A player is obviously not limited to just seven notes on a brass instrument. Each valve combination or slide position offers a fundamental pitch that has a unique series of resonant frequencies, or harmonics, above it. In essence, each valve combination or slide change represents a different harmonic series. Using embouchure and air to change pitches, a player has access to all of the notes in a particular harmonic series. Before the advent of valves, brass players (trombonists excepted) were limited to playing the notes of a single harmonic series—not a lot of flexibility or choice unless playing in the extreme upper registers, which is at least one reason why Bach’s trumpet parts are so high!   To clarify terminology, a harmonic series comprises the fundamental pitch and all of its resonant frequencies that follow a particular pattern. A harmonic or partial describes each of these pitches, so if we count, the fundamental would be “1” (the first partial) the next frequency or partial above that would be “2” and so on. In comparison, overtones comprise only the resonating frequencies above the fundamental, so the first overtone is actually the second partial. A full harmonic series will always contain the same patterns of partials. As seen in Example 2, the second partial is an octave above the fundamental. Moving through the rest of the series, the intervals are Perfect 5th, Perfect 4th, Major 3rd, Minor 3rd, Minor 3rd, Major 2nd, Major 2nd, Major 2nd, Major 2nd, Minor 2nd. Theoretically, the series would continue to infinity with intervals becoming closer and closer. In practice, few brass players need to worry about going too much further than those depicted here!   An interesting exercise to hear the harmonic series in action involves sympathetic vibrations on a piano. Choose a fundamental pitch to play. Press down one or more pitches in its harmonic series without actually letting the hammers touch the strings. Then play the fundamental; the pitches of the depressed keys will ring. Experiment with notes not in the series and various notes within the series to discover the tendencies of the various overtones and non-series pitches.

Catch #1: Fundamental Problems Although they theoretically exist, the fundamental pitches on brass instruments do not speak well as a result of instrument design. For example, although the technical fundamental on a Bb trumpet is Bb2, Bb3 is the first pitch that sounds on the instrument. But the harmonic series continues as if Bb2 were the fundamental. Non-transposed, the series of a Bb trumpet would look like this:

https://sites.lib.jmu.edu/brasspedagogy/files/2016/05/Screen-Shot-2020-08-25-at-10.15.36-AM.png

In their respective ranges (non-transposed) brass instruments’ harmonic series would look like this:

Players can produce the fundamental pitches as pedal tones, but they are not included in the typical playing ranges, especially of high brass.

Catch #2: Out-of-Tune (Naturally) The harmonic series, as a natural acoustic phenomenon, is logical in and of itself and reflects just temperament. Unfortunately, it doesn’t always allow for the best intonation in the world of equal temperament—a compromise system in which almost everything is slightly out of tune with respect to mathematical ratios. Any work with a pianist will require specific tuning adjustments by a brass player. Bands and orchestras typically utilize more flexible intonation, yet the demands constantly change due to the number of players involved and a director’s conception. As a result, brass players always need to remain aware of pitch tendencies of an instrument even on pitches that are considered as in tune in the harmonic series. The natural fingerings or positions of certain pitches should generally be avoided without modifications of some sort (see The Practical Applications):

Catch #3: Instrumental Shortcomings Instrument limitations compound intonation issues. On a trombone, each slide position gets further away from the previous position, i.e. the length of tubing does not increase at a consistent rate. Moving from first to second position requires a slight wrist motion. Changing from second to third requires traveling a longer distance, and each subsequent position requires more and more arm length, challenging young students (and some adults) by the time they reach seventh position.

Valves provide players easier physical access to the seven options, yet valves do not represent a perfect solution. Depressing the second valve lowers the sound by a half step, the first valve by a whole step, and the third valve by a minor third. But these are not precisely in tune since the instrument must also function well when combining valves. The math does not add up acoustically. As a player adds more valves, an instrument gets progressively sharper since, unlike a trombonist, he can’t continually increase tubing length. For those of you interested in such things, you can read about the mathematical issues here.

Instrument manufacturing becomes a study in compromise to build something that produces intonation close enough to allow a player to compensate for the deficiencies. All trumpets should have a moveable third valve slide that a player can move out for sharp 1-3 and 1-2-3 valve combinations (this slide requires some maintenance and attention to ensure its movement). Some trumpets have a moveable first valve slide that can also be used for these combinations as well as sharp 1-2 combinations. Euphoniums and tubas will often have a fourth valve that acts as an “in-tune” option for the 1-3 valve combination, i.e. its tubing is slightly longer than the first and third valve tubing combined. Horns have a significant number of slides that need to be tuned and balanced to achieve acceptable intonation tendencies.

Compensating System. Even with a fourth valve, the problem of sharp intonation persists in lower ranges, especially in low brass instruments that are often required to provide a preferably in-tune musical bass for ensembles. Players are forced to finger pitches a half step lower than written in their lowest ranges to compensate for the sharpness. To avoid this, some professional horns feature a mechanical compensating system.   Not surprisingly, instruments with a compensating system have extra tubing to counter the sharpness in lower registers. The mechanisms can look different on different instruments. On a four-valve instrument, each valve has its own tubing that works independently as previously described. But when the fourth valve is depressed in combination with one or more of the other valves, it reroutes the air back to those valves and into additional tubing. Tubas could feature five or even six valves for further compensation. Regardless, its extra lengths of tubing coming out of the valves will identify a compensating instrument. (Need more information?)