# Impedance and damping- part 3

In part 1 and part 2 of this 5-part blog series, we discussed the features of quarter wavelength resonators (brass instruments) and half wavelength resonators (stringed instruments and most woodwind instruments). This part has to do, not with what the frequencies of the resonances are, but the magnitude or amplitude of the resonances.

We have all learned about impedance when it came to learning about tympanometry. We learned that impedance is a complex (actually that’s not a joke from a mathematical point of view!) mixture of the reactance component and the resistive components. Specifically, impedance equals the square root of the reactance2 + resistance2.   If our eyes have not yet glazed over, reactance (being the mathematically complex term) is actually a function of frequency and is made up of the mass and stiffness components. And for those of us with great memories, reactance is made up of the difference between the mass and stiffness components such that when these two components are equal, they add up to 0. And when reactance is 0, we are merely left with resistance which has nothing to do with frequency.

So when is reactance equal to 0? And the answer is, at the resonant frequency. Now we can significantly simplify things.   At the resonant peaks that we talked about in parts 1 and 2 of this blog, which are so important for making a violin sound like a violin, and a trumpet sound like a trumpet, we see that we are dealing with pure resistance. And if it is pure resistance, then the effect of damping or resistance at 200 Hz should be identical to the effect of damping at 2000 Hz or 20,000 Hz. Damping is an equal-opportunity employer- it is the same for every frequency region.

Now it’s not that simple because in next week’s blog we get to learn about the acoustic transformer effect which alters the amplitude of the higher frequency resonances, but as far as simple damping is concerned, it is the same for all frequencies.   Any frequency dependence is due to a different principle of acoustics (but see the next blog entry!)

Virtually all musical instruments that I know about are hard walled and do not contain gooey, soft substances that should alter the damping (pun intended) characteristics. Subsequently, the height of the various resonances of musical instruments is fairly similar- there may be subtle differences because of some tubing dimensions, but these are secondary. Therefore, we can talk about the difference between the average (or root mean square, RMS) of the musical signal and the peak amplitude. This difference in decibels is called the crest factor. For most musical instruments the crest factor is roughly 18-20 dB. That is, the peaks are roughly 18-20 dB greater magnitude than the average playing level. Musical instruments tend to be quite peaky.

This is certainly something that can be assessed clinically. And actually whether a musical instrument is quarter or a half wavelength resonator, can also be assessed clinically. Using a real ear measurement system, disable the reference microphone and turn off the loudspeaker. In the Audioscan device set the stimulus level to 0 dB and in the Frye system, set the device to “off”. Once this is done, your real ear measurement system is a spectral analyzer and it can be used to measure the output of musical instruments. You can perform the measurements at 1 meter from the player, or in their own ear canals.

But back to crest factors. Musical instruments are hard walled instruments with minimal damping- hence a large crest factor of 18-20 dB. In contrast, humans articulate speech through a highly damped structure called the vocal tract. Because of the soft tongue, cheeks, lips, soft palate, nasal cavity, nostrils, and … various fluids, the vocal tract is a highly damped structure. Our formant resonances of our own vowels, nasals, and liquids (i.e, the sonorant sounds) are squatter and lower amplitude than those of a musical instrument. Subsequently the crest factor for human speech is on the order of only 12 dB- about 6-8 dB lower than those of musical instruments.   This has ramifications for fitting hearing aids for speech and music- a topic that is indeed dear to my heart. If you check my website and go to the articles section, there will be many articles there about the effect of crest factor and the fitting of hearing aids for music.

Music tends to be played (and listened to) at a much higher level than speech, and then add in the crest factor, the peak levels of the resonances are even higher. Music can easily overdrive the front end of modern hearing aids and cause unresolvable distortion… but that’s a different blog as well.

Suffice to say that because of the lower level of internal damping of musical instruments, the crest factor is higher than that of speech and this has ramifications for setting the OSPL90 output levels in modern hearing aids for music, as well as for ensuring that a non-distorted signal can be processed by modern hearing aid technology.