An examination of the resonances in modern flutes with ergonomically angled headjoints Open Access

The positional challenges of playing the modern Boehm flute (and its transverse predecessors) have motivated many attempts to design more ergonomic instruments. Past efforts^1–9 include the vertically-held Pfaff,¹ Guenther,² and Giorgi³ flutes [Figs. 1(a)–1(c)] and their modern derivatives,^4–7 often a conventional flute body used with an angled headjoint [the part of the flute containing the embouchure hole, Figs. 1(d)–1(g)]. A question of interest is how the acoustics of these ergonomic instruments compare to those of modern flutes outfitted with conventional headjoints, and whether the deficiencies of the ergonomic instruments (presumably responsible for their not being more widely played) are intrinsic to their basic design or amenable to correction. Ergonomic flutes present an ideal case for study in that they are simple enough to model as a quasi-one-dimensional branched waveguide, yet complicated enough not to have been thoroughly optimized.

This paper investigates the passive resonance (acoustic power) spectra of various combinations of flute bodies and headjoints using the pressure-based (vs the conventional impedance-based) experimental method and modeling technique introduced in Ref. 10. The method uses a localized sound source placed just outside the flute's embouchure hole to generate forward-going and backward-going pressure waves (with amplitudes $P+$ and $P−$⁠) inside the flute. Power spectra (⁠$P++P−2$ vs frequency) are obtained by Fourier analysis of the acoustic pressure recorded by a microphone placed inside the flute headjoint at the position upstream from the embouchure hole normally occupied by the cork stopper. The resulting spectra are modeled with a transfer matrix method that extracts the input impedance at the embouchure hole (Z_in) from the computed values of $P+$ and $P−$ at the measurement position.

With the exception of a commercial U-shaped, recurved headjoint, the ergonomic headjoints examined were all home-built from off-the-shelf plastic pipe Tees. Each Tee headjoint had a play-tested embouchure hole cut into its top and was connected to a standard flute body by means of a modular jointed neck. For an acoustically transverse configuration, the neck was attached to one of the Tee's two arms; for an end-blown configuration similar to the Guenther flute of Fig. 1(b), the neck is attached to the Tee's base. Unused Tee arm and/or base sockets were sealed with adjustable stoppers which could be positioned some distance away from the embouchure hole (as is typical for the modern transverse flute, which contains a volume of air or “cork cavity” between the cork and the bottom of the embouchure hole) or very close to the embouchure hole [for a flute like the end-blown Giorgi of Fig. 1(c), which does not have a cork cavity]. A sampling of the flute geometries possible with these Tee-based ergonomic headjoints and jointed necks is shown in Fig. 2.

The modern Boehm flute has a cylindrical body and a tapered headjoint having a bore profile that is narrower at the end of the headjoint containing the embouchure hole and wider at the end connecting to the flute body. It is known from the substantial literature on flute acoustics^11–27 that the intonation and playing characteristics of a modern Boehm flute are sensitive to both headjoint taper and stopper position.^{11–13,17,18} An advantage of the present Tee-based headjoint design is that these key geometrical parameters (as well as the flute connection direction, to the Tee arm or Tee base) can be varied without changing the embouchure hole. This feature makes it easier to study how these geometric factors affect the playability of the Tee-headjoint flutes, since a flute's playing characteristics are extremely sensitive to minor variations in the cut of the embouchure hole whereas its passive resonances are not.

Expected intonation (i.e., how the played notes of the instrument's scale relate to a standard reference scale) was assessed by using the spectral analysis described above to compare the resonance frequencies for a given set of fingered notes measured on the ergonomic test flutes to those measured on a flute with the conventional headjoint. (This avoids the problem of needing to know how the resonances are affected by the player's lips, since it can be assumed that the lip position for a given note would be the same on both flute types.)

Harmonicity (the tuning of the upper resonances of a given flute note relative to the note's fundamental, or lowest, resonance) was evaluated for two types of flute bodies: the standard flute body (with keys and toneholes) and non-standard flute bodies comprising plain metal tubes of the same acoustic length (to eliminate tonehole lattice effects). It was found that flute bodies paired with Giorgi-like end-blown Tees often produced harmonicities (and subjectively judged playabilities) quite similar to those of a conventional flute, a somewhat surprising result given that a clearly defined cork cavity has long been viewed as an essential component of the modern flute headjoint. This suggested that a better understanding of flutes with these non-standard headjoints might provide insights relevant to conventional headjoints as well.

The present comparison of conventional and ergonomic headjoints also provided a reminder of some of the known anomalies in the spacing of the conventional flute's resonances. For example, a ∼20–35 cent (0.2 to 0.35 semitone) stretch of the expected octave spacing between the frequencies of the fundamental [f₁] and next-lowest resonance [f₂] in the lowest octave of the flute's range is thought to be desirable,^12,19 whereas the typical flatness of the flute's low C (C4) is usually considered to be a problem.^16,17 This left open the possibility that some differences in harmonicity and tuning might actually favor the ergonomic headjoints.

