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Play a simple sine tone at 44100 sample rate and 16 bit. These are the parameters used for compact disks. The period of the tone is 2*pi*10. Playing at sample rate 44100 Hz results in a tone of 44100 / (20*pi) Hz, that is about 702 Hz. This is simple enough to be performed in real-time, at least on my machine. For playback we use SoX.

Now the same for a stereo signal. Both stereo channels are slightly detuned in order to achieve a stereophonic phasing effect. In principle there is no limit of the number of channels, but with more channels playback becomes difficult. Many signal processes in our package support any tuple and even nested tuples using the notion of an algebraic module (see C). A module is a vector space where the scalar numbers do not need to support division. A vector space is often also called a linear space, because all we require of vectors is that they can be added and scaled and these two operations fulfill some natural laws.

Of course we can also write a tone to disk using sox.

For the following examples we will stick to monophonic sounds played at 44100 Hz. Thus we define a function for convenience.

Now, let's repeat the sine example in a higher level style. We use the oscillator static that does not allow any modulation. We can however use any waveform. The waveform is essentially a function which maps from the phase to the displacement. Functional programming proves to be very useful here, since anonymous functions as waveforms are optimally supported by the language. We can also expect, that in compiled form the oscillator does not have to call back the waveform function by an expensive explicit function call, but that the compiler will inline both oscillator and waveform such that the oscillator is turned into a simple loop which handles both oscillation and waveform computation.

Using the oscillator with sine also has the advantage that we do not have to cope with pis any longer. The frequency is given as ratio of the sample rate. That is, 0.01 at 44100 Hz sample rate means 441 Hz. This way all frequencies are given in the low-level signal processing.

It is not optimal to handle frequencies this way, since all frequency values are bound to the sample rate. For overcoming this problem, see the high level routines using physical dimensions. For examples see Synthesizer.Dimensional.RateAmplitude.Demonstration.

It is very simple to switch to another waveform like a saw tooth wave. Instead of a sharp saw tooth, we use an extreme asymmetric triangle. This is a poor man's band-limiting approach that shall reduce aliasing at high oscillation frequencies. We should really work on band-limited oscillators, but this is hard in the general case.

When we apply a third power to each value of the saw tooths we get an oscillator with cubic polynomial functions as waveform. The distortion function applied to a saw wave can be used to turn every function on the interval [-1,1] into a waveform.

Now let's start with modulated tones. The first simple example is changing the degree of asymmetry according to a slow oscillator (LFO = low frequency oscillator).

It's also very common to modulate the frequency of a tone.

A simple sine wave with exponentially decaying amplitude.

The ping sound can also be used to modulate the phase another oscillator. This is a well-known effect used excessively in FM synthesis, that was introduced by the Yamaha DX-7 synthesizer.

One of the most impressive sounds effects is certainly frequency filtering, especially when the filter parameters are modulated. In this example we use a resonant lowpass whose resonance frequency is controlled by a slow sine wave. The frequency filters usually use internal filter parameters that are not very intuitive to use directly. Thus we apply a function (here parameter) in order to turn the intuitive parameters "resonance frequency" and "resonance" (resonance frequency amplification while frequency zero is left unchanged) into internal filter parameters. We have not merged these two steps since the computation of internal filter parameters is more expensive then the filtering itself and you may want to reduce the computation by computing the internal filter parameters at a low sample rate and interpolate them. However, in the list implementation this will not save you much time, if at all, since the list operations are too expensive.

Now this is the example where my machine is no longer able to produce a constant audio stream in real-time. For tackling this problem, please continue with Synthesizer.Generic.Tutorial.