Most analog synthesizers use subtractive synthesis to get the sounds they produce.  That is, start out with sounds that have lots of harmonics and then selectively remove them to get the desired end result.  One of the key components for subtractive synthesis is a filter.  A filter often defines the characteristic "sound" of a synthesizer.  Perhaps, filters are the single most important element of an analog synthesizer.  That is why slogging through the upcoming verbiage is probably worthwhile.

Before we jump into filters, it is only appropriate to first discuss harmonics.  See, without a good understanding of harmonics, the discussion of filters is only a bunch of gibberish.  (It might be gibberish anyway ... guess it depends on how well I can explain harmonics!)

HARMONICS

Different, real instruments sound different from each other even when playing the same note ... why?  Harmonics.  The principal frequency (pitch) that you hear when a note plays, is called the fundamental.  The additional "stuff" that makes different instruments sound different from each other are called harmonics.  The pitch (frequency) of the fundamental determines what note is heard.  A sinewave is the fundamental tone you associate with a given note.  Note A1 has a fundamental frequency of 55Hz and nothing else (if it is a sinewave).  Harmonics are nothing more than exact, whole multiples of the fundamental pitch added in.  That is very important to remember.

Let's 'do the math'.  Note A1 fundamental is 55.000Hz.  If A1 has a so-called "second harmonic", that means another tone at a frequency of exactly 110.000Hz is mixed in with the fundamental.  Why 110Hz?  The SECOND harmonic is exactly two times the frequency of the fundamental (55Hz X 2 = 110Hz).

If our A1 note also has a "third harmonic", what is the frequency of that harmonic?  The THIRD  harmonic is exactly three times the frequency of the fundamental (55Hz X 3 = 165Hz).  Yes, the fourth harmonic is exactly four times the fundamental frequency.  Does this go on forever?  Theoretically, yes.  But when harmonics are at a frequency beyond human hearing, their existence is of no practical consequence ... you can't hear 'em!

"You keep using the word exactly in the above two paragraphs."  "Do you really like typing that much, or is there some point?"  There is a point!  If say, the second harmonic was not "exactly" twice the frequency of the fundamental, you would not hear a different sound of the fundamental note ... you would hear two notes!  The second harmonic of note A1 happens to be the fundamental frequency of note A2!!!  Well, why then when I play A1 and A2 at the same time, do I hear two notes?  A1 and A2 are theoretically related by a factor of two, but will be detuned slightly on purpose.  Analog synthesizers cannot be exact in the frequencies they produce.  They are very close, but not exact, and are "detuned" by the fact that analog circuits are inherently imperfect.  But harmonics ARE exact or they are not harmonics.  A violin cannot produce "exact" note pitches.  It will always be a wee bit high or a wee bit low.  But a digital synthesizer (not an analog one) CAN produce EXACT pitches.  It turns out, producing pitches of exact frequencies does not sound all that good.  They  "sound" like a machine.  So digital synths have a built-in detuning to make them sound more "natural".  (Or they are designed in such a way so as to not produce "exact" pitches in the first place.)

"Okay, so a fundamental is the pitch my brain perceives as the note being played, and the presence of various harmonics makes different instruments playing the same note, sound different, right?"  Yes.  "How does my brain know which is the fundamental and which are the harmonics?"  That is an excellent question!  And the answer is: The amplitude of the fundamental pitch will always be greater than the sum of the amplitudes of all the harmonics that are present.  That is how your brain knows which note is playing ... the fundamental is the loudest bit!

"So a VCO that puts out different 'waveforms' as you called them, are actually made up of different harmonics mixed in different strengths, and added to a common fundamental?"  Yes, yes, yes!!!  The electronic circuits in a VCO can hack up the waveform in various ways, and the result is different "sounding" waves of the same perceived pitch.  By jove I think you've got it!

FILTERS

A filter is a module that attenuates signals, depending on their frequency.  They are in other words, a frequency selective piece of wire.  Some frequencies might be attenuated out of existence, other frequencies might pass through unaltered, and still other frequencies might actually be amplified as they pass through.  But the "magic" of filters is that most of them can change the frequencies blocked, passed through, or amplified, under voltage control!  Change the control voltage and the filter changes what gets through and what is blocked.  Brilliant! And these changes can be done slowly or extremely fast, depending on what the source of the control voltage might be.

Very slowly, pronounce this word: "beeeooooowwww".  You just made your vocal tract perform a very common filter action ... a low-pass filter.  In fact, early speech synthesizers only had a couple of noise sources and some filters.  Together, they could create human speech!  I think you are starting to get the idea that filters are a very important part of analog synthesis.

All filters have a few things in common.  First of all, they pass some audio frequencies and attenuate others.  They have a signal input, one or more signal outputs, and a control input which varies the frequency at which the filter operates.  Filters are generally classified by their characteristic response.

The Low-Pass Filter

The following "graph" shows the frequency response of a typical low-pass filter.  The vertical axis represents the relative amplitude of an output signal.  The horizontal axis represents the frequency at which that amplitude is obtained.  Once there is a frequency presented to the signal input, that is greater the frequency of the filter's "knee", the frequency will be reduced in amplitude.  The higher the input frequency, the less of that frequency will appear at the output:

The High-Pass Filter

The following shows the frequency response of a typical high-pass filter.  The vertical axis represents the relative amplitude of an output signal.  The horizontal axis represents the frequency at which that amplitude is obtained.  Once there is a frequency presented to the signal input, that is less than the frequency of the filter's "knee", the frequency will be reduced in amplitude.  The higher the input frequency, more of that frequency will appear at the output.  The high-pass filter functions as a mirror-image of a low-pass filter:

The Band-Pass Filter

The following shows the frequency response of a typical band-pass filter.  Notice that only a small band of frequencies are passed and many others are simply ignored.  A guitar "Wah-Wah" pedal uses a filter similar to the response of a band-pass filter.  In that case, the "wah" sound is created when the "knee" point of the filter is shifted up and down the audio spectrum.  This causes some notes and harmonics to be emphasized while others are attenuated.

The Resonant Low-Pass Filter

Filters often include the ability to increase resonance at the knee frequency.  When that happens, the filter response starts looking like that depicted below:

Notice that the average response has dropped but around the knee it remains the same.  If the resonance is increased further, the low-pass filter can oscillate as a sinewave oscillator.

Other filter types often have a resonance control that will create a similar hump in the response curve.  That makes the filtering sound much more pronounced if carried to an extreme.

The most useful aspect of filters is that the frequency where the "knee" occurs, is voltage controllable.  Like with a VCO, the filter's knee range can often span the entire audio spectrum and beyond.