It should be noted that this paper is examining the passive resonances of these flutes, i.e., the resonances present without the air jet and resulting dc airflow normally provided by the player. Air flow patterns (as well as energy losses due to turbulence at sharp edges) could potentially be different in the headjoints having different steps and/or bends. Consequently, the assumption underlying this paper is that a similar passive resonance structure is a necessary condition for two flutes to have the same playing characteristics, rather than the logical inverse of this statement, that two flutes having the same resonance structure would be expected to have the same playing characteristics.

It should also be noted that his paper does not address differences in sound quality, perceived by both the player and bystanders some distance away, that can arise from a changed separation between the instrument's two sound sources (one at the embouchure hole, the other at the first open hole). These effects were investigated by Coltman²⁶ and Laszewski and Murray²⁷ for the case of the recurved headjoint, and found to be quite strong.

The passive resonance spectra and resonance frequencies for the flute body/flute headjoint combinations of this paper are determined using the method and model introduced in Ref. 10. The experimental set-up is summarized in Sec. II A; the flute bodies and notes examined are described in Sec. II B; and details of the headjoints used are presented in Sec. II C. Section II D describes the model and the fitting parameters used.

The experimental set-up is shown schematically in Fig. 3. A flute containing an embedded microphone positioned at the place normally occupied by a cork headjoint stopper (e.g., stopper #1 in Fig. 3) is excited by a localized sound source positioned 1 cm in front of the flute's embouchure hole. The acoustic pressure at the microphone position is recorded and Fourier analyzed to produce a raw power spectrum. A reference power spectrum for calibration/normalization is obtained in the same manner by replacing the flute with a reference microphone assembly. The flute's power spectrum at the cork, i.e., I_cork $≡Pcork++Pcork−2$⁠, is obtained by dividing the raw power spectrum by the reference power spectrum.

As described in more detail in Ref. 10, the localized sound source is provided by a computer-driven audio speaker connected to the input end of a gasline delivery system. The delivery system comprises various tubes and plastic fittings and terminating in a short length of 1-cm diameter pipe. The speaker played an Audacity-programmed sound clip consisting of 150 s of pink noise.²⁸ For sound isolation purposes, the speaker and computer were in one room and the test flutes were in another.

Acoustic pressure was measured with a 10-mm-diameter pre-polarized, cardioid-pattern condenser microphone (Audix ADX10-FLP) in combination with an XLR-to-USB adapter (Shure X2U). The microphone was embedded in a pair of rubber grommets and squeezed into place to make an approximately planar air-tight seal at the headjoint stopper position mentioned earlier.

The reference microphone assembly used for obtaining the reference power spectra comprised a nominally identical microphone (corrected for differences in frequency response) tightly embedded in a thick-walled Delrin tube that was press-fit into a 22-mm outer-diameter plastic holder tube. The microphone face was flush with the end of the holder tube and positioned to be the same (1.0 cm) distance away from the sound source as the resonator.

Microphone signals were collected at a 44.1 kHz sample rate and saved in *.wav format. Spectral analysis on the acquired *.wav files was done by Audacity's fast Fourier Transform (FFT) algorithm using a Hann window on contiguous sample segments 16 384 points in length (resulting in a frequency resolution of 2.7 Hz). A fresh reference microphone spectrum was typically collected within an hour or two of every flute measurement, with every effort made to perform the flute and reference microphone measurements at the same temperature (no more than 0.3 °C apart).

Maxima in the I_cork spectra were determined (with one exception, see supplementary material,²⁹ Sec. A) by a parabolic fit to a given peak's highest three points. Reproducibility was assessed using the stability of the f₁-f₂ octave stretch, since that measure eliminates the issue of peak position variability due to temperature. Typical reproducibility was better than ±5 cents; more detail can be found in the supplementary material,²⁹ Sec. A.

The flute bodies and headjoint-to-body connection geometries are shown schematically in Fig. 4. The keyed C flute body used in these experiments was a 1980 “standard model” solid silver, open-hole, B-foot Muramatsu [Fig. 4(b)]. Open-hole (perforated French-style) keys were plugged with plastic key plugs (Yamaha style caps) and the desired keys (French, plateau, and/or register) were clamped closed with flute key clips (Ferree's Tools, Inc. Battle Creek, MI).

Five fingered flute notes were examined for each test case: three with non-forked fingerings, with fundamentals nominally at 261.6 Hz (C4, low C), 392.0 Hz (G4, low G), 523.3 Hz (C5, medium-low C); and one or two forked fingering notes in the third register nominally sounding at 1568.0 Hz (G6, high G) and 2094.0 Hz (C7, high C, using the gizmo key fingering to close the lowest footjoint tonehole). No-lattice flute bodies [Fig. 4(c)], created by replacing the “with-lattice” keyed flute body by an appropriate length of a nickel-silver tube having the same diameter and wall thickness as that of the flute body tube, were also examined for G4 and C5. Some additional low register notes were also examined for the case of the conventional reference headjoint.

