poomshanka wrote:"Input anomalies" notwithstanding, why are some tubas very well in-tune, while others are almost unmanageably out-to-lunch? I'm curious about this as it relates to a design/manufacturing perspective.
It's all in the impedance.
Acoustic impedance is the ratio of sound pressure (i.e., loudness) to particle velocity. It describes how the amplitude and phase of a source sound becomes an amplitude and phase of a resulting sound.
The tuba has an impedance that varies by frequency. Some frequencies have a high ratio of sound pressure to particle velocity, and some frequencies are low. Thus, some frequencies are magnified and others diminished. The ones that are magnified are the resonant peaks of the impedance curve. The peaks are sharp, and the peaks generally line up on musically valuable frequencies. How they line up is the result of the taper (which is a broad description that includes anything that upsets the intended taper).
The mouthpiece also has an impedance that varies with frequency. Most tuba mouthpieces peak at similar values. You can hear the peak of the mouthpiece resonance curve by popping the rim on the palm of your hand. But unlike the spiky resonances of the tuba, the mouthpiece resonance is broad and flat, meaning that when you buzz into it, you get a discernable pitch but also quite a lot of noise and a range of frequencies around it. The mouthpiece shapes the sound of the buzz a bit but not much. A tuba mouthpiece impedance curve makes it weaker as you get away from the popping frequency, which is typically in the range of about Ab at the bottom of the staff. Making pitches some distance away requires more energy from the buzz.
The tuba filters out some of the non-resonant noise, and leaves the resonant frequencies. If you overload it with non-resonant noise, that noise will leak into the sound, or it will serve to cancel some of the resonances.
The player's lips also have an impedance, and that impedance also varies with frequency. It is controllable based on the tension of the lips as balanced against the air flowing through them. If the peak resonance of the lips matches the peak resonance of the tuba, there will be no attenuation in the sound. But the impedance curve of the lips is even broader and flatter than with the mouthpiece. If the noise produced by the buzz contains enough energy and tonal variety, it will excite all the resonant frequencies of the tuba. That's why a noisy buzz with lots of energy ends up making better sound than a buzz that seems to ring a single pitch more clearly, especially if it does so weakly. More air is the secret to a bigger buzz, assuming there are no fundamental faults in the structure of the embouchure. I was quite surprised to discover that great players have a fairly noisy, but loud, buzz. Listen to Jacobs's buzz on the recording of his TUBA lecture from 1973--it's quite noisy but it is also very loud and filled with energy.
If the buzz lacks energy or the impedance curve does not match the instrument (i.e., buzzing the wrong pitch), then you'll get noise and attenuation as the sound is reflected back into the instrument rather than out the bell. Also, a buzz that is off pitch will not get reinforced by returning pulses reflecting back from the bell opening, making it harder to maintain the buzz.
So, the sound that comes out is the sum of the impedance curves of the tuba, the mouthpiece, and the lips.
The resonance peaks in the impedance curve of the tuba are controlled by the length of the tubing and the shape of the taper. The shape of the taper can be affected significantly by disturbances in the tubing (i.e., leaks, edges where parts don't fit well, solder blobs, curves in the branches, and even blobs of grease). It might be subtly affected by the vibration and impedance of the brass itself, but I think this effect is very subtle indeed. The player might, however, hear the ring of the brass and mistake that for acoustic energy that is in the sound coming from the bell and getting "out front".
A straight tube has a well-defined set of resonance peaks, but a tapered tube is all over the place. It's quite possible to have tapers that kill upper overtones, and others that amplify them. If the overtones are not well-tuned, they will affect the apparent pitch of the sound. As the player forces the tuba into different modes of vibration by buzzing the higher partials, those overtones are mixed differently. Dominant but poorly tuned overtones will therefore make some notes worse than others.
Since the objective is an instrument that plays an equally tempered scale accurately, and then provides the flexibility to line up the pitches to the more resonant natural pythagoran scale when playing in an ensemble, the instrument has to be designed to counter the natural resonance harmonics to some extent. Figuring this out ain't easy. There is a computer program used by the German makers that will optimize a taper design for a particular intonation objective, but it does not optimize the taper for a particular sound objective. There is no indication that these objectives are always the same.
So a tuba that sounds great, responds well, and plays in tune is a dynamic balance between competing objectives, each of which can be affected by the taper design, the location of bends and curves in the tubing, production faults that disturb the propagation of sound waves, the match of the impedance curve of the mouthpiece, and the match of the impedance curve of the player's lips.
Thus, Dan is right. Great tubas mostly result from experimentation, because modeling all these effects in a design process is just about impossible. That's why we keep copying the great instruments of the past, making incremental changes to correct or improve this aspect while not doing too much damage to that aspect, and hoping that the production techniques are adequate not to negate those little design tweaks. Very occasionally, someone makes a breakthrough, and creates a new archetype to be copied in the future.
Rick "who can describe it but not model it" Denney