Review the ways in which a string of a musical instrument can vibrate.
- Two nodes and one anti-node: in this case you produces a sound of frequency ν1, that corresponds to a certain pitch of the note (for example, C). Call ν1 the fundamental frequency; the corresponding sound is said fundamental or first harmonic.
- Three nodes and two anti-nodes: as you can see from the figure, it is as if the string were separated from the central node into two equal parts that vibrate independently; then the string vibrates as if he had half-length. The corresponding frequency is ν2=2ν1 and the corresponding sound, said second harmonic, is the C immediately next to the fundamental tone (the one an octave higher). To distinguish the two notes you will call C1 and C2 the first two harmonics.
- Four nodes and three anti-nodes: the string vibrates as if it were divided into three equal parts; the frequency is ν3=3ν1 and the sound (G2, ie the sound a perfect fifth above C2) is called the third harmonic.
- Five nodes and four anti-nodes: the frequency is ν4=4ν1 and the sound (C3, the one an octave higher than C2) is the fourth harmonic.
- Six nodes and five anti-nodes: the frequency is ν5=5ν1 and the sound (E3, the one a major third higher than C3) is the fifth harmonic.
and so on.
If you take, as the unit of measurement of frequency, ν
1, the frequencies of the harmonics are 1, 2, 3, 4... You can
listen the first 16 (size of musical example: 63 K). In other words a string is capable to emit frequencies multiple of the fundamental one. The sequence of the first eight harmonics is, if the fundamental note is C1 (but you can build it from any other),
| HARMONIC |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
| FREQUENZA |
ν1 |
2ν1 |
3ν1 |
4ν1 |
5ν1 |
6ν1 |
7ν1 |
8ν1 |
| NOTA |
C1 |
C2 |
G2 |
C3 |
E3 |
G3 |
Bb3 |
C4 |
In reality a string vibrates in a way that is the sum of the configurations corresponding to various sounds of the harmonic series; for this reason a real sound is the sum of some harmonic sounds (for example, the first harmonic with amplitude 3.2, the second with amplitude 2.5, the third with amplitude 1.9, and so on), whose frequency is multiple of the fundamental one. By changing the amplitudes of the individual harmonics also changes, of course, their sum; In other words, you get a sound with a different timbre.
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| String that emits simultaneously more harmonic sounds (1) |
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| String that emits simultaneously more harmonic sounds (2) |
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