No-lattice configurations were useful both for evaluating harmonicity without the complications of closed-tonehole volume issues, and for extending the harmonicity measurement to resonances that would normally be masked by (open) tonehole lattice effects. However, the keyed flute was typically preferred for comparing the tuning of the main resonances associated with each flute note because (i) it was more realistic and (ii) the fingerings could be changed without dismantling the flute between measurements.

The conventional headjoint used in this study, shown schematically in Fig. 4(a), came with the Muramatsu flute body. The headjoint had the conventional taper (long viewed as a necessity to keep the upper register notes from being flat¹¹), namely a bore diameter that gradually increases from ∼17 mm at the cork position near the embouchure hole to ∼19 mm at the start of the tuning slide.

The Tee headjoints were constructed in two makes of ½ in. polyvinyl chloride (PVC) slip-fit Tees, denoted Charlotte or Genova, depending on the supplier (Charlotte Pipe and Foundry Company or Genova Products). As indicated in the caption of Fig. 5, the Charlotte and Genova Tees differed in their internal dimensions. The parts of a Tee headjoint, including embouchure hole, stoppers, and a representative modular jointed neck, are shown in Figs. 5(b) and 5(c). Embouchure holes were rounded rectangles of dimensions 10 × 12 mm² (as measured from the top of the Tee) with 5/16 in. diameter corners. The embouchure hole walls were typically angled at 7° (widening with increasing depth), with sidewalls further undercut by hand. Stoppers for the Tee's side-arms were typically fabricated from a short stack of slightly modified beveled styrene-butadiene rubber (SBR) faucet washers sandwiched around rubber O-rings and strung on a spanner pan head screw and squeezed together with a nut. Transverse Tee headjoint configurations requiring a plugged base socket were fitted with stoppers of a similar construction, though typically modified so that the (normally countersunk) washer in contact with the screw head was replaced with a Delrin fixture machined to match the curved contours of the Tee's interior body. Though the two arm stoppers in the end-blown Tees could easily be configured (and modeled) to be asymmetric, with non-identical distances from the embouchure hole, the stopper spacings examined in this paper were always set to be as similar to each other as possible.

A modular jointed neck was used to connect the Tee to the flute body. An effective taper was implemented in the jointed neck by using a “stepped bore” consisting of a narrower tube on the Tee side of the joint and a wider tube on the flute body side of the joint. The effective taper could be varied by changing the relative lengths of the two sections, allowing headjoint taper effects to be investigated in both the ergonomic headjoints and in transverse-Tee versions of the conventional flute (realized by using a straight joint in place of a bent one).

The jointed necks were fabricated from a collection of ¾ in. copper-tube-size (CTS) chlorinated polyvinyl chloride (CPVC) Nibco slip-fit connectors [mostly the 45° elbows shown in Fig. 5(d)] and connecting segments of ¾ in. CPVC or nickel-silver tubes. The CPVC tubes, with a nominal inner diameter (ID) of 17.7 mm, were used for the Tee-to-joint connection (“neck 1”) and the elbow-to-elbow connection (“neck 2,” needed only for double elbow joints). Neck 1 was turned down at one end to fit into the receiving socket of the PVC Tee. The sockets of the CPVC joints were sometimes shortened to allow a closer spacing between adjacent fittings.

Connection to the flute body was provided by a metal “neck 3,” comprising a length of the same 19-mm-ID nickel-silver tubing that was used for the no-lattice flute bodies. The tuning-slide end of the neck 3 extended into the flute body barrel by a distance of 1 to 4 cm, where the distance was selected to give the best tuning for a played G4. The other end of neck 3 was inserted into a CPVC sleeve (a standard ¾ in. CPVC tube modified to have a 25/32″ ID) that fit into a socket of the CPVC receiving joint.

Some of the same nickel-silver tube stock was also cut into short (4–6 mm) lengths for optional use as “tuning rings.” As illustrated in Fig. 4(d), one or more rings could be placed in the wider-diameter bore region of the barrel (present whenever the tuning slide end of the headjoint is not fully inserted) to provide a uniform-diameter bore region over the barrel's entire length. (For completeness, it is now noted that analogous plastic rings were also used to maintain a constant-diameter cork cavity for cases in which the Tee stoppers were less than fully inserted.)

