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Reference : harmonic series

When we hear a sound produced by a musical instrument, we are actually hearing a multiplicity of sounds that form the harmonic series. In the 17th century, the Frenchman Joseph Sauveur (1653-1716) and the Englishman Thomas Pigot (1657-1686) noticed that the strings vibrate in sections, a phenomenon that explains why one string produces this multiplicity of sounds.

The video below shows the spectral analysis of a C two octaves below the middle C (C2):

Notes:

We have written the musical notes corresponding to the first 16 harmonics of the series. It is obvious that many harmonics follow these first harmonics.

The first harmonic or fundamental is not necessarily the strongest harmonic.

The balance between harmonics is constantly changing. This, together with the immense number of harmonics, explains the difficulty we encounter in synthesizing sounds.

Importance of the Harmonic Series

The harmonic series defines many of our intervals. Listed below are the octave, fifth, fourth, major third and minor seventh:

We can calculate mathematical ratio (or size) by dividing the frequencies of notes. Here we use the frequency of some harmonics to calculate the size of intervals:

Interval

Ratio

From the harmonics

Octave

130 / 65 = 2

1 and 2

Fifth

195 / 130 = 1.5

2 and 3

Fourth

260 / 195 = 1.33

3 and 4

Major third

325 / 260 = 1.25

4 and 5

Minor seventh

455 / 260 = 1.75

4 and 7

Interestingly we can calculate the values using harmonic numbers:

Interval

Ratio

From the harmonics

Octave

2 / 1 = 2

1 and 2

Fifth

3 / 2 = 1.5

2 and 3

Fourth

4 / 3 = 1.33

3 and 4

Major third

5 / 4 = 1.25

4 and 5

Minor seventh

7 / 4 = 1.75

4 and 7

Calculating frequencies

The mathematical ratios can be used to calculate the frequency of notes. From an A 440 we calculate the frequency of C#, E and G:

A

C# (major third)

E (perfect fifth)

G (minor seventh)

440

440 x 1.25 = 550

440 x 1.5 = 660

440 x 1.75 = 770

If we divide by the mathematical ratio we obtain descending intervals. Here we calculate the frequency of an F, a major third below A: