MaryAnn wrote:For example: I have an old Eb tuba which plays, with the mouthpiece I have for it (small shank,) sharp in the high register compared to the low register. I can't get them both in tune at once, so the tuba never gets played and just sits in its bag in the closet. If I wanted to, for example, lower the high register response via a different mouthpiece, what dimensional difference would be wise to try? I know it's possible but don't know how to get there without the concepts.
You are ready for Benade's
Fundamentals of Musical Acoustics (get the second edition) and Fletcher and Rossing's
The Physics of Musical Instruments, though I'm surprised you don't already have them.
I know what you do for a living, so I won't try to define my terms below, particularly impedance.
The intonation effect of a mouthpiece is a result of combining the impedance curves for the mouthpiece, the player, and the instrument. The mouthpiece has a very broad impedance curve. The resonance curve (which is the inverse of the impedance curve) has a broad hump of resonance, centered in the vicinity of A on the bass clef. There is so much inharmonic noise in the buzz through a mouthpiece (which is why the resonance peak is broad and flat) that it does not affect intonation that much in and off itself. Benade suggests that the resonance of the space in the mouthpiece (determined by the popping pitch, which you can hear if you slap the rim opening against your palm), if high, will cause the upper register to tend high in terms of intonation with respect to the lower register. I have a feeling that's a subtle effect in the midst of less subtle effects.
Benade also suggests that mouthpieces with a large throat provide a broader resonance curve for more pitch flexibility. But the larger throat also increases the popping frequency, while greater volume decreases it. So, a smaller mouthpiece with a larger throat might have the same popping frequency as a larger mouthpiece with a smaller throat, though the former will have more pitch flexibility. The shape of the cup affects the mix of overtones. These are generalities, of course.
The resonance peaks of the instrument are pointed, and include all the harmonic overtones. The pointed peaks indicate that the instrument is far more resonant at these frequencies than is the mouthpiece. The inverse impedance curve will show humps with pointed slots at the resonance peaks. The impedance shown in those humps damp the buzz coming out of the mouthpiece at those frequencies, filtering out the inharmonic overtones in the buzz to produce a musical sound. Thus, I think the instrument has a much greater effect on the perceived pitch of the system than the mouthpiece, in general.
Then, there is the player, whose buzz may be on pitch, high, or low. The buzz may provide the frequencies necessary to produce a full range of harmonics, or it may not. A buzz that is off frequency will be damped--it will be invading one of those humps in the instrument's impedance curve--and so will its important overtones. The resulting sound will be thin. The only reason an instrument can produce a sound at all is because the resonance peaks have a little width to them, making it possible to create some sounds that are not centered on those peaks.
Some instruments have resonance peaks that are a little broader, which provides more pitch flexibility. And some instruments have that breadth without undermining important overtones, and those instruments are highly regarded because they can be steered without ruining the sound.
A buzz that is on frequency but that lacks harmonic energy will also starve the resulting sound of important overtones. The buzz must include all those upper frequencies that go into the color of the sound--if they aren't in the buzz, nothing on the instrument is going to produce them. That's why a buzz with a lot of harmonic noise energy is better than a buzz with more apparent tonal purity. That purity means that the upper harmonics are weak. It's the instrument's job to do the filtering, and the player's job to provide enough harmonic energy so that something useful is left after the filtering is done.
I suspect the best science on these topics has been done in the electrical world more than in the acoustic world. I'm thinking a feedhorn antenna might be an interesting analogue. The frequency domain is the frequency domain.
Rick "who knows a lot more about impedance than he used to" Denney