Chapter Four: Synthesis
2. Waveforms
Waveforms determine the spectral content output by an audiorate oscillator. In subtractive synthesis, an oscillator is normally at the beginning of the audio chain. The waveform output by the oscillator is selected for its spectral characteristics. Filters farther down the audio chain then 'subtract' or modify frequencies in that spectrum. Additive synthesis relies on many oscillators chained together, each normally producing a sine wave (which produces only the fundamental frequency) with their outputs being summed, to build up a timbre from scratch. Both methods are still very much in use in digital synthesis applications.
Early modular synthesizers, such as the Moog modular models, c. 1967, provided the user a choice of four basic waveforms on their oscillators, believing they offered a good variety of choices for the subtractive synthesis process. It is important when studying the characteristics of these waveforms to take into account not only the partials each form produces (see Chapter One) , but also the strength of those partials. These two factors combined form what we consider timbre and together are called the waveform's spectrum. As discussed in Chapter One, the fact that these are periodic waveforms means that they will have harmonic partials. Below is a chart indicating these spectra for the four basic waveforms (p# = partial number, 'relative strength' is relative amplitude in relationship to the fundamental).
Waveform 
Shape/Sound 
Spectrum 
Relative Strength 
sine wave 

fundamental only, no additional harmonics 
n/a 
triangle wave 
odd partials only (1,3,5,7...) 
1/p#^{2} (one over the partial # squared) (3rd partial=1/9, 5th partial=1/25, etc.) 

sawtooth wave 
all partials 
1/p# (one over the partial number) Sawtooth, sometime called a ramp wave, can be an 'up' or 'down' ramp, which doesn't effect spectrum unless mixed with another wave 

pulse wave 
duty cycle = x:y with 'x' = length of the shortest phase and 'y' = the length of the entire cycle (NOT the other phase). With x = 1, all integer multiples of y are missing from the spectrum. A square wave has a duty cycle of 1:2 and has only oddnumbered partials. While a pulse wave can use a variety of impulse shapes, analog synthesizers tend to use the rectangular function, pictured on the left. 
1/p# (one over the partial number) 
Current software synthesizers and music programming languages may offer additional waveforms to those listed above. While primarily used in the digital domain, they are listed here for completeness.
bandlimited pulse train, bandlimited impulse train or sinc wave 

With the advent of digital synthesis, strong upper partials of a sawtooth or pulse wave may noticeably alias as they spread above the Nyquist frequency. Therefore, a waveform with a finite number of partials that cease at a certain frequency, known as bandlimited, is often used either directly or as part of additional synthesis techniques. The sinc function, not to be confused with sync (which coordinates the phases of multiple oscillators of a synth), is one such form. In digital synthesis languages, a function that utilizes this waveform is often called BUZZ, because of its strong, buzzysounding complete spectrum of equalamplitude partials. NI Absynth, for example has Midbuzz and Highbuzz waveforms...explore their sound and partials. 
stairstep or staircase wave 

The cycle of this stepwise waveform can contain 2inf number of steps, and while its spectrum is similar to a buzzy pulse wave, it is frequently used as a control signal where stepwise modulation is desired. 
1  2