The recurved headjoint was manufactured by the Emerson company. The tubing of the 180° bend had a diameter of ∼18.1 mm (the same as that of the cylindrical tube segment containing the embouchure hole), a center-axis radius of curvature of ∼17.0 mm, and a distance of ∼5.6 cm from the embouchure hole to the beginning of the bend. Somewhat surprisingly, L_cork [as shown in Fig. 4(d), the distance between the cork and the center of the headjoint's embouchure hole] had to be decreased to 10 mm for the headjoint to play in tune (in contrast to the 17 mm typical for the conventional modern flute). Incidentally, the Emerson headjoint studied by Coltman in Ref. 26 had fairly similar external dimensions, but the tube segment containing the embouchure hole was slightly tapered and the L_cork value was the traditional one.³⁰ 

For simplicity and better reproducibility, all measurements were performed with a “no-lips” configuration of the embouchure hole (i.e., just the bare hole, without the foam-tape strip previously shown¹⁰ to be a reasonable approximation to the player's bottom lip when attached to the long side of the embouchure hole). In cases where a “with-lips” configuration was of interest, the modeled spectra were recalculated with the lip parameters previously identified for the artificial lip just mentioned, namely a reduced embouchure hole area and a hybrid unflanged/infinite flanged end correction term modified to account for the hole's more elongated shape (12 mm × 5 mm instead of 12 mm × 10 mm).¹⁰ 

Table I lists the geometrical parameters for the different configurations of ergonomic headjoints investigated in this paper; details on the reproducibility of the listed dimensions can be found in the supplementary material,²⁹ Sec. A.

Modeling provided two essential components of this investigation: (i) a way to determine the positions of the resonances from the measured I_cork spectra (since the |1/Z_in| spectra from which the resonances would ordinarily be determined are not directly measured, but rather derived from fitting parameters used to generate the I_cork simulations) and (ii) a way of predicting how a given change in headjoint geometry would affect the positions of resonances (a useful aid for optimizing tuning and harmonicity). However, as will be discussed in Sec. III A, the maxima in the computed I_cork and |1/Z_in| spectra were nearly coincident over a wide range of headjoint geometries. Consequently, the positions of the resonances were instead inferred from the maxima in the experimental I_cork spectra, rather than by extraction from the potentially imperfect |1/Z_in| spectra derived from the I_cork simulations.

The I_cork spectra for each flute configuration were modeled with a pressure-based formulation of a transfer matrix¹⁰ method which calculates the amplitudes of the forward-going and backward-going pressure waves $P+$ and $P−$ at each point in the resonator. Power spectra at the cork (i.e., $Pcork++Pcork−2$ vs frequency) for the various flutes are computed and fit to the experimental I_cork spectra; input impedance is derived from the computed values of $P+$ and $P−$ at the flute's embouchure hole.

Schematics of the flute body geometries and tonehole positions were shown in Fig. 4; fitting parameters for the flute bodies (taken from Ref. 10 with only minor modification) are listed in Sec. B of the supplementary material.²⁹ Schematics of the conventional headjoint, an end-blown Tee headjoint, and a 45° elbow are shown without the fitting parameters in Fig. 6, and with the fitting parameters in Sec. C of the supplementary material.²⁹ A schematic of the recurved U-shaped headjoint with fitting parameters is shown in the same section. Fitting parameters for the 45° elbow were determined in a two-step process: a transverse Tee was first paired with a straight joint (e.g., case S) to get the fitting parameters for the Tee; after this, the same Tee was paired with the elbow (e.g., case Q) to get the fitting parameters for the joint.

The reflection coefficients and end correction factors for the input and output orifices were taken from Ref. 10, which used unflanged values³¹ for the tube ends, infinite-flanged values³² for open toneholes, and a hybrid of the unflanged and infinite flanged values for the embouchure hole (supplementary material,²⁹ Sec. D).

Good playability requires an instrument to have a good scale (with satisfactory placement of the main resonance used to produce each note) as well a harmonicity providing suitably aligned upper resonances;¹⁹ this section shows how both are (i) evaluated from the positions of the passive resonances and (ii) affected by various features of the headjoint design. Section III A presents the measured and simulated I_cork spectra for a note of the reference flute (the keyed flute body with a conventional headjoint) along with the |1/Z_in| spectra derived from the same simulation to illustrate the technique and provide a plausibility argument for using the experimental I_cork maxima as a stand-in for the position of the resonances. Results on the tuning of the resonances are shown in Secs. III B (baseline data for the reference flute) and III C (test flute data relative to the reference flute). Section III D compares the harmonicities measured for the reference flute to those found for the end-blown Tee headjoints judged to be the best and the worst matched to the reference; Sec. III E examines the I_cork spectra for similarities and differences related to headjoint geometry. Finally, Sec. III F shows two examples of how calculations predicting the effect of various changes in the headjoint bore profile can both improve understanding of the existing data and be a useful guide for further optimization.

Experimental I_cork spectra, along with the calculated I_cork fits, I_cork difference spectra Δ (fit vs measurement), and |1/Z_in| (input admittance) spectra derived from the same simulation, are shown in Fig. 7 for two cases of a C5 played with a conventional headjoint. In the first case [Fig. 7(a)], the flute body is a plain tube (without a tonehole lattice); in the second [Fig. 7(b)], the flute body is the conventional keyed one (which, for C5, has all the full-sized toneholes open). Note that here and elsewhere, the |1/Z_in| spectra are plotted as |Z_c/Z_in|, where 1/Z_c is the input admittance of an infinitely long tube having the same cross-sectional area as the flute's embouchure hole. Note also that the dB scale is amplitude-based for |Z_c/Z_in| (so the plotted quantity is 10*log[|Z_c/Z_in|]) and intensity-based for I_cork (so the plotted quantity is 10*log[I_cork] ≡ 20*log[$Pcork++Pcork−$]).

