Harmonic series and Chopin’s 2nd concerto
In the study of acoustics we learn that a single tone is composed of a fundamental frequency and an array of harmonic overtones frequencies (also known as harmonics) which are integer multiples of the fundamental frequency. Our neural system assembles the fundamental and the overtones into what we perceive as a single note with a particular timbre and dynamics. Here is the overtones series in traditional notation:
The small arrows on the 7th, 11th and 14th overtones indicate frequencies that are out of tune with respect to the familiar equal temperament frequencies. In fact, the deviation numbers above the staff show that all of the overtone frequencies except the octave are out of tune to a certain extent. The same 16 overtones are written in pitch bracket notation below. You can see that the pitch brackets reduce in interval the further you go up the series.
The overtone series is important in music because our concept of tonality depends on a hierarchy of related pitches found in the overtone series. Tonality is a vast and complex subject which I will not attempt to talk about here. Rather, I will give a fine example of the overtone series as a compositional element. The second movement of Chopin’s 2nd piano concerto begins with:
Below, I have written the first seven notes in pitch bracket notation. If you compare this to the melody line above you will see that Chopin’s melody almost exactly follows the harmonic series. As a result, Chopin achieves a great sense of calm stability and satisfaction in just seven notes.