Players of the modern bassoon—Heckel or Buffet—are accustomed to using ornate fingerings to stabilize and correct certain notes that are difficult to tune when simple fingerings are used. These “auxiliary” fingerings open or close tone holes far below the first open tone hole. This paper describes laboratory experiments and computer simulations, and the author’s interpretation of the results to explain why auxiliary fingerings are effective.
I will use the system of naming pitches approved by the Acoustical Society of America. Each note is described by its letter name and a subscript denoting the octave. Middle C on the piano is C4, and A-440 is A4. The conjunction of all the fours makes it easy to remember. The subscript changes between B and C, so the semitone below C4 is B3. The lowest note on the bassoon is Bb1.
Plain fingering: One in which all the open tone holes below the first open tone hole are normally open; that is, finger holes or holes covered by open-standing keys. Most bassoon fingerings up to and including D4 are plain fingerings. Small vents opened to facilitate octave overblowing are not considered tone holes in this definition.
Forked fingering: A fingering in which the tone hole below the first open hole is closed to effect a semitone flattening of a plain fingering. Eb3 is normally played this way on the bassoon. Forked fingerings were an essential part of the technique on baroque and classical woodwinds.
Cross-fingering: Sometimes misleadingly called harmonic cross-fingering. A fingering in which, typically, several tone holes below the first open hole are closed in order to produce an irregular tone-hole lattice. Cross-fingerings are often used in conjunction with a vent or speaker key. The effect is to misalign the lower air-column resonances while strengthening the third, fourth or fifth resonance so that a high note is produced. The bassoon employs cross-fingerings from Eb4 upwards. The various “long fingerings” for C#4 are also cross-fingerings.
Auxiliary fingering: A plain or forked fingering modified by opening or closing a tone hole far below the first open tone hole. Auxiliary fingerings are used to modify tuning, resonance and tone color, and some are mandatory. For example, Heckel players customarily add the low D key to the plain fingering for C#3 and press the low Eb key and/or the low C# key when playing G3.
I have been interested in auxiliary fingerings for a long time. I first read about them in Baines1, and learned how to use them from my bassoon teacher. My interest was intensified when I began to play a replica of a baroque bassoon, which had a different set of auxiliary fingerings from my modern bassoon. In an article in the IDRS Journal, Gerald Corey2 presented a comprehensive list of auxiliary fingerings for the modern French bassoon, further whetting my curiosity. Why are these fingerings necessary? How do they work? I lacked facilities for laboratory experiments, but I was encouraged by the 1979 publication of a paper by Plitnik and Strong3 describing a numerical method for calculating the input impedance of an air column of arbitrary shape and arbitrary placement of tone holes. At that time, however, I didn’t have access to a sufficiently powerful computer. In 1982 IBM introduced its PC with an empty socket for a “numerical coprocessor.” By 1984 the Intel 8087 coprocessor was available, and Microsoft’s FORTRAN compiler could generate code for the 8087. I knew the computer could perform the complex calculations fairly quickly. I had everything I needed, or so I thought.
(From here on, “data” will refer to the results of measurements using an actual bassoon, and “model” will refer to descendants of the IMPED program of Plitnik and Strong and the output of these programs.) I wrote the first of several versions of IMPED in Microsoft FORTRAN and ran them on my 8088-based PC4. The initial results were pleasing, and the predictions of the model can be summarized as follows:
Disappointingly, the model results did not clarify the mystery of the auxiliary fingerings. For example, looking at the output of the program, it was not clear that the plain fingering for C#3 was unstable, nor that adding the low D key should make an improvement. Without access to laboratory equipment, it was impossible to validate the model or confirm its accuracy, so I put the model aside, intending to return to it when I had fresh ideas (and a faster computer).
Late in 1994 the IDRS announced the Special Project Grants. I saw an opportunity to go forward with this work. About a year earlier, I had met Doug Keefe at a meeting at the Center for Computer Research in Music and Acoustics (CCRMA, pronounced karma) at Stanford University. Keefe was professor in the school of music at the University of Washington. With his help I made some refinements in my model which would prove important. I applied for and received a Special Project Grant that allowed me to spend two months in Seattle working in Keefe’s well-equipped acoustics laboratory.
