Posted by Rick Denney on April 17, 2003 at 10:46:17:
In Reply to: A question for Rick Denney posted by jlb on April 15, 2003 at 12:30:56:
Okay, I've gone back and reread a section in Fletcher and Rossing that had led to my response to Sean, and I probably need to back up a bit. Nothing is ever as simple as we hope.
Fletcher presents an equation for flow through the mouthpiece. It is rather complicated, as one might expect, but even so it had been simplified in ways that I didn't realize on my first reading. I based my statement on the notion that this equation assumed only that the note being blown was resonant on the instrument. There was a constant resistance value that seemed to be constant because the note was fully resonant, but there are no other terms that could be linked to the instrument in question. Thus, it seemed to me that flow depended only on the pitch being produced and its intensity.
But there are two complicating factors. One is that the resistance isn't constant, but is rather described by a complicated impedance curve of the instrument and mouthpiece. That curve is the sum of two other curves--the impedance curve of the instrument (which has peaks at the resonant overtones, the fundamental being slighly stronger than the higher overtones), and the impedance curve of the mouthpiece (which is a broad hump between a group of upper overtones--for trumpets between about the third and sixth overtones and for tubas probably an overtone lower with respect to the fundamental). Thus, the overall impedance is weak at the fundamental and the highest overtones (where the mouthpiece impedance makes it so) and strong in the middle. This exactly matches my frequency response measurements in my web article. But the factor here is perhaps a factor of two (on the logarithmic dB scale) for pitches that are resonant within a reasonable range, which is within the range of useful approximation, I think.
The other complication is the efficiency of the instrument and the player. Brass instruments are most efficient when played loudly, but the pneumatic power required ranges up to 10 watts or so. But the maximum acoustic energy from a brass instrument is about 1 watt, so at their best, the instrument and player are about 10% efficient in converting pneumatic to acoustic power. At soft dynamics, the efficiency is MUCH lower, perhaps only 0.1%. The pneumatic power is the product of the flow and the pressure, so that's how flow enters the picture.
Thus, for the acoustic sound pressure level to be the same between a tuba and a trumpet playing the same absolute pitch, the low-range efficiency of the trumpet player has to be similar to the high-range efficiency of the tuba player if the impedance is the same, or the relative efficiency has to offset the difference in impedance.
It seems to me, therefore, that Jacobs's results are probably as much as statement about the efficiency of the players he tested over a wide pitch range as the basic truth that the same note at the same volume requires the same flow. But if we assume highly skilled players (who can be efficient on the same absolute pitch, which no two beginners could do), and if we assume resonant notes of approximately similar impedence, then the flows on the same absolute pitch between trumpet and tuba are broadly similar.
On the same instrument, the flow required for a note an octave lower is four times the flow required for the note in question. It is an inverse-square relationship. But it assumes that the resistance is the same for the two notes, which probably holds across the middle two octaves of the instrument. More air is required to achieve the same output in extreme octaves because the instruments don't resonate those notes as easily.
Rick "who thinks Jacobs's experimental results depend as much on player skill as on the instruments" Denney