The “frequency” of a note is the number of repeated cycles per second. If the waveform of a sound with definite pitch has any shape other than that of a perfect sine curve, then that wave shape is the sum of many sine or cosine curves with frequencies that are integer multiples of its fundamental frequency, i.e. a “harmonic” series. If the fundamental frequency is “f”, then the harmonic series of frequencies is f, 2f, 3f, and so on. Any sound with a wave-shape that repeats periodically can be shown to be the sum of a harmonic series of sine-shaped waves. (That’s **Fourier’s theorem**.) For example, if you record one hand-clap, put that recording in to your computer, copy and paste it over and over again to create series of evenly-spaced claps with a reasonably high frequency you get a sound with definite pitch. Most of that sound will be in the higher harmonics, but the pitch that you hear will be that of the fundamental.

Brass instruments utilize the familiar “bugle-tone series”, which closely resembles the harmonic series. If “f” represents the frequency of the imaginary “pedal” note, then the useful frequencies are 2f, 3f, 4f, and so on. The sound is generated by periodic puffs of air from the lips into the mouthpiece. Each puff forms a sound pulse which travels through the instrument to the bell, where it is partially reflected back. If the returning pulse arrives at just the right time it triggers the lips to release the next pulse. In that way we build up a strong, steady air vibration in the instrument. The process resembles how we push a child on a swing, building up large swinging motion with a series of small pushes at just the right frequency.

If you repeat that pushing motion with just half of the swing frequency you can still maintain a large swinging amplitude. That series of pushes is the sum of a harmonic series of cosine curves with a fundamental frequency just half of the swing’s frequency. (The lower-frequency members of that series will have rather small amplitude, but they are present.) A graph of the swing’s motion will then resemble a cosine curve with alternating cycles of larger and slightly smaller amplitude. That graph is also the sum of a harmonic series with a fundamental frequency just half of the swing’s frequency. In this case the second member of that series (corresponding to the swing frequency) will have large amplitude, but there will still be a small component at half the swing frequency.

That’s how “pedal notes” work on a brass instrument. The player sends puffs of air from well-trained lips at just half the frequency of the lowest “good” note in the bugle series. Since the instrument resonates at twice that frequency, return pulses arrive at twice the frequency of the puffs. Half of those return pulses arrive at the right time to trigger the next puff, and the other half don’t. The result is tone that has a small component of fundamental frequency but also many other components that are at integer multiples of the fundamental frequency. We hear it as a tone at the “pedal” frequency.

If we produced the sound with a pure sine or cosine generator instead of a series of puffs the result would be quite different because the fundamental resonant frequency of the air in the bugle (as described by Benade in *Horns, Strings, and Harmony*) is actually much lower. But the harmonic series formed by puffs of air at the instrument’s true fundamental resonant frequency does not match up at all with the frequencies of the instrument’s higher resonant frequencies. That’s why the true fundamental is virtually unplayable.

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