t through the values 0.0 to 2*pi (which is approximately 6.283), and it outputs values ranging from -1.0 to +1.0. The reasoning for using such a seemingly odd parameter range is mathematical and related to the circle, traditional home of both sine and cosine. A circle of radius 1 has a circumference of exactly 2*pi.

Triangle

A triangle wave is an attempt to mimic the sine wave more closely than you could with square, but still using rather simple calculations. It is just a rising and falling ramp of values. The result does sound very much like a sine wave, so in that sense it's useful. Still sounds boring though, naturally.

Sawtooth

This is half a triangle wave, resetting back to the starting value right when it reaches the top. From that jump in signal level you get a harsh quality similar to the square wave, but there's only one such jump in a period (compared to the square, which has two - both up and down) and the rest of it is smoothly shaped, so it tends to sound a bit warmer and more organic than the square.

Noise

If you want more chaotic sounds, resembling explosions or general thrashing, you won't have much luck with simple repeating patterns (unless you vary them a lot, and then they cease to be simple repeating patterns). A straight-forward way of getting chaotic patterns is to employ a random number generator. Most programming languages have one. In C you would do something like this:

next_number = (float)(rand()%20000-10000)/10000

...to get a random number between -1.0 and +1.0. Look up rand() and srand() for details.

Despite this pattern being completely random and without order, you can still control the apparent "frequency" or lightness of the sound. This can be done in several ways, but the simplest is to delay the switch to a new random value based on whatever quality you're after. It is very similar to the square wave in that respect, except you will have to keep the period lengths much shorter for noise.

General waveforms (for programmers)

To try out different kinds of waveforms using the same basic control code, you can keep a variable that will pass from 0.0 to 1.0 and wrap around for each full period of the sound. This period length would be determined by your desired frequency. To get the actual sample values, you can use different functions (depending on what waveform you want) to look up a suitable value based on the current period position.

For square, you would just check if it's less than or greater than 0.5. If it's less, you output -1.0, else +1.0 (or the other way around). To get a sawtooth, you just output something like (-1.0+period_pos*2.0), which will result in a signal ramp from -1.0 to +1.0 over the {0.0 - 1.0} period position range.

One thing I've found useful for noise is to regenerate a buffer of random samples each time the period counter wraps around, and continuously read out values from that buffer based on the current position within a period. You can choose the size of the buffer to be some suitable value, based on which frequency values you want to use. It doesn't have to be long but a few hundred samples is probably a good idea to avoid recomputing it all the time.

To change sound frequency you just set a new value for period length (though this will only affect the rate at which period position increments, it will always span between 0.0 and 1.0). For controlling volume you multiply the output sample by a volume value (say between 0.0 for silence and 1.0 for maximum volume). You can also mix several independent waveforms by adding their samples together, though make sure that you're not exceeding the allowed range of your soundcard (which would probably be the familiar -1.0 to +1.0 range). In other words, you'll have to use a lower volume for each waveform (or "channel") if you are going to mix (add) several of them together... or you can just multiply the final sum with a master volume value if you don't need volume control over each individual channel. This saves some multiply operations.

Another small heads-up worth considering is that you shouldn't use floating point numbers as if they were integral datatypes. That is; don't increment them in small steps over a large range. They have limited precision and will misbehave if you do that. It can manifest itself as a slight but noticable change in pitch or worse. You can increment floats as long as you keep them within a reasonable range, like between 0.0 and 1.0. The problems occur when the size of increments is much smaller than the current value (for example 50000.0 + 0.1).