Harmonics

Waves

Every sound wave consists of a left-going wave and a right-going wave. When these add, they produce what physicists call a standing wave. This means that the wave does not move. It can swing back forth, but it will always return to its original position. There is no “net” movement. A standing wave on a string simply vibrates the material of the string back and forth. Standing waves in wind instruments involve areas in and outside the instrument that flip back and forth between having more air (compression) and less air (rarefaction).

Waves on a String

Strings and the columns of air in wind instruments support standing waves with specific wavelengths. These are called harmonics. The harmonics on a string are easiest to understand. The first harmonic, called the fundamental and notated as f1, vibrates with a wavelength twice the length of the string. Only one half of the wave has to exist. The second harmonic (f2) vibrates with a wavelength equal to the length of the string. Here, the string contains the entire length of the wave. The third harmonic (f3) vibrates with a wavelength of two-thirds the length of the string. The fourth (f4) needs only half the string. Theoretically, this pattern continues with no end with the wavelength of each harmonic being equal to twice the length of the string, divided by the number of the harmonic (first, second, third, etc.) The set of all these harmonics is called the harmonic series. As the wavelength decreases, the pitch one hears gets higher.

wavelength

string length

wave on string

Waves in a Saxophone

Standing waves and harmonics in saxophones are more complicated. In the saxophone, the reed produces standing waves by sending compressions down the tube that add to opposite rarefactions coming back up the pipe (See Sound Production). Only certain wavelengths that “fit” are sustained. There are two main characteristics of the saxophone that govern what wavelengths fit. First, the saxophone is closed at one end (by the mouthpiece and musician). Second, the saxophone is shaped like a cone. The fact that the saxophone is closed at one end and open at the other requires there to be a maximum pressure difference at the mouthpiece and a minimum at the bell. Based on this requirement, the saxophone would only support the odd harmonics, f1, f3, f5, etc, as demonstrated below and they would produce much lower tones (with wavelengths twice as long as normal).

instrument body

pressure wave in tube

wavelength

effective wave

However, any saxophonist familiar with playing harmonics (often called overtones) knows that this is not true. The saxophone supports a complete set of harmonics. This is where the second characteristic becomes important. Due the conical shape of the instrument, the sound pressure waves spread out even before leaving the instrument. Logically, it follows that there will always be a maximum pressure at the mouthpiece and a minimum at the bell. This allows the saxophone to produce the complete harmonic series as shown above.

instrument body

pressure wave in tube

wavelength

Waves in a Trombone

normal cylindrical tube

with mouthpiece, flare, and bell

The harmonic series in trombones is very similar to that of saxophones. However, important differences remain. Like the saxophone, one end is closed and one end is open on the trombone, but the tube is mainly cylindrical. This presents the same effects discussed earlier: doubly long wavelengths and skipped even harmonics. Once again, any trombonist knows that this is not the case. The trombone can only produce wavelengths twice its length and it supports a full harmonic series. This is due to the flare (conical portion along near the end), bell, and mouthpiece. The flare and bell raise the pitches of all the harmonics (especially the lower ones). Conversely, the mouthpiece lowers the pitches of some of the higher harmonics. Together, they work to squeeze the odd harmonics to fit into a complete harmonic series.

Harmonic Series

These sound files demonstrate the pitches produced in a harmonic series. The first file is the harmonic series of the lowest string on an acoustic six-string guitar. The length of string was not changed, but the pitch increased by isolating higher harmonics. The second recording is the harmonic series of the lowest note of an alto saxophone. Again, the length of pipe was not changed but the harmonics isolated. The last recording is the harmonic series of a trombone in first position. The pipe length did not change but the pitch changed by exciting different harmonics. The staffs represent the notes in each harmonic series. 

Guitar Harmonics - William Sims
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Saxophone Harmonics - William Sims
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Trombone Harmonics - William Sims
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