Filters

The waveforms chapter showed that a sawtooth contains every harmonic of the note it is playing. That makes it the usual starting point for subtractive synthesis, because a filter can only reduce harmonics that are already present. A sine wave has no harmonics above its fundamental, so a filter has nothing to reduce except the fundamental itself. The Source control in the tool below switches between a sawtooth, a square, a triangle and a sine, all at 110 Hz, so you can hear how much less a filter changes a source with fewer and quieter harmonics.

The filter below has controls for its type, slope, cutoff and resonance. The curve shows the filter's response, which is how much of each frequency gets through, drawn over the harmonics of the source. Drag the cutoff to move the curve across the harmonics.

Cutoff and slope

The curve shows that the filter reduces frequencies gradually rather than all at once. With no resonance, a lowpass filter reduces a frequency at the cutoff by 3 decibels, which is roughly where the reduction starts to be heard, and reduces higher frequencies by more. The slope sets how many decibels the reduction increases by for each octave above the cutoff.

At 12 decibels per octave, a harmonic one octave above the cutoff is reduced by 12 decibels, to a quarter of its amplitude, and is still clearly audible, so the sound keeps much of its brightness. At 24 decibels per octave the same harmonic is reduced by 24 decibels, to about a sixteenth of its amplitude. The Moog ladder filter is 24 dB per octave and the Oberheim SEM filter is 12, which is one reason those two instruments sound different. At 36 and 48 decibels per octave a harmonic one octave above the cutoff is reduced to about a sixtieth and a two-hundred-and-fiftieth of its amplitude, so each harmonic goes from clearly audible to nearly silent over a smaller movement of the cutoff. Switch between slopes with the cutoff near the middle of its range to hear the difference.

What resonance does

Resonance boosts the frequencies around the cutoff. At low settings it adds an emphasis that makes a slow sweep audible as a moving peak. At higher settings the peak becomes narrow and loud enough to pick out individual harmonics as the cutoff passes them, and the sweep sounds vocal. On the lowpass and highpass types in this tool, the Resonance control sets how many decibels the peak rises above a flat response, so at 0 there is no peak and at 20 the peak is 20 decibels high. On the bandpass type it sets how narrow the band is.

At the highest settings many filters oscillate on their own: the feedback that produces the peak becomes self-sustaining, and the filter produces a sine wave at its cutoff frequency with no input. A self-oscillating filter can be played as a sound source, and on instruments whose oscillators have no sine wave it is one way to get one. The filter in this tool stops at a 20 decibel peak and does not self-oscillate.

The slopes in this tool go up in steps of 12 because the filter is built from two-pole sections, each 12 decibels per octave, so 24 is two sections in a row and 48 is four. The bandpass type uses one section whatever the slope, because each extra bandpass section narrows the band further, so the slope control is hidden while it is selected. Resonance is applied to the first section only. Applied to every section, the boosts multiply into a peak far louder than the rest of the signal.

Highpass and bandpass

A highpass filter does the opposite of a lowpass: it reduces the low frequencies and keeps the high ones. Sweeping its cutoff upward thins the sound, reducing the fundamental first and then the lower harmonics while the upper harmonics stay. A highpass filter is often used to remove low frequencies from a sound so that it does not overlap with a bass part, for example on a pad below about 200 Hz.

A bandpass filter keeps a band around the cutoff and reduces the frequencies on both sides. Narrowed far enough, it leaves a band of a few harmonics that moves as you sweep, and the sound loses both its low fundamental and its high harmonics. Two bandpass filters set to the two lowest formant frequencies of a vowel, about 700 Hz and 1100 Hz for the vowel in "father", give an approximation of that vowel, which is one way a synthesiser can produce vowel sounds. Try the bandpass with the resonance low and then high. At 0 the band is wide and passes many harmonics, and at 20 it is about a twentieth of the cutoff frequency wide, so at 2 kHz it passes little more than the one harmonic of the 110 Hz note nearest the cutoff.

What moves the filter

In a subtractive voice the cutoff is one of the settings most often modulated, because moving it changes the brightness of the sound continuously while the pitch stays the same. An envelope can move it so that each note starts bright and darkens, an LFO can sweep it up and down repeatedly, and the pitch control signal from a keyboard or sequencer can raise it with each higher note so that high notes are brighter than low ones, which is called keyboard tracking. On analogue hardware a filter whose cutoff is set by a control voltage is called a voltage-controlled filter, or VCF. Later chapters cover each of these connections.