Harmonic Series

We need to start by explaining a phenomenon known as the harmonic series. When we hear a sound produced by a musical instrument, we are actually hearing a multitude of sounds that constitute 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 spectrum analysis of a C two octaves below middle C (C2). Each peak represents a harmonic of the series:

Notes:

Importance of the harmonic series

The dialogue between human beings and nature, to which we refer in the subtitle of this article, is closely related to the harmonic series. Nature speaks with the harmonic series, and humans answer with tuning systems.

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

We can calculate the 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 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

It is interesting to note that we can calculate the values using harmonic numbers:

Interval Ratio From 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:

These concepts are essential for the comprehension of the different tuning systems ...

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