Why Is The 5th Partial Flat?

On many BBb tubas (especially the fat ones) the open "D" in the middle of the bass clef staff tends to be flat. (On CC tubas it's the open "E", on Eb tubas it's the open G, and so on.) That note is called the "fifth partial", as explained below.

Although it is really more complicated, a brass instrument’s behavior is somewhat similar to a vibrating string. You can play familiar bugle calls on the harmonics of a guitar string. The fundamental mode ("first partial") is what we call the pedal tone. The second partial is an octave higher, the third partial is a fifth above that, the fourth partial is a fourth above that, (i.e. two octaves above the pedal note) the fifth partial is an untempered major third above that, and so on.

If you allow rust and crud to build up on some parts of a guitar string and use a file to remove metal from other parts of the string then the pitches of the harmonics will be altered. Similarly, the non-uniform taper of the tubing in a brass instrument leads to a set of bugle tones somewhat different from the ideal. Every make of tuba is different in that respect.

But even if the bugle tones were exactly like the harmonics on a brand-new guitar string, the fifth partial would still be flat because its frequency would be precisely 5 times the fundamental frequency. (F5 = 5 × F1) The "untempered major third" between the 4th and 5th partials is a frequency ratio of F5/F4 = 5/4 = 1.25. The major third in an equal-tempered scale is four semitones, so its frequency ratio is S4, which comes out to approximately 1.25992. The fractional difference between those two frequencies is:

(1.25992/1.25 - 1) = 0.007936.

To convert that result to semitones, we divide by (1 - S), where "S" is the semitone ratio, or 12th root of 2:

0.007936 ÷ 0.0595 = 0.1333 semitones, or approximately 13 "cents".

To convert that result into cents more correctly, 3986 × log(1.25992/1.25) = 13.7 cents.

So if an instrument did behave like a perfect guitar string then its 5th partial would be about 13 cents flat. If you want to use your first valve on the 5th partial (to play C above the staff on a euphonium or "C" in the staff on a BBb tuba, for example) and if that 5th partial is about 13 cents flat, you can tune your first-valve tubing to be 13 cents sharp to compensate on that note, and pull it out for other notes.

The seventh partial is hardly ever used on valved instruments because it is very flat for similar reasons. But trombonists use it all the time, adjusting their slide positions for correct pitch. Calculating how flat it should be and how much slide adjustment would be needed to use it is left (for now) as an exercise for the reader. If your instrument has a 5th valve you may be able to find some exotic alternate fingerings using the 7th partial.