Task 1 at the lab was to validate the model. A probe apparatus consisting of a tiny sound source and an equally tiny microphone was connected to the bassoon bocal by a brass tube of sufficient length to account for the actual and equivalent volume of the reed5. The probe was connected to a desktop computer that provided the signal to the sound source and recorded the microphone response6. With the computer both conducting the experiment and recording the data, data collection was rapid. Measurements of input impedance vs frequency were taken for fingerings for every note from Bb1 to C5. Figures 1–3 show comparisons of data and model results for three notes, Bb1, Bb2 and C5. In addition to showing the sort of agreement between data and model achieved, these plots of absolute impedance vs. frequency illustrate certain characteristics of the bassoon and other woodwinds.
The reed is a pressure-driven flow controller; that is, the reed opening (and thus the flow of air through it) is controlled by the pressure inside the reed cavity, or more accurately, the difference between the reed cavity pressure and the player’s breath pressure. Oversimplifying somewhat, regeneration (oscillation of the flow controller) is favored at those frequencies where pressure is a maximum for a given flow; that is, when the input impedance of the air column is a maximum. If the impedance maxima are suitably located at integral multiples of a fundamental, then regeneration can occur at harmonics7 of the fundamental, stabilizing and enriching the sound. This has been called a regime of oscillation by Benade8, who gives a much more detailed explanation. Now let’s look more closely at the figures.
In all the figures that follow, which show air-column resonances vs. frequency, vertical lines mark the harmonics of the relevant note. In Figure 1, the vertical lines mark the harmonics of Bb1, the bassoon’s bell note. The impedance peaks march in regular progression out beyond 1000 Hz, the result of strong reflections at the bell. Note that the first three impedance peaks are well aligned with the first three harmonics, but the higher peaks run increasingly sharp9. This seems to be characteristic of the bassoon. Now check out Figure 2, which illustrates very well the cutoff frequency phenomenon mentioned earlier. Except for Bb1, notes on the bassoon involve one or more open tone holes. The series of open tone holes is called a tone-hole lattice, and is different for each note (excepting notes that overblow at the octave). Associated with the tone-hole lattice is a cutoff frequency above which reflection does not occur at the first open hole10. The bassoon’s cutoff frequency is about 400 Hz over most of the range of the instrument. Notice how the character of the impedance peaks in Figure 2 changes above 400 Hz. Below 400 Hz there are three resonances aligned pretty well with the harmonics of Bb2. The tall peak near the fourth harmonic (466 Hz) actually lies above cutoff. It is there thanks to one of the bassoon’s “automatic” auxiliary fingerings. When the bassoonist presses the middle Bb key, tone holes to both bores of the boot joint are opened. If the holes to the large bore, which are far below the first open tone hole, were not opened, there would be an impedance minimum near 466 Hz, and the quality of the note would suffer. Above cutoff, the peaks are lower in height and are irregularly and more closely spaced. The peaks below cutoff are the result of reflections at the first open tone hole—the standing wave does not “visit” the lower part of the bore, and opening or closing holes far below the first open tone hole has no effect on these peaks. Above cutoff, the standing wave is not reflected at the first open tone hole, but is free to propagate down the lattice of open tone holes until it reaches a discontinuity at the bell. The irregular impedance peaks represent the sum of weak reflections at the open holes and bell. Therefore, it is possible for auxiliary fingerings to alter the input impedance at frequencies above cutoff.
Next, consider Figure 3, which shows input impedance vs. frequency for C5, a cross-fingered note near the top of the bassoon’s range. The vertical line at 523 Hz marks the first harmonic, and both data and model indicate a strong impedance peak close to this frequency. That the bassoon should play the note C5 based on the fourth peak is, perhaps, not obvious from the graph, but let the reader note that the first three impedance peaks are close to the notes G3, F4 and G#4, respectively, and these are not in any harmonic relationship. It appears to be the case that in the low register, at least, the reed requires two or more well-aligned air-column resonances in order to produce a clear and stable note, but satisfactory high register notes can be played using only one strong resonance. Why this should be so, and whether it is a function of the reed, the air column with its tone-hole lattice, or both, will need to be determined by further investigation.
