Second-Valve Tubing Length:
Lowering the pitch of any note in an equal-tempered scale by one semitone is equivalent to dividing its frequency by "S", which is the semitone ratio. We accomplish this in wind instruments by increasing the effective length of the instrument by that same factor. If "Lo" represents the overall effective length of a horn, then we can make it play one semitone lower by increasing its length by a factor of S, so its new length is S × Lo. The amount of extra tubing required to do this is the difference between those two lengths. I call that extra length "L2" because it is normally the length of the second valve tubing.
Therefore, L2 = S × Lo - Lo, or L2 = (S - 1)Lo = 0.0595× Lo.
On most tubas it is easy to measure the length of the second valve tubing, so it is convenient to describe the lengths of all the other valve tubings as multiples of that quantity. For example, as a first guess one might try making the first valve tubing have a length of 2 × L2. But that would be like making the frets on a guitar or the positions on a trombone evenly spaced; the result would be a little sharp.
First-Valve Tubing Length:
An equal-tempered whole step is two semitones, so we need to multiply Lo by S twice.
Thus the first-valve tubing length required is L1 = S2 × Lo - Lo = 0.1225 × Lo.
If you divide that by the L2 value found above, you find that L1 = 2.059 × L2, which is about 3% more than the 2 × L2 mentioned in our first guess above.
Some people prefer to make the L1 value smaller, so that using the first valve on the fifth open tone or "partial" will be in tune. (On a CC tuba that would be the "D" in the middle of the bass clef staff, played with the first valve.) Most German tubas seem to be built with that in mind. Older 3-valved American tubas such as Kings, Martins, and Yorks often had first valve tubing that was a bit long, so that 1 & 3 would not be so sharp. On such instruments alternate fingerings (1 & 3 or 4) may be necessary in lieu of the first valve on the fifth partial.
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