Basic Fingering Chart
For a 4-valve Non-Compensating BBb tuba or Euphonium
With the fourth valve tuned to a perfect fourth,
as in the original tuba patented in 1835 by Wieprecht & Moritz,
Starting from middle (open) F and going down chromatically:

F.......open or 4
E.......2
Eb.....1
D.......12+ or 3-
Db.....23
C.......4 or 13+
B.......24+ or 123++
Bb.....open
A.......2
Ab.....1
G.......12+ or 3-
Gb.....23
F.......4 or 13+
E.......24+ or 123++
Eb.....124-
D.......234
Db.....134
C.......1234++

The "+" signs indicate that you will probably need to pull a slide out some or lip the note down.
The "-" signs indicate that you will need to lip it up or push a slide in. I like to keep the 4th valve slide on my BBb tuba set to give me a perfect fourth when used alone, as described below in more detail. I pull either the 4th or the second slide about 2 inches (5 cm.) when I play a long low B or E. (On a euphonium of course the pull would be half as much.)
Note: On some BBb tubas and euphoniums G or D can be rather sharp when fingered 1&2. Using the third valve instead often solves that problem.

Tuning the 4th Valve:
As the name implies, the fourth valve is usually tuned to a perfect fourth, correcting the inherent sharpness of the 1 & 3 combination. To do that, it must increase the overall length by a factor of 4/3, because a perfect fourth is by definition a 4/3 frequency ratio. The 4th valve tubing length is then 1/3 of the overall effective length of the instrument, or 5.602 times the length of the second valve tubing, i.e. 5.602 × L2. If we want to be very picky and insist on a tempered fourth then it should be 5 semitones, so that it increases the overall length by a factor of " S" to the 5th power, which is 1.3348. The difference between a tempered fourth and a perfect fourth is only about 2.5 cents. (That's about 4 millimeters of slide adjustment on a BBb tuba.)

In some situations one may wish to pull the 4th or 2nd tubing to get a good 6 semitones when using the 2 & 4 combination.
Before pulling, L2 + L4 = (0.3333 + 0.0595) × Lo = 0.3928 Lo. The value needed for a good 6 semitones is:

(S6 - 1) × Lo = 0.4142 Lo.

Subtracting, we find that the slide pull needed is 0.0214 Lo. Dividing by L2, we find that the amount of pull needed is about 0.36 L2. On a BBb tuba that's 0.36 times about six inches, or a bit more than 2 inches. You can pull the second valve slide on most contrabass tubas by this amount. The one exception is the big Jupiter, where that range of adjustment was unfortunately sacrificed in favor of “open wrap”.

In a T.U.B.A. Journal article (vol. X, no. 1, summer 1982) Chester Roberts strongly advocated keeping the 4th valve pulled out to this position on a five-valve tuba, so that you get a good 6th semitone with 2 & 4, "as God and Reiner and Szell intended". With this "long fourth valve" you have to use 1 & 3 or a fifth-valve combination for the 5th semitone. You can't use 4 alone or 234 with this system because they are too flat, but other fingerings using the 5th valve can work out to be better. In this case, L4 = 0.3546 Lo.

In some situations you also may want to shorten the 4th valve tubing so that you can get a perfect fifth with 12&4. To lower the pitch by a perfect fifth we must increase the effective length of the instrument by a factor of 1.5. Therefore:

Lo + L1 + L2 + L4 = 1.5 Lo, so

L4 = 0.5 Lo - (L1 + L2) = 0.5 Lo - (0.1225 Lo + 0.0595 Lo) = 0.3181 Lo.

I call that a "short 4th valve" to distinguish it from the "standard" and "long" fourth valves described above. Subtracting the short L4 value from the standard L4, we find that we need to push it in by (0.3333 - 0.3181) Lo = 0.0152 Lo = 0.256 L2. (On a BBb tuba that's about 1.5 inches.) Most tubas that I have seen allow that much adjustment and more.

Next: The Fifth Valve

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