In the early days of brass instruments the third valve was often tuned to lower the pitch by a major third. Apparently this was still common in Sweden until quite recently. Today everybody prefers a minor third, but many instruments still have third valve tuning slides long enough to reach a major third. If you are using a shortened-whole-step first valve you may want to use the third valve alone instead of 1 & 2 on some notes. In this case (for an equal-tempered minor third) the length of the third valve tubing will be:
(S3 - 1)(Lo) = 0.1892 × Lo = 3.18 × L2
(Remember that "Lo" represents the effective length of the instrument with no valves depressed, and "L2" represents the length of the second valve tubing.) Also remember that "S" represents the semitone ratio, as explained in section 3.
The more conventional approach is to live with the slight sharpness of 1 & 2, and to make L3 somewhat longer, so that the 2 & 3 fingering will give you a good major third, i.e. four semitones: In that case the overall length with valves 2 & 3 must be:
(S4) × Lo = 1.299 Lo.
The combined lengths of the second and third valve tubings lengths is found by subtracting Lo:
L2 + L3 = 1.299 Lo - Lo = 0.2599 Lo
The third valve tubing length is then found by subtracting L2:
L3 = 0.2599 Lo - 0.0595 Lo = 0.2005 Lo = 3.369 L2.
The difference between these two values for L3 is:
(0.2005 Lo - 0.1892 Lo = 0.0113 Lo.
The amount of extra tubing that would be needed to bring the pitch down one semitone when the third valve is depressed is:
(1 - S)(S3)Lo = 0.0707Lo.
(That's like the difference between 4th and 5th positions on a trombone.) Dividing the difference above by that amount, (0.0113 ÷ 0.0707) = 0.1598. This tells us that the last two L3 values described above differ in pitch by roughly 16 "cents" . In other words, we normally tune the third valve to be roughly 16 cents flatter than a tempered minor third, so that the 2 & 3 combination will give a tempered major third.
It is interesting to note that a "perfect" or "just" minor third (with a frequency ratio of 6/5) is about 15 cents more than a "tempered" minor third (i.e. S to the third power) so the third valve alone ends up being pretty close to a perfect minor third.
Next: Tuning the Fourth Valve
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