Notch Filters, Resonance and Oscillation
A notch filter is the opposite of a band-pass filter. Instead of passing a band of frequencies, it attenuates just those frequencies and allows all others to pass through unaffected. Notch filters are used to remove or attenuate specific ranges of frequencies and narrow 'notches' can be used to remove single harmonic frequencies from a sound. Notch VCFs usually provide control over both the cut-off and the bandwidth (or 'stop-band') of the filter (Figure 3.3.13).
FIGURE 3.3.13 A notch filter is the opposite of a band-pass filter, which it attenuates a band of frequencies. It can also be formed from a series combination of a low- and a high-pass filters, provided that the low-pass cut-off frequency is lower than the high-pass cut-off frequency. If not, then no notch will be present.
If the keyboard pitch voltage is connected to the cut-off frequency CV input of a VCF, then the cut-off frequency can be made to track the pitch being played on the keyboard. This means that any note played on the keyboard is subjected to the same relative filtering, since the cut-off frequency will follow the pitch being played. This is called pitch tracking or keyboard scaling (Figure 3.3.14).
FIGURE 3.3.14 Filter scaling, tracking or following is the term used to describe changing the filter cut-off so that it follows changes in the pitch of a sound. This allows the spectrum of the sound produced to stay the same. In the example shown, the filter peak tracks the changes in the pitch of the sound when two notes two octaves apart are played – the peak coincides with the fundamental frequency in each case. With no filter scaling then the note with a fundamental of 4f two octaves up would be strongly attenuated if the filter cut-off frequency did not change from the peak at a frequency of f.
Low-pass and high-pass filters can have different response curves depending on a parameter called resonance or Q (short for 'quality', but rarely referred to as such). Resonance is a peaking or accentuation of the frequency response of the filter at a specific frequency. For band-pass filters, the Q figure is given by the formula:
Q = Center frequency/Bandwidth (or pass-band)
This formula is often also used for the resonance in the low-pass and high-pass filters used in synthesizers. For these low-pass and high-pass filters, the resonance is usually at the cut-off frequency and it forms a 'peak' in the frequency response (Figure 3.3.15).
FIGURE 3.3.15 Resonance changes the shape of a lowpass filter response most markedly at the cut-off frequency. The result is a smooth and continuous transition from a low-pass to something like a narrow band-pass filter.
In many VCFs, internal feedback is used to produce resonance. By taking some of the output signal and adding it back into the input of the filter, the response of the filter can be emphasized at the cut-off frequency. This also means that the resonance of the filter can be made voltage controllable by varying the amount of feedback with a VCA. See Section 3.3.5 for more on VCAs and see Section 3.6 for more information on the implementation of filters.
Most subtractive synthesizers implement only low-pass and band-pass filtering, where the band-pass is often produced by increasing the Q of the lowpass filter so that it is a 'peaky' low-pass rather than a true band-pass filter. This phenomenon of a peak of gain in an otherwise low-pass (or high-pass) response is called 'corner peaking'. Some models of analogue synthesizer also have an additional simple high-pass filter, whilst notch filters or band-rejects are very uncommon.
There are two types of filters: constant-Q and constant bandwidth. Constant-Q filters do not change their Q as the frequency of the filter is changed. This means that they are good for applications where the filter is used to produce a sense of pitch from an unpitched source such as noise. Since the Q is constant, the bandwidth varies with the filter frequency and so sounds 'musical'.
Constant-bandwidth filters have the same bandwidth regardless of the filter frequency. This means that a relatively narrow bandwidth of 100 Hz for a filter frequency of 4 kHz, is very wide for a 400-Hz frequency: the Q of a constant-bandwidth filter changes with the filter frequency. Most analogue synthesizer filters are constant-Q.
The effect of changing the cut-off frequency of a highly resonant low-pass filter in 'real time', with a source sound rich in harmonics, is quite distinctive and can be approximated by singing 'eee-yah-oh-ooh' as a continuous sweep of vowel sounds.
If the resonance of a peaky low-pass or a band-pass VCF is increased to the point at which the filter plus its feedback has a cumulative gain of more than one at the cut-off frequency, then it will break into self-oscillation. In fact, this is one method of producing an oscillator – you put a circuit with a narrow band-pass frequency response into the feedback loop of an amplifier or operational amplifier (op-amp) (Figure 3.3.16). The oscillation produces a sine wave, sometimes much purer than the 'sine' waves produced by the VCOs!
FIGURE 3.3.16 If a filter with a strong resonant peak in its response is connected around an amplifier, then the circuit will tend to oscillate at the frequency with the highest gain – at the peak of the filter response. This can be easily demonstrated (perhaps too easily) with a microphone and a PA system.
Coming up in Part 3: Envelopes.
Printed with permission from Focal Press, a division of Elsevier. Copyright 2009. "Sound Synthesis and Sampling" by Martin Russ. For more information about this title and other similar books, please visit www.elsevierdirect.com.
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