"False" Pedal Notes

Most tuba players have noticed that on some BBb tubas and most BBb sousaphones it is possible to play Eb below low Bb using no valves, sometimes with passable tone quality. What follows is my conjecture about how that works:

Let “f” represent the frequency of the (imaginary) pedal note. (On a BBb tuba that’s approximately 29.13 cycles per second, but we don’t need that number for this argument.) The tuba doesn’t actually resonate well at that frequency, but it does at frequencies 2f (“low BBb”), 3f (middle F), 4f (middle Bb), 5f (D in the staff), 6f (high f), and so on.

What happens if the player sends puffs of air into the mouthpiece with a frequency that is a “perfect fifth” below BBb, corresponding to the “false pedal” Eb? Going down a perfect fifth means multiplying the frequency by 2/3, so the puff frequency is 2f times 2/3, or 4f/3. The harmonic series based on that fundamental will then include frequencies of 4f/3, 8f/3, 12f/3, 16f/3, 20f/3, 24f/3, and so on. Notice that every third member of that series is a fraction that can be reduced: 12f/3 = 4f = middle Bb; 24f/3 = 8f = high Bb; and so on. So now every third pulse reflected back to the mouthpiece arrives at just the right time to trigger a new pulse. It’s like giving a child on a swing a push every third time she comes back to you. The air vibration in the tuba then resembles a series of pulses with the frequency of Bb, but every third pulse is bigger that the two in between. Like the series of claps described earlier, this pattern of pulses can be shown to be the sum of a harmonic series of cosine curves with a fundamental frequency corresponding to the low “false pedal” Eb. Even though the strongest overtones in that series are at Bb frequencies in upper octaves, a Korg tuner recognizes the false pedal note as “Eb”.