Pythagoras and the Harmonic Series

Pythagoras: more than just that one Greek guy with the triangles.

If you’ve heard of Pythagoras at all, it’s probably because of his most famous contribution to basic geometry: a2+ b2= c2. But did you know that Pythagoras did more than just write a cool formula and come up with a fun math theorem? As the story goes, he may be the father of both music theory and composition. Pythagoras, already a well-known scholar and teacher, was visiting a friend on a neighboring island when he passed the village blacksmith. Inside, the blacksmith was pounding away with his hammer. The noise was so loud it made Pythagoras’ ears ring. Instead of being upset or moving away from the sound however, he investigated it and noticed something peculiar. Each blow of the hammer on the anvil was one impact, but the sound he heard wasn’t just one note! What Pythagoras was experiencing was what we now call the harmonic series. Through experimentation with the blacksmith and his three assistants, Pythagoras was able to draw out a series of ratios between physical dimensions and pitch, which we still use today to describe how specific sounds are related on our instruments. As brass players, you may know the various intervals of the harmonic series as “partials.”

Harmonic series for open F horn.

Above are all the notes that are playable on the open F horn. Valves help fill in the notes in between to allow a fully chromatic scale. Chromaticism is an artificial construct within the natural overtone series. When you use a different valve combination, the harmonic series has the same intervallic relationships because the interval sequence is always the same. For instance, the interval from the 2nd to 3rd harmonic is always a fifth. As you ascend, the intervals become narrower. As brass players, you might recognize some of your favorite lip slurs as a part of the harmonic series above. Lip slurs are a basic fundamental you likely began with when you started your horn. You can slur through the partials in order, or you can skip through them when playing larger intervals. Some of the partials on the harmonic series are naturally out of tune. When Pythagoras was hearing different notes ringing at the same time at the blacksmith’s, he was hearing the harmonic series. As musicians, our sound concept is rooted in the harmonic series. When you play a single note on your instrument, the other notes in the harmonic series will resonate. Listen closely to your favorite artists, teachers, and players and you can often hear the overtones ringing. These are the other notes in the harmonic series! Professional musicians use the harmonic series to their advantage, sometimes choosing fingerings or positions that emphasize the higher overtones or the lower overtones, depending on the piece, character, or key of the music. Pythagoras discovered that music was scientific and could be reduced to mathematical rules and relationships, and with this information, we can better understand our modern instruments with valves and make more informed performance decisions.

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