Before proceeding further, it is necessary to introduce some advanced concepts. No doubt the reader remembers Ohm’s law of electric current from high-school physics:
E = iR,
where E is voltage, i is current and R is resistance. In this form, Ohm’s law holds for direct current circuits, but in the case of alternating current, the voltage varies sinusoidally with time:
E = E1coswt,
where E1 is a constant, w is the circular frequency (2p x frequency in Hz) and t is time. We might reasonably suppose that the current is also a sinusoidal function of time, but—trust me on this— we must allow the possibility that the current is not in phase with the voltage; that is,
i = i1cos(wt + f),
where f is the phase angle. What, then, becomes of Ohm’s law? Standard electrical engineering practice is to express voltage and current as vectors rotating in the complex plane11. In this case,
E = E1e jwt,
where j is the square root of –1, or unit imaginary number. Ohm’s law can now be expressed in a generalized form as
E = iZ,
where Z, the complex impedance, is now a quantity with real and imaginary parts:
Z = R + jX,
where the real part of Z, ReZ = R is the same resistance as in the dc case, and the imaginary part ImZ = X is called reactance. Acousticians have adopted an analog in which pressure p corresponds to voltage, and air flow u corresponds to current. The acoustic impedance Z is then defined by
Z = p/u.
The unit of acoustic impedance is the acoustic ohm, or dyne-sec/cm# in cgs units. The admittance Y is the inverse of Z; that is,
Y = 1/Z.
Admittance is useful for the display of certain types of data. The real part of Y is called conductance, and the imaginary part is called susceptance. The unit of admittance is the mho (ohm spelled backward— get it?). I hope it is clear that these quantities are real and imaginary only mathematically. Physically they are all quite real.
In the case of the input impedance of an air column, the real part (resistance) is associated with the dissipation of energy by friction inside the bore and radiation to the surroundings. It is always positive, since no energy is generated within the air column. The imaginary part (reactance) is associated with energy storage and release. In a vibrating air column, energy is stored as the potential energy of compressed air (compliance) and as the kinetic energy of moving air (inertance). If the input reactance is positive, the flow lags behind the pressure, and the air column looks like an inertance. If the reactance is negative, the flow leads the pressure, and the air column looks like a compliance. The sign of the reactance varies with frequency, and at resonant frequencies the reactance is zero.
We now proceed to the investigation of auxiliary fingerings and try to answer the question: Why is the plain fingering for C#3 so bad, and why does adding the low D key help? In Figure 4, the solid line corresponds to the plain fingering. The first two impedance peaks, which are below cutoff, are well aligned, but there are two spots that look like trouble. Both the third and fourth harmonics (416 Hz and 554 Hz) fall on impedance dips. Now look at Figure 5, which plots the real part of the admittance (conductance) vs. frequency. The power dissipation at any frequency is proportional to the conductance. In Figure 5, high values of conductance at the third and fourth harmonics of C#3 mean that excessive power is drained from the oscillation. The dotted lines in Figures 4 and 5 show what happens when the low D key is closed, as in the standard fingering for this note. There is a slight improvement at the third harmonic, and things look much better at the fourth harmonic. In Figure 4, the impedance dip at 554 Hz has been replaced by a modest peak, and in Figure 5, the conductance is reduced by a factor of three at this frequency.
Let us now take a brief look at the note G3. The plain fingering is slightly unstable, and tends to sharpness. Most bassoonists correct by pressing the low Eb key and/or the low C# key. We show only the effect of the low Eb key. In Figure 6, the first and third peaks near 133 Hz and 324 Hz, respectively, would normally support the low-register note G2 at 98 Hz and 294 Hz, but they have been shifted upward in frequency by the partial opening of the highest finger hole. The second peak at 196 Hz supports the fundamental of G3, while the fourth peak at 412 Hz is disturbingly sharp. The third harmonic of G3 (588 Hz) lies dangerously close to an impedance minimum at 594 Hz. In Figure 7, the conductance, and hence, the power dissipation, at the third harmonic is relatively high, and the auxiliary fingering reduces it by a factor of two. While this may seem slight, it seems to be in line with the slight improvement needed. We should not leave this note without saying something about the sharp fourth peak. At about 412 Hz, it is closer to G#4 than to the desired G4 (392 Hz), and it is not altered by the auxiliary fingering. Although the peak in Figure 6 is misaligned, Figure 7 shows that the conductance is near a minimum. We may conclude that a misaligned impedance peak in this frequency region (near cutoff) is tolerable, provided that the conductance is not too large.
