Pitched musical instruments are usually based on some sort of harmonic oscillator, for example a string or a column of air, which can oscillate at a number of frequencies. Taken in order from lowest to highest, these frequencies make up the harmonic series.
The lowest of these frequencies is called the fundamental or first harmonic. This is the note you get when with normal bowing of a stringed instrument or from the lowest octave of a woodwind instrument. All of the other frequencies in the harmonic series are integer multiples of the fundamental.
The second harmonic (or first overtone) is twice the frequency of the fundamental, which makes it an octave higher. On most wind instruments, for example the saxophone, oboe, or bassoon, there is an octave key which opens a small hole in the tube, prompting the instrument to oscillate at the second harmonic and giving the second octave of the instrument. On brass instruments, the second harmonic is the lowest playable note. The fundamental is called a pedal note or pedal tone and can be faked.
The third harmonic (or second overtone), at three times the frequency of the fundamental, is a fifth above the second harmonic. Similarly, the fourth harmonic is four times the frequency of the fundamental; it is a fourth above the third harmonic (two octaves above the fundamental). Note that double the harmonic number means double the frequency, which in turn means the pitch is an octave higher. For example, the 6th harmonic g is an octave higher than the 3rd harmonic G.
After that the harmonics come thick and fast, getting closer and closer together. Some harmonics correspond exactly to named pitches; others, for example the 7th harmonic, lie between the semitones.
For example, given a fundamental of C', the first 16 harmonics are:
An illustration of the harmonic series above, as musical notation. Not all the "wrong" notes are marked as such  see text for more details.
These notes are exact in just intonation. In modern equal temperament they are approximate, so that music can be played in any key without retuning; see musical tuning.
Not all musical instruments have overtones that exactly match the harmonics as described here. Piano overtones are increasingly sharper than harmonics because the strings are stiff, leading to nonlinear effects.
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