As expected, the spectra for these two cases are nearly identical below the cutoff frequency (∼2000 + –300 Hz for a conventional modern flute^12,22) with both showing strong, uniformly-spaced resonance peaks. Above the cutoff frequency, the peaks for the keyed flute show a much reduced spectral contrast (the dB spread between adjacent minima and maxima) as well as an increased irregularity in spacing (the details of which will be discussed in Sec. III E, where the I_cork spectra are shown up to 16 kHz). Some reduction in high-frequency contrast is also seen in the no-lattice spectrum, an effect attributed to wall and radiation losses, which increase with frequency.¹³ The fits (thin light line) are quite good, as can be seen more quantitatively from the plotted difference spectra (thin dark line), though it should be noted that the fits (i) tend to be less accurate above the cutoff frequency for fingerings with a large number of open toneholes (presumably due to the cumulative effect of small errors in the tonehole fitting parameters) and (ii) can be as much as 2 dB off in fitting the minima and maxima of resonances below the cutoff frequency.

Additional spectra and fits can be found in the supplementary material²⁹ for both the reference flute (Sec. D) and test flutes (Sec. E), along with a version of Fig. 7 plotted to 8 kHz (Fig. S7, Sec. D). The simulated spectra generally provided a good match for the positions of the observed resonances and the minima between them, though the model tended to underestimate the depths of the minima between the lowest few resonances for the test headjoints with the strongest bends. Some discrepancies between model and measurement for the non-standard headjoints might be expected, given the inadequacies of approximating a bent tube as a collection of stepped tube segments, each with its own cross-sectional area S and length L (vs the more sophisticated treatments described in Ref. 33). However, the possibility that some discrepancies arise from experimental artifacts and/or losses that are not well accounted for in the model cannot be definitively ruled out.

Before comparing the input admittance (|1/Z_in|) and I_cork spectra in the plots of Fig. 7, we first note that the |1/Z_in| spectrum of Fig. 7(b) is in good qualitative agreement with results obtained by others^20,23 through direct measurement. However, as shown in supplementary material,²⁹ Sec. F, exact comparisons can be difficult due to differences in how the embouchure hole coverage is taken into account.

The |1/Z_in| and I_cork plots show markedly different spectral contrast vs frequency behaviors, a fact noted without explanation in Ref. 10. These differences are most obvious at high frequencies (especially easy to see for the case of the no-lattice flute), where the spectral contrast is markedly larger for I_cork compared to |1/Z_in|. At lower frequency, the differences are more subtle: the contrast in the |1/Z_in| spectra monotonically decreases with increasing frequency, whereas the contrast in the I_cork spectra first increases with frequency, reaches a plateau, and then (above 2 kHz or so) starts to decrease. This low frequency behavior can also be seen in the simpler case of open-input cylindrical tubes having a measurement point close to the input aperture; for this case, the behavior results from a shift in the standing wave pattern (different for each resonance) that brings one of the pressure antinode closer to the measurement position (supplementary material,²⁹ Sec. G).

From Fig. 7 it can be seen that the frequencies of the maxima in the computed I_cork spectra are quite close to the minima of the input impedance spectra calculated with the same fitting parameters. Some agreement is not surprising, since the intensity of the (pressure amplitude) standing wave pattern at each resonance, clearly a maximum at the pressure antinodes, will also tend to be a maximum at all points in the resonator except those exactly situated at a pressure node. In the absence of losses, the frequencies of the I_cork maxima are exactly coincident with those of |1/Z_in| (though the I_cork spectrum would be missing the peaks for which there is a pressure node at the measurement position); with losses, the situation is more complicated (supplementary material,²⁹ Sec. G). I_cork maxima calculated for the four lowest resonances of various test and reference flute cases (details in supplementary material,²⁹ Sec. H, Tables S-III to S-VI) were 2–4 cents lower than the |1/Z_in| maxima for G4 and C5, and 6–9 cents lower for C4. For a given instrument, even smaller discrepancies between I_cork and|1/Z_in| were found in the frequency ratios of these resonances. This important result is the justification for taking the positions of the resonances to be those of the experimental I_cork maxima since a small systematic error was viewed as preferable to the more random (and potentially larger) errors that might be introduced if the impedance minima were extracted from an imperfect fit to the I_cork spectra.

Figure 8 compares the passive resonance frequency associated with a given fingered flute note to the nominal frequency of the note meant to be played. The resonances of interest, extracted from the measured I_cork spectra, are the fundamental for C4 through C5; the 2nd resonance for G5 and C6; and the strongest resonance for G6 and C7.