While this paper has concerned itself, for space reasons, only with the two most common auxiliary fingerings on the modern German bassoon, there were other investigations carried out under this grant, such as looking into some of the numerous auxiliary fingerings used on baroque bassoons, using a baroque bassoon replica made by the author. Other measurements were taken in an effort to determine the resonant frequency of baroque and modern bassoon reeds, and the effect of the embouchure on resonance and damping. Much work remains in this area. It was discovered that the computer model and the experimental impedance measurements are both incorrect in calculating/measuring the effect of small-diameter register holes, such as the bocal “pin hole.” It is thought that an experimental technique using higher sound pressure levels would be more accurate, since the nonlinear behavior of register holes is not apparent at extremely low levels of excitation12.
I think that we are well on our way to understanding the operation of auxiliary fingerings on the bassoon. The results of our investigations can be summarized as:
It remains to be determined what constitutes “excessive” power dissipation. We don’t yet know how to calculate or estimate the level of conductance that begins to cause trouble. There is still a lot to learn about bassoons!
1. Anthony Baines, Woodwind Instruments and their History, Faber and Faber, London, 1967, pp. 152, 159, 161.[return]
2. Gerald Corey, “How to Make the French Bassoon ‘Work,’” Journal of the International Double Reed Society, No. 1, 1973, pp. 34–39. [return]
3. George R. Plitnik and William J. Strong, “Numerical method for calculating input impedance of the oboe,” Journal of the Acoustical Society of America, Vol. 65, No. 3, Mar. 1979, pp. 816–825. [return]
4. My implementations of the model have incorporated certain extensions to the Plitnik-Strong method, such as the inclusion of viscous and thermal boundary-layer effects. Doug Keefe provided an improved tone-hole model and showed how the numerous cylindrical segments of the Plitnik-Strong model could be replaced by a smaller number of conical segments. My first program running on a 4.73 MHz 8088-based PC took 2 hours to calculate the bassoon input impedance for 1000 frequencies. My latest program running on a 90 MHz Pentium takes 10 seconds for the same task. [return]
5. The resonances of a well-designed conical reed instrument resemble those of a complete (i.e, including the apex) conical air column. Real conical musical intruments are truncated so that a reed or mouthpiece can be attached. The proper operation of conical reed instruments requires that the small end of the air column be terminated by a reed or mouthpiece-and-reed assembly which imitates the acoustical properties of the missing part of the cone. The fundamental quantity here is the acoustic volume of the reed, consisting of the sum of the actual, geometric volume of the air contained inside the reed, and the equivalent volume due to the reed motion. A value for the equivalent volume of the bassoon reed was determined using the model, as that value which gave correct tuning. This value, approximately 1 cm3, was found to be suitable for experimental data collection. [return]
6. Douglas H. Keefe, Robert Ling and Jay C. Bulen, “Method to measure acoustic impedance and reflection coefficient,” Journal of the Acoustical Society of America, Vol. 91, No. 1, Jan. 1992, pp. 470-485. [return]
7. Harmonics are simply integral multiples of a fundamental. If you play the note A2, the fundamental or first harmonic is 110 Hz, the second harmonic is 220 Hz,etc. [return]
8. Arthur H. Benade, Fundamentals of Musical Acoustics, Second, Revised Edition, Dover Publications, New York, 1990, pp. 394-396 and 439 ff. [return]
9. The player can coax the bassoon into producing a fundamental based on each of the first seven or so impedance peaks using suitable embouchures and “leaking” certain holes to initiate the sound. A tuning meter will confirm that the higher notes produced are sharp to the true harmonics of the low Bb. [return]
10. Benade, pp. 447-462. [return]
11. C. R. Wylie, Jr., Advanced Engineering Mathematics, McGraw-Hill, New York, 1960, Chapter 6. [return]
12. The measured sound pressure level inside the bassoon reed under playing conditions is on the order of 160 dB. The sound pressure level produced by the measurement probe assembly does not exceed 90 dB. Thus the pressure amplitude employed in the measurement is a factor of 3000 less than that associated with playing conditions. [return]