What Is Acoustic Impedance and What Does It Mean To A Trumpeter
The other night I got into an intense discussion about mouthpiece gap and acoustic impedance just before a gig with Jeff Hedberg and C11 at the Jazz Showcase in Chicago. The discussion was with Roger Ingram, trumpet and Lisa Taylor, French horn. We were trying to come to terms with the gap between mouthpiece and receiver and the impact on the way a horn plays and how the concept of acoustic impedance fit into an explanation.
So, I told Roger I'd jot some notes down and share them with him. I hit me the next morning that this was going to be my comeback blog after missing some time due to a flood in my home on top of a very busy gig rehearsal schedule. What follows are those notes.
My attempt here is to discuss this in "laymen's terms" and still keeping faith with the math. I might end up having to write another blog down the road to clarify. Let' see what happens.
The formal definition of acoustic impedance is as follows:
Z = p/U where Z is acoustic impedance, p is the sound pressure at the point in question and U is the VOLUME VELOCITY (in liters per second) of the air. The point in question would be a cross sectional area of the tube. For most trumpet players this would probably the bowl of the mouthpiece.
For plane waves propagating in a tube, Z = (rho)(v)/S where rho is the density of the air and v is the speed of sound and S is the cross sectional area of the tube in question.
Acoustic impedance is analogous to electrical impedance which is R = V/I. OK, for those of you in the know, this is actually Ohm's law, and R is resistance which is the real part of the complex quantity of electrical impedance, so this is a simplification, but let's work with it, for now.
The title includes the idea "...and what does it mean to trumpeters?" These last two paragraphs contain some oversimplified physics and I'm sure it still seems daunting. As we continue, I'll try to put things in concrete terms that should make more sense.
If one looks at the impedance versus frequency curves for a trumpet we immediately encounter something counter intuitive.
These peaks are the resonant note - the notes we PLAY on a trumpet. So far so goo, but note this is an IMPEDANCE versus frequency (notes) graph produced by the legendary Arthur Benade, the guru of musical acoustics. Well, if you think about it, this you can see that this means when the horn is played on a resonant note the impedance is HIGH. Why might this be counterintuitive? Let me explain.
Early in these notes I used some ideas discussed by Tom Rossing that IMPEDANCE can be analogous to ELECTRICAL RESISTANCE. This might trigger a slightly confused way of thinking when it comes to RESONANCE which is an AC (alternating current) concept. Resistance is a DC flow idea. Yes there is a DC flow in playing trumpet, which is simply the number of liters per second of air we blow through the horn (the U in the equation above), and there is resistance, largely caused by the throat size. If you play on a 29 hole, there will be more resistance to the flow than if you play on a 19 hole. That is going to be the subject of another blog, but, for now, we need to see that impedance is a bit trickier.
Clearly the possible confusion stems from the idea that we might tend to think that when we are playing a resonant note that this means the IMPEDANCE should be LOW because it is some much easier to play a note in the "center of the slot" than not. In other words, it is easier to play a low C open horn perfectly in tune (the center of the slot) than to play it with all three valves depressed (the wrong fingerings, so playing a low C would mean you were not playing a resonant note, hence off center). That is not the case however. Let's use the math to try to make sense out of it.
In the original definition of acoustic impedance we have p/U. If U is very LOW this will mean the impedance is quite high. When you play a resonant note, it is very easy to do. You don't have to blow real hard, so the U is LOW meaning the Z is high. Usually when you hit a resonant note, the sound pressure will be higher since the horn is resonating. This will also drive Z up.
So when you play a note on a TRUMPET the acoustic IMPEDANCE is higher, thus "requiring" less air flow to do it.
Now, our discussion last night was triggered by the idea of "gap versus no gap" on a trumpet. How would one use acoustic imnpedance to discuss that. Let me attempt to here.
This is a sketch of the idea Roger, Lisa and I were discussing. This is the gap that the Bob Reeves company manages with their specialized mouthpiece back bores. That gap between the mouthpiece back bore and receiver venturi constitutes a tiny little resonator. Certainly this will not be largely relevant on low notes as the wavelength of the notes will be to long to "see" this little chamber, but in the extreme upper register (or for higher harmonics of a given note) this can have some impact. There are some folks who feel that there should be no gap and that the mouthpiece back bore should but up flush with the end of the receiver venturi. I am of the school that this gap can be exploited for use in the upper register.
Slapping a ruler on my trumpet with a mouthpiece in it, I find that the distance to the end of the mouthpiece is about 3 and 3/8 inches. If I do a calculation of the location of the end of the first resonance pressure node of a double high C using a quarter wave model, I find that should be right around 1.8 inches, or back in the mouthpiece back bore. Now, the point is that at those frequencies the gap could become quite relevant.
If I do a backwards calculation and figure out what note would produce a first quarter wave model that would drop roughly where that gap is, and I get right in the neighborhood of a trumpeters high C or C#. So the gap should have some impact.
Trumpeters tend to judge things from an imperial basis - i.e., the play test things. The impact of a gap like this could be quite subtle. However, tweaking it could contribute to the way your horn plays.
Now, to conclude, how could we use the idea of "acoustic impedance" to discuss the gap? A little resonator like the gap has its own acoustic impedance. Whenever we add an acoustic impedance to a system, either in series or parallel, the entire acoustic impedance of the original system has been altered a bit. So, changing the gap slightly alters the acoustic impedance of the system. By doing so, you have also introduced alterations of the impedance versus frequency graph. I.e. the resonances have been tweaked.
So there could be some merit to the idea that changing the gap changes the way the horn plays. How much it changes it is open to discussion.
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