The data of this work (dark circles) are in excellent agreement with previous measurements by Nederveen¹⁷ for a similar flute with a bare embouchure hole (light circles). However, both data sets are as much as 50–100 cents sharp compared to the intended frequency of the fingered note, an expected consequence of doing the measurements without an obstruction at the embouchure hole to simulate to the player's lips. Lowering the present data by a uniform 65 cents (dark dots) brings it into rough agreement with data from Coltman¹⁶ for a similar flute with a fixed artificial lip positioned to cover half of the embouchure hole (light squares).

More precise estimates of the flattening effect of a given lip coverage can be straightforwardly calculated from lip parameters given in Ref. 10. For simple tubes cut to lengths to give C4, G4, and C5 with a standard headjoint, one finds (Ref. 29, Sec. I) that a 50% physical lip coverage reduces the frequencies of the respective fundamentals by 33, 50, and 68 cents, with nearly negligible effect (−2 to +3 cents) on the f₁–f₂ octave width.

With or without correction for a fixed-position lip, the frequencies of the resonances clearly deviate from the flat line that might be expected for a flute designed to be in tune with itself. However, previous work has established that the embouchure hole coverage varies with the pitch of the played note; more precisely, the optimum ratio of v (the jet velocity) to h (the length the jet travels before reaching the far side of the embouchure hole, controlled by embouchure hole coverage), increases with the frequency of the played note.^12,14 The fact that most flute players manage to play fairly well in tune suggests that these apparent flaws in the positioning of the resonances are actually built-in features of the flute design, intended to compensate for the expected variations in v and h during playing. That said, it is generally agreed that lip positioning is quite variable from player to player.²⁴ 

The tuning of the test flute resonances relative to those of the reference flute is shown in Fig. 9 for four transverse headjoint configurations and in Fig. 10 for seven end-blown configurations; the headjoint geometries are listed in Table I. A configuration with tuning exactly in agreement with the reference flute would have all its points on the horizontal zero-cents line; configurations with points on a line offset from, but roughly parallel to, the zero-cents line for frequencies <1200 Hz can also be considered to have fairly good tuning, since the overall pitch can be adjusted with the tuning slide. There is clearly a range of behaviors: some are quite similar to that of the reference flute (cases G, M, and O, for example), while others (cases I, N, and V, for example) are not. Several of the headjoints trend quite sharp for the two highest notes (a feature likely arising from a stopper-to-embouchure hole distance Lcork slightly smaller than optimum).

The harmonicity of the various headjoint/body combinations (i.e., the tuning of their upper resonances relative to the fundamental) is expected to affect both tone quality and instrument responsiveness.¹⁹ Of particular interest is the low register f_1–f₂ octave spacing, which is usually around 15–35 cents wider than an exact octave.^12,19 Harmonicity data [f_n/(n*f₁) vs f_n, with f₁ being the frequency of the first (lowest) resonance and f_n being the frequency of the nth resonance] is shown in Fig. 11 for three notes of a flute with a conventional headjoint and a body that is either conventional (with the tonehole lattice) or a plain tube (without a tonehole lattice). The deviation from perfect harmonicity (a case for which the plotted points for each given note would all fall on the horizontal cents = 0 line passing through the note's f₁) first increases with frequency and then decreases. The no-lattice curves are quite similar to the with-lattice ones for C4 (a case for which the last tonehole of the B-foot is the only one open), but consistently sharper (higher in pitch) than the with-lattice ones for G4 and C5. Interestingly, the same differences between the no-lattice and keyed flutes are seen in the simulated spectra as well, suggesting that harmonicity is affected by both closed and open tonehole lattices in ways that are difficult to predict without numerical modeling.

Figure 12 shows the harmonicities of the same three notes for two configurations of an end-blown Tee headjoint, along with the conventional flute data from Fig. 11 for reference (shown as dots). The two Tee headjoint configurations differ only in stopper position, and were selected to illustrate the extremes of good and bad harmonicity. The best case (G) had a physical embouchure-to-stopper spacing of 11 mm, leaving a lateral space of ∼1 mm between the face of the stopper and the vertical wall of the base outlet bore; the worst case (I) had an embouchure-to-stopper spacing of 18 mm. The best-case data show harmonicities that are remarkably close to the reference case; the worst-case data shows upper resonances that are somewhat flatter than ideal for C4, but unacceptably flat (including a 0-cent f₁–f₂ octave stretch, when 20–35 cents is thought to be ideal^12,19) for G4 and C5.

Incidentally, it should be noted that harmonicity plots of this type provide an alternative (and perhaps more intuitive) way to think about a well-studied^15,17,18 problem in flute acoustics, namely, why L_cork = 17 mm appears to be the optimum stopper position in the conventional headjoint. With rare exceptions,²⁵ previous approaches to this problem have been computational and require calculation of L_emb, a frequency-dependent parameter defined as the acoustic length of the embouchure hole/cork cavity combination. The optimum cork position is then taken to be the one that leaves L_emb with the least sensitivity to frequency. The present approach, in contrast, allows the effects of stopper position on harmonicity to be determined by measurement as well as computation; the results are clearly related to physically meaningful parameters (e.g., the f₁–f₂ octave stretch) and are readily visualized. That said, there is no universally agreed upon figure of merit for harmonicity, though one could easily adopt some specific features of a reference flute's harmonicity to use as a standard.

