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We now know that the speed at which a waveform oscillates is its frequency, but when this oscillation is periodic or repeats at the same oscillation speed over a series of cycles, we perceive the sound as a specific pitch. The faster the periodic oscillation, the higher the pitch we will perceive. Think of it this way. I'm sure everyone is played with a rubber band stretched between two fingers. Plucking that rubber band will yield an oscillation at a specific speed, where the rubber band quickly moves back and forth, cycling so many times per second.
Now if that rubber band is stretched more tightly across two fingers, striking it again yields a much faster oscillation, and if you listen carefully, you can probably hear those vibrations as sound waves of a specific pitch. This same basic principle applies to all stringed instruments. A string of a certain tension is fixed at two points. The length of that string and its tension combined to create a specific frequency at which it will oscillate when struck. When that string is placed under a higher tension, tightening the tuning peg, or when the string length is reduced, fretting at a higher position of the neck, the string now oscillates much faster when struck, and thus produces a higher pitch.
When an instrument like our guitar generates a sound wave, the frequency at which the entire wave vibrates is known as the fundamental. The fundamental of a waveform contains the most power or amplitude and thus defines the perceived pitch of the note. For example, playing an open A string on a guitar tuned normally produces its strongest vibration at 110 Hz. (music playing) In other words, the entire waveform moves back and forth at 110 times per second.
If I were to pluck that same A twelve frets up the neck, effectively halving the string length, I would produce a fundamental of 220 Hz. (music playing) This would be referred to as one octave higher, and represents a doubling of the fundamental frequency. But why does a guitar's open A sound different than a piano playing that same note, since they both vibrate at the exact same fundamental of 110 Hz? Unless it's a pure tone like a sine wave other higher-frequency waveforms or overtones generally travel along with this fundamental waveform.
All of the frequency components that make up the total waveform, including the fundamental and the overtones, are called partials. Think of these partials as different frequency pieces at different amplitude strings, all adding up into one complex sound wave. So that open A string with the fundamental at 110 Hz would also excite the first octave at 220 Hz with slightly less amplitude than the fundamental, the octave and the fifth at 330 Hz, the second octave at 440 Hz, and so on.
Together the fundamental and the overtones add up to form the harmonic series. A note's fundamental frequency along with its overtones and other non-pitch- related resonant frequency energy combine together to create the pitch and complex timbre or tonal character of that instrument. So a piano playing an A note-- (music playing) --in the same octave as a guitar-- (music playing) --sounds different because the piano excites the harmonic series differently than the guitar does.
In fact, two pianos playing the same note would likely sound at least a little different for the same reason, as many complex factors, including the shape, material, and resonate qualities of the unique instrument factor into its frequency content and how it excites the harmonic series. Even the way an instrument is performed can change these qualities. I doubt anyone but B.B. King could make his guitar Lucille sound just so. Most instruments have a frequency range and harmonic series that live within the human range of hearing, and there are many charts that you can reference to determine the general frequency range of a specific instrument.
These charts can be excellent references when learning to work with the frequency domain of your recordings, because for the most part, EQs and filters will not reference pitches like middle C or D-flat specifically, but will use frequencies measured in hertz to define the area of interest. It's worth mentioning that in the end you should always use your ears, not a frequency chart, to tell whether or not an instrument is sitting correctly in the mix, but there's no shame in referencing these charts to help point yourself in the right direction.
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