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cylinder (about 2L/c ~ 2 ms), it returns to the input where this now negative pressure pulse makes it easy to inject the next pulse of air, whether it come from a device to measure Z or from the jet of a flute. So, a standing wave with a period of ~2 ms or a frequency of ~500 Hz produces a minimum Z and it can drive a jet oscillating near that frequency.
But what happens instead if that returning negative pulse meets an input nearly closed by a clarinet reed? The negative pressure pulls the reed more closed, and the pressure pulse is reflected this time with no phase change. So, on its second round-trip, a negative pulse now travels down the bore where it is reflected at the open end with a π phase change and returns as a positive pulse. This time, it can push the reed open, let more air in, and thus amplify the next injected pulse. Thus, after two round-trips (about 4L/c ~ 4 ms), a positive pulse returns and the clarinet cycle repeats. An oscillation with frequency c/4L ~ 250 Hz sees an impedance maximum, and the clarinet in Figure 6, middle, fingered to give an effective length of ~32 cm, plays C4, an oc- tave below the flute with the same effective length. To under- stand the high-frequency behavior in Figure 6, we must under- stand more about tone holes.
Tone Holes, Register Holes, End Effects, and the Cutoff Frequency
On keyboard instruments, the octave is usually divided into 12 equal semitones with a frequency ratio of 21/12 = 1.059. The name octave means eight notes, and a major scale has seven unequal steps of 2, 2, 1, 2, 2, 2, 1 semitones; the scale of C major is played on just the white keys of a piano. Some woodwinds, including some baroque instruments, have no or few keys but a tone hole for each of the three long fingers on each hand, plus one for the left thumb. Suitably placed and sized holes allow a major scale to be played by lifting one digit at a time. After the seventh hole is opened, the player can switch from the first to the second resonant mode and begin the next register, repeating nearly the same fingerings.
At a low frequency, an open tone hole roughly approximates an acoustic short circuit, so the effective end of the bore is close to the first of the open tone holes. But the open tone hole is not exactly a pressure node (nor are the open ends of the pipes in Figure 4) because the air in and near the tone hole has inertia and must be accelerated by the sound wave. Consequently, the standing sound wave in the bore extends some distance past the first open tone hole. This effect in- creases with increasing frequency because the accelerating force for a given flow is proportional to frequency.
This effect allows a simple instrument with seven tone holes to “fill in” at least some of the remaining semitone steps in a chromatic scale using what players call cross fingering. If clos- ing the first open tone hole lowers the pitch by two semitones, then a one semitone descent is achieved by leaving that hole open but closing the next one or more open holes. A fingering chart for a folk or baroque instrument provides examples.
To aid the production of the second register, a register hole is often used. For example, in the recorder (Figure 2a), the thumbhole is half-covered to provide a leak at a position where the first mode of oscillation would normally have substantial pressure. This weakens (and detunes) the first resonance and thus allows the second resonance to deter- mine the pitch; the instrument plays its second register. (The frequency dependence of the inertial effect of air near tone holes means that, in a simple cylindrical bore, the same fin- gering would play somewhat less than an octave between two registers because the higher note would have a longer end effect. For this and other reasons, real instruments de- part from cylindrical shape to improve intonation.)
At a sufficiently high frequency, the inertia of nearby air vir- tually seals the tone holes. So, above a value called the cutoff frequency (fc), the standing wave is little affected by open tone holes. For the clarinet in Figure 6, middle, fc ~ 1.5 kHz. Below fc, the maxima (or minima) are spaced about 500 Hz apart and open tone holes determine the effective length of the bore. Above fc, their spacing is roughly half this frequen- cy because now the effective length is almost the entire bore, despite many open tone holes.
The cutoff frequency therefore limits the range of harmonic extrema in Z(f) and that limits which high harmonics are radiated efficiently. From baroque to romantic to modern instruments, more and successively larger holes raised fc, contributing to increased power in higher harmonics and thus to increased timbral brightness and loudness.
From 1831 to 1847, Theobald Boehm revolutionized the flute. In his system, a dedicated tone hole opened for each ascending semitone. This meant that, for most fingerings, open holes were more closely spaced. The tone holes them- selves were also larger, which required keys with pads to close them. A system of axles and clutches allowed the keys to be operated by eight fingers and one thumb and made playing all the keys relatively easy. The modern oboe, clari- net, and saxophone use some of his ideas.
Woodwinds have from thousands to millions of possible fin- gerings, of which only a small fraction are regularly used.
54 | Acoustics Today | Spring 2018

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