Figure 13 compares the I_cork passive resonance spectra measured for a standard keyed flute body with a variety of headjoint configurations to the spectrum for the same flute body with a conventional headjoint. The spectra, all for C5, are divided into three groups, each of which includes a copy of the reference spectrum (heavy line). The top group, in Fig. 13(a), shows data for three transverse headjoints: the recurved one (case U) and an end-blown Genova Tee configured with two different necks (case Q, with a single 45° elbow; and case T, with a double one). All three spectra show good alignment with the reference flute fundamental and spectral contrast similar to that of the reference for the 3rd and 4th resonances. However, the recurved headjoint shows a contrast for the fundamental that is more than 10 dB higher than the other spectra, due to its lower minimum. This unexpectedly large contrast was present for the no-lattice C5 tube as well as for the lowest resonances of the G6 and C7 (notes having the same first open hole as the C5), but was absent for the G4 and C4. A wide variety of trial neck profiles (differing in bulge dimensions and placement) were tested in an attempt to reproduce this feature, but without much success. However, since it is the maxima of these spectra that are of the most musical interest, this anomaly is perhaps best saved as a challenge for more sophisticated models of bent tubes (e.g., Ref. 33).

The middle group of spectra in Fig. 13(b) is for the end-blown, double-elbow headjoints M and O, configurations, which Fig. 10 shows to have similarly good alignment with the reference flute fundamental. The spectra both show ∼5–10 dB stronger contrast on f₁ and f₂ compared to the reference spectrum, and similar or weaker contrast on f₃ and f₄. The very close similarity of the case M and case O spectra is at first surprising, since the headjoints were made from Tee types with very different internal dimensions (e.g., a core diameter of 19.3 mm for Charlotte and 16.7 mm for Genova). Apparently, the difference in core diameter for these end-blown headjoints can be compensated for by an appropriate choice of stopper position (e.g., an L_cork of 11 mm for Charlotte and 13 mm for Genova). Incidentally, the core diameter of the Tee appears to be more critical for the transverse Tees, in that good headjoints could easily be made with the narrower bore Genova Tee, but were a struggle with the Charlotte.

The bottom group of spectra in Fig. 13(c) was selected to shed some light on some secondary effects of large tuning pull-outs, hinted at by some of the peculiar differences between cases P and V in the tuning data of Fig. 10. The primary effects of headjoint/tuning slide pull-out are well known: pulling out the headjoint makes the flute tube longer and flattens all the notes (though, somewhat inconveniently, not by identical amounts,¹¹ since the degree of flattening is less for a note with all the toneholes closed and more for a note with all the toneholes open). However, the secondary effects resulting from the enlarged bore left behind can be significant when the pull-out is large. Cases P and V were configured with exactly the same headjoint, a single-elbow version of the end-blown double-elbow Tee used for case M, and differ only in whether or not tuning ring inserts are used to fill the unusually long (∼2.0 cm) pull-out gap.

An examination of the peak positions in the spectra of Fig. 13 suggests the reason for the Fig. 10 differences in the case P and case V tuning data, in particular, an f₁–f₂ octave stretch for C5 (medium-low C) that is 20 cents narrower than the reference headjoint for case V and 5 cents wider than the reference for case P. Adding the tuning ring insert (i.e., going from P to V) sharpens the fundamental f₁ with relatively little effect on f₂, resulting in a decrease in the f₁-f₂ stretch (in this case, from 30 cents to 5 cents for a 2-cm gap length, or about 12 cents per cm of gap). As will be discussed in the next section, this ∼12 cents/cm is somewhat larger than the 8 cents/cm calculated for a headjoint wall thickness of 0.016 in. (though exactly the calculated value for a headjoint wall thickness of 0.025 in.). However, these results suggest a way for fine-tuning the octave stretch (something many flutists probably prefer to be wider than it is): cut off 5–10 mm from the bottom of the headjoint and then pull the headjoint out (from its fully inserted position) by the same amount. This creates a gap without increasing the physical length of the flute (and contrasts with the normal situation where the player pulls the headjoint out a bit to deliberately physically lengthen the flute and flatten the pitch).

Though there are subtle differences between the various spectra of Fig. 13, they are qualitatively quite similar over the 0–2.5 kHz frequency range shown. As a sanity check to see if more distinctive differences would appear at higher frequencies, the spectra were replotted in Fig. 14 to show data up to 16 kHz. On this frequency scale, several headjoint-geometry-correlated differences in the spectra above 4 kHz are apparent. First noted are the characteristic “dead spots” (localized regions of severely reduced spectral contrast) in the transverse headjoints of Fig. 14(a). These dead spots, previously described in Refs. 12 and 21, move from ∼4.5 kHz for the conventional headjoint (L_cork = 17 mm, case D) to ∼7.5 kHz for the recurved headjoint (L_cork = 10 mm, case U), and correspond to the cork cavity resonance (treating the cork cavity as a waveguide) at roughly (1/4)*c/L_cork, with c being the speed of sound. Next to be compared are the end-blown Tees of Figs. 14(b) and 14(c). Headjoint combinations O, P, and V, all Charlotte Tees with an L_cork of 11 mm, differing only in neck geometry or tuning slide pull-out, all show a characteristic step at ∼7.5 kHz. This step shifts down to ∼6.2 kHz for case M, a Genova Tee with a 13 mm cork position. All four of these Tees lack the characteristic dead spot, possibly an advantage for notes having harmonics falling in that frequency range.

Plots for C4 and G4 analogous to the C5 plot of Fig. 14 (i.e., supplementary material,²⁹ Sec. J Figs. S21 and S22) show that the dead spots and steps are not affected by the flute note being fingered; Fig. S23 for G4 and C5 of the conventional flute (case D) and an end-blown Tee (case P) show that these dead spots and steps are not affected by the replacement of the keyed flute body with a no-lattice one. This suggests that these features are related to the geometrical details of the cork cavity and adjacent spaces near the embouchure hole, rather than to any tonehole lattice effects.

Starting at 4–5 kHz (about a factor of two above the expected cutoff frequency), the keyed flute body spectra of Figs. 14, S21, and S22 all show ∼3-kHz-wide spectral regions that are populated with sequences of relatively low contrast peaks spaced apart by about 250–260 Hz. Similar spectral features seen previously for the case of a conventional C-foot flute fingered to play C#5/C#6 were attributed to the fact that the flute's effective length in this frequency range now encompasses the whole length of the instrument, rather than being limited to the portion of the flute's length containing the closed toneholes.²¹ The present spectra (as well as the supplementary material²⁹ Fig. S24, an expanded version of a plot from Ref. 10 comparing the I_cork spectra for the G4 of a conventional keyed flute with and without the footjoint) provide further, though still anecdotal, support for this explanation.

This section highlights two examples of how calculations predicting the effect of various changes in the headjoint bore profile can both enhance understanding of the existing data and guide further optimization. The first example examines the effect of a tuning slide gap on the f₁–f₂ octave stretch. This situation occurred to a greater extent than usual for the flutes studied here, since the as-constructed test headjoints had acoustic lengths that could be too short by as much as 2 cm. Some flutists consider the tuning slide pull-out gap to be enough of a concern that they use a tuning ring insert [illustrated in Fig. 4(d)] to restore the bore to its original diameter; others think it is nonsense that such a small change in bore diameter could have a perceptible effect. Calculations were done for three no-lattice flute notes (C4, G4, and C5) with a conventional headjoint, a tuning gap of 1 cm, and a barrel with a diameter exactly sized to fit a flute tube with an inner diameter of 19 mm and a wall thickness of 0.016 in.; the results are summarized here, with more detail in the supplementary material,²⁹ Sec. K.

The addition of the gap consistently decreases the frequency of the first (lowest) resonance (by an amount that increases from 0.5 cents for C4 to 4.6 cents for C5), but has a non-monotonic effect on the f₁–f₂ octave stretch (–1.8 cents for C4, 4.7 cents for G4, and 7.9 cents for C5). It seems reasonable that an ∼8 cent change in octave stretch is large enough to affect the playing characteristics of the flute (for better or worse, depending on the initial value of the stretch) and that a change in octave stretch by double this amount (e.g., for the 2-cm pull-outs sometimes required to make the test headjoints play in tune) is more than enough to reduce the accuracy of test headjoint comparisons when one headjoint has a gap and the other does not.

The second example shows how the harmonicity is affected by changing the effective headjoint taper. As stated earlier, conventional headjoints for the modern flute are typically tapered to keep the flute's upper octave from being flat.¹¹ For the Tee headjoints of this work, the tapered tube was replaced with a modular jointed neck, an example of which is shown schematically in the Fig. 15 inset. The jointed neck shown comprises a narrower first neck segment (neck 1, with an ID = 17.7 mm) that connects to the Tee containing the embouchure hole; a connecting joint (in this case, a single 45° elbow); and a wider final segment (neck 3, typically metal, with an ID = 19.0 mm) that connects to the flute body. While the total length of the modular neck is constrained to be roughly the same as the headjoint flute tube it is replacing (to avoid changing the pitch of the fundamental), the relative lengths of the narrower neck 1 and wider neck 3 can be adjusted to simulate various tapers.

The effect of neck 1 length on the harmonicity [f_n/(n*f₁) vs f_n] is shown in Fig. 15. Calculations were performed for a (no-lattice) C4, G4, and C5 configured with either the (conventional) reference headjoint or an end-blown Tee headjoint configured with one of three different neck 1 lengths. The harmonicities for all three neck 1 lengths are roughly similar to that of the reference headjoint, though with significant differences in the details. For the neck 1 parameters examined, the f₁–f₂ octave stretch always increases as neck length increases; for C4 and G4, the octave stretch is less than that of the reference