Dr. Sloan wrote:No matter what convention you chose, the bell end is the opposite of the mouthpiece end. Brasses are open at one end and closed at the other. Open ends behave differently than closed ends.
Oh, and when you build the spreadsheet...the taper is only part of the story - you might find the effects due to the flared bell and the mouthpiece wreak havoc with the theory you've put forth so far.
One more time...my understanding (see the links I posted, above) is that the effective length of the bugle contains 1/4, 3/4, 5/4, 7/4, ... complete waves (see the diagrams). This gives you a harmonic series with a, 3a, 5a, 7a,... The flared bell changes the effective length of the bugle as a function of wavelength - this affects one end of the spectrum; the mouthpiece affects the OTHER end of the spectrum. Combined, these two effects compress the series of resonances to (close enough for musical work) create a sequence x, 2f, 3f, 4f, 5f,... (where 'x' has no simple relationship to f, or a). If you buzz 'x', you get no help from higher harmonics (which won't resonate) so it will sound and feel "different". If you buzz 'f' (along with it's harmonics), the tuba will emit only 2f, 3f, 4f,... - which will work out OK because your ear and brain will fill in the phantom 'f'. [Doug Elliot claims that conical instruments resonate at 'f' - I would *really* want to see confirmation on that, but note that ears don't count].
Ken is right and that explains some nagging results of my simplistic arithmetic, for one thing why my bugles always seemed to be shorter than the wavelengths I was measuring. When playing a low C in free air (in standard conditions), the pulses associated with the fundamental will be 17.23 feet apart. If they were a different distance apart, it would not be a low C. When playing middle C, the pulses will 4.31 feet apart. I know that much. But how to square that with the requirement that there be an odd number of quarter waves in the bugle (as shown on the diagram below), so that the bell is always a quarter wave out of phase with the lips? If the speed of sound was constant inside the tuba (with respect to the frequency of interest), the pulse from the lips would be 17.23 feet behind the previous pulse from the lips. That would put a pulse a little beyond the bell at the same time the next one is emerging from the lips.
We assume the speed of sound is the speed of sound is the speed of sound. But in a conical bore, pressure, and therefore density, is not constant, and the wavelength therefore changes as you go through the instrument.
The math for this is the same as it would be for, say, a microwave antenna that uses a feedhorn reflector, with an important exception--the feedhorn is usually many multiples of the wavelength. That doesn't mean the math is easy. Dr. Sloan is a math professor, and I am not. I'll defer to him, in hopes that he is willing to explain rather than just challenge my attempts at explanation. Note also that the conical taper in question is straight. The taper is not straight in a tuba (or other brass instrument)--it widens at an accelerating rate as you near the bell. Fletcher and Rossing described a taper shape that was mathematically easy to describe (a Bessel function), and they also tied that to a cylindrical section. But the resulting math still depended on a lot of partial derivatives. Even I have a point where my eyes glaze over.
But I did get this right: The interaction between the fundamental and the overtones is the story. The resonance just cannot be described in terms of the fundamental only.
There is a symbiotic relationship between the sound and shape of a tuba. The shape affects the sound, but our definition of good sound results from the shape. Those two migrate from design to design over the course of many years. I suspect few of us would be interested in performing using a Wieprecht F tuba, except maybe for historical purposes. I have played of that design one for a few minutes. Without considering any possible defects in that instrument, it sounded to me like a euphonium played with a tuba mouthpiece, the fourth valve down, and with all the water key corks removed. Nothing matched really well. After 170+ years, the mutual interaction between performers and makers has resulted in tubas that do indeed resonate very openly and clearly and seem well matched. But one characteristic of a feedback mechanism is that with a little nudge it can go off in unpredictable directions. There have been times when tubas have just hit the mark, and those are the designs that persist, and bring that feedback process back to a given path. Personally, I think the Symphonie-style B&S F tuba was one such mark.
I sat down and played my F tuba last night, needing soothing after being drubbed by Dr. Sloan. If the low C on that instrument is hard to manage, I've gotten past it, and I think I can play it to the same level of mediocrity I play all other notes. It doesn't feel any different, and it doesn't sound any different. And this is a classic, German rotary F tuba. Why was it different before?
Because it was not what I expected it to be.
On the subject of computer modeling, etc. There are two basic approaches to dealing with challenging arithmetic. One is to work through all the challenges while remaining in the domain of abstract math (by abstract, I mean using variable names instead of actual numbers). That requires having a mathematical model of every important effect. Once the model is constructed, supply it with givens and solve for the unknowns. The problem with this is that the math may turn out to be extremely difficult once you include all those effects, such that solving for the unknowns or optimizing over the range of important values becomes intractable.
The other approach is to approximate the math using numerical methods. Instead of trying to work out the math completely in the abstract, we turn it over to arithmetic at some point and use computers to run so many numbers that we can see and optimize the result even if we can't solve for it in the abstract.
There are computer programs that calculate the relationship between intonation and taper design, and the instrument manufacturers do use them. The problem is that intonation is not the only objective. Also, the taper design is subject to so much distortion to make a practical instrument that the model becomes difficult to manage. So, they start with something, and then proceed to a range of experiments with performers to tweak the results. Another problem with the computer program is that the taper that produces the best intonation or the most resonance on the notes of interest may not produce the characteristic sound--the sound that has developed from the feedback between performers and makers for 170 years. The characteristic sound has two components--harmonic content and articulation--that make it characteristically a tuba sound.
That leads us back to Bloke's comment. It may turn out that the characteristic sound we prefer for an F tuba is actually unsuited to an instrument that is 12 feet long, especially if we are going to use the same mouthpiece we might use for an instrument 18 feet long. So, in attempting to accommodate that desired sound, issues emerge. Repairing those issues causes other issues to pop up. We are trying to achieve an instrument with the agility of a euphonium with the depth of sound of a contrabass. The instrument is halfway in between, but we use a contrabass mouthpiece, a contrabass-sized bell, and a contrabass-sized valve machine. The old F tubas that really were halfway in between a contrabass and a euphonium were just not asked to play low.
I have a tuba that came to me as an Eb high-pitch instrument from the late 1800's. Even after fixing the leaks and using heavy oil on the valves, the range below low Eb is stuffy and unresonant. It would pretty happily play any pitch you buzz, without a sense of clear resonance at all. It sounds and feels like a water key is open. A recent discussion on Ken's similarly aged Eb helicon has revealed that this was not uncommon. I played a Distin Eb tuba at the Army conference and while it wasn't quite as bad, it still had that characteristic. That was the state of the art at the time.
Rick "who has avoided this trap in the past and should have avoided it this time" Denney
