above: Current model King 2341 Tuba
Note: All fingerings discussed in this will be for BBb Tuba.
As students progress, they upgrade from 3 valve Tubas to 4 valve Tubas. Why? What is the fourth valve for? It is mostly for two uses: extending the lower register and helping with tuning.
Tuning - Tubas, like all brass instruments share a fundamental problem (in non-compensating systems): you can tune the individual valves, but when used in combination the pitch can fluctuate depending on the partial. David Werden does a good job of explaining the math behind this in his article about compensating Euphoniums:
"Usually unnoticed by the musician, a great deal of mathematics is going on under his fingertips. Based on acoustical theory, each time the player wishes to lower the pitch by 1/2 step, he must increase the working length of the instrument by approximately 6%. For the sake of mathematical convenience, imagine an instrument with a basic length of 100" when no valves are depressed (this is only slightly longer than a Bb euphonium). If the 2nd valve is to lower the pitch by 1/2 step, its tubing would then have to be 6" long (6% x 100"). The total working length is now 106". To lower the pitch another 1/2 step below this level, the instrument needs an additional 6.36" of tubing (6% x 106" = 6.36"). Therefore, in order for the 1st valve to be capable of lowering the pitch of the basic instrument by 1 step, its tubing should be 12.36" long (6" + 6.36"). With just the 1st valve depressed, the total working length is now 112.36". For another 1/2 step below this level, an additional 6.74" of tubing (6% x 112.36") is necessary. Therefore, the 3rd valve's tubing should be 19.1 " long (12.36 + 6.74) in order to produce the desired 1-1/2 step change.As seen in the preceding paragraph, changes of 1/2 step, 1 whole step, and 1-1/2 steps, require adding 6%, 12.36%, and 19.1% respectively to the working length of the instrument. While each of the 3 valves can be designed to provide exactly the length needed when itis used alone, a conflict arises when 2 or 3 valves are used at the same time. Using the example of a 100" instrument, the working length with the 3rd valve depressed would be 119.1". To lower the pitch by 1 step from this point, 12.36% must be added to its length, which in this case would be 14.7" (12.36% x 119.1 " = 14.7"). Since the 1st valve's tubing is only 12.36" long, the 1 & 3 combination will be quite sharp. Because of similar discrepancies, 2 & 3 will be slightly sharp and 1, 2 & 3 will be a full 1/4 step sharp."
Dave Werden does continue this article, which will be linked at the end of the article.
To partially help correct this problem, you can use 4th valve for 1+3 combinations like second line C below the staff or 2+4 for 1+2+3 combinations like B natural 2 lines below the staff. As you go lower in the Tuba register, the tuning gets a bit more difficult but this can be corrected with "false" fingerings. On some Tubas, there are tuning issues within the staff. Some Tubas have a problem with second space C (1st valve). You can sometimes correct this also by using the 4th valve.
Remember, Tubas don't play or tune themselves, Tuba players do and not all Tubas have the same pitch discrepancies. It is important to understand pitch tendencies of your Tuba (this article is coming in the future).
Extending the lower register or making it easier to play the low notes-
You can play in the lower register with a three valve Tuba, but it becomes difficult for younger players to play below pedal F (1+3 or 4).
Using a four valve Tuba you can begin to more easily play the notes below this F. It isn't an automatic skill. It has to be developed just like any other musical skill. On the fingering chart below, you can see the extension of the range on the lowest notes. In my opinion it is possible to reach those notes on a three valve Tuba, it is just more difficult without a fourth valve.
(click to enlarge)