Apple Logic Express 7 - Manual (Page 78)

78 Chapter 12 Synthesizer Basics
Fourier Theorem and Harmonics
“Every periodic wave can be seen as the sum of sine waves with certain wave lengths
and amplitudes, the wave lengths of which have harmonic relations (ratios of small
numbers)”. This is known as the Fourier theorem. Roughly translated into more musical
terms, this means that any tone with a certain pitch can be regarded as a mix of sine
partial tones. This is comprised of the basic fundamental tone and its harmonics
(overtones). As an example: The basic oscillation (the first partial tone) is an “A” at
220 Hz. The second partial has double the frequency (440 Hz), the third one oscillates
three times as fast (660 Hz), the next ones 4 and 5 times as fast, and so on.
You can emphasize the partials around the cutoff frequency by using high resonance
values. The picture below shows a sawtooth wave with a high resonance setting, and
the cutoff frequency set to the frequency of the third partial (660 Hz). This tone sounds
a duodecima (an octave and a fifth) higher than the basic tone. It’s apparent that
exactly three cycles of the strongly emphasized overtone fit into one cycle of the basic
wave:
The effect of the resonating filter is comparable to a graphic equalizer with all faders
higher than 660 Hz pulled all the way down, but with only 660 Hz (Cutoff Frequency)
pushed to its maximum position (resonance). The faders for frequencies below 660 Hz
remain in the middle (0 dB).
If you switch off the oscillator signal, a maximum resonance setting results in the self-
oscillation of the filter. It will then generate a sine wave.