The Harmonic Series

            The harmonic series is something that every Brass player should know, but very few really understand it.  Knowing the series gives insights into the intonation tendencies of your instrument as well as useful alternate fingerings to use in fast passages.  It is also the raw material for playing natural instruments.  It can be easily played on any brass instrument with any valve or slide combination.  It is the reason we can play different notes with the same fingering, the reason we need so few valves compared to the number of keys woodwinds have, and also the reason we miss notes (one of the reasons anyway).

            The series is shown below as written for the open fingerings of the Horn in F, Bb Trumpet, Trombone and Euphonium (they're the same length), and BBb Tuba respectively.  The individual notes in the series are labeled below.

            Notice that all the instruments share the same overtone series, merely transposed depending on the length of the tubing used.  A tube of any given length will produce a fundamental pitch.  The different harmonics, also known as partials or overtones, are multiplications of the frequency of that fundamental pitch.  In other words, the 2nd partial is twice the frequency of the fundamental, the 3rd is three times the fundamental, 4th - four times, and so on (please refer to the chart below).  Every time the frequency doubles, the pitch is an octave higher, so partials 2,4,8,16, 32, ad infinitum are octaves of the fundamental.  As you can see, there are an increasing number of multiplications of the fundamental between the different octaves, which is why the notes are closer and more plentiful the higher you go in the series to the point that after the 16th partial they come in intervals of less than a semitone.

            Many teachers will tell their students while teaching the harmonic series that there are several partials that are out of tune.  These intonation "problems" are, in fact, caused by our interpretation of them with regards to equal temperament.  The equal tempered scale was originally conceived so that the space between the twelve chromatic notes of the scale would be the same and keyboard instruments would be able to modulate at will between the different keys without having to retune.  This tuning system is at odds with the frequencies nature wants to produce which are simplified in the chart below.  For the sake of mathematical ease, let us say the fundamental pitch is 100.  The frequencies of the series would then be as follows (the fundamental and it's octaves are shown in bold blue type):

If we take, for example, the 3rd, 6th, and 12th partials, which are all octaves of the same note, we will see that the frequencies are 300, 600, and 1200 respectively.  According to the equal tempered scale, that 3rd harmonic is slightly high.  Again for mathematical ease, let's say the proper frequency for that note in equal temperament should be 290 instead.  This means that the 6th should be 580 and the 12th should be 1160, so each octave is a little bit further away from the equal tempered scale.  The same can be said for any other "out of tune" harmonics, such as the 5th and 10th, which are a little low compared to equal temperament, the 7th and 14th which are so flat they are unusable, and the 11th, which rests neatly between two notes.