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 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 = or 1.25992. The fractional difference between those two frequencies is: