This will be the first of a three part blog on the acoustical bases of musical instruments. The other two parts will be called Part B flat, and Part C sharp… a bit of music humor there… Understanding how some instruments work help us to understand some of the comments and complaints from our hard of hearing clients. It also provides insight into why some music composers do things the way they do.
Like speech acoustics from our first year speech and physiology class, we need to turn to the wonderful world of quarter wavelength resonators. We see them everywhere (or at least I do, but then again I am a bit nerdy) from the way that our vocal tract works when we utter the low back vowel [a] as in ‘father’ to the first “1000 Hz” earmold tubing resonance in behind the ear hearing aids. Essentially a quarter wavelength resonator occurs whenever a tube is closed at one end and open at the other, such as our mouth when we say [a] or the sound emanating from a trumpet or any other brass instrument.
Above we see the first “two modes of resonance” of a quarter wavelength resonator. This example is from the human vocal tract while the vowel [a] is uttered. There is some unsubstantiated rumor that the Martian vocal tract is similar. The first mode of resonance is shown on the left and, as the name implies, is one quarter of the entire wavelength. The second mode of resonance is shown on the right, and is three quarters (or for our American readers, three fourths) of a wavelength. If we had enough space to show the third mode of resonance, it would be at five quarters… In other words, when a tube is functioning as a quarter wavelength resonator, the peaks are at odd numbered multiples of the first mode. In the vowel [a] the resonances (which in the vocal tract are called formants) are at 500 Hz, 1500 Hz, 2500 Hz, and so on. In a hearing aid earhook/tubing combination for a behind the ear hearing aid, the resonances are at 1000 Hz, 3000 Hz, 5000 Hz, and so on. There are of course other non-quarter wavelength resonances and these occur at other frequency locations, but more on that next time.
Clarinets, and brass instruments are the typical quarter wavelength instruments. In a clarinet, for example, the first mode above middle C is not a higher C (one octave higher) but an octave and a half higher (a G). That is why the clarinet has a “register” key and not an octave key. The register key increases the sound by a factor of 3 (to the second mode of resonance) and not 2 (which would only be an octave…. and that’s for next week).
Other instruments that look like the clarinet such as the saxophone, oboe, and bassoon, are also quarter wavelength resonators…. right? Well, it turns out that I lied a bit to you. These clarinet look-a-likes all have a gradually flaring tube (unlike the clarinet that seems to have a uniform diameter tube). And upon closer examination, the saxophone, …, and its flared friends appear to have an “octave” key and not a register key. Pushing the key merely doubles the frequency such that when pushed we can easily move from middle C to the octave above that, also a C.
You can easily verify this for your selves. Using any commercially available real ear measurement system, disable the reference microphone and the loudspeaker, and you have an in situ sound level meter where the probe tube can be inserted into any orifice (such as the earcanal… another quarter wavelength resonator assuming no significant ear drum perforation). While your favorite musician plays middle C on his clarinet (or trumpet) you will measure the fundamental note that is being played and the next resonant peak will be at 3x and then at 5x the frequency of the fundamental.
If you like learning about this further check out these two articles:
Chasin, M., “A Re-examination of the Etiology of the REUG- did we get it completely right?” Hearing Journal, Dec. 2005.
Chasin, M., “What your mother never told you about earmold acoustic formulae”, Hearing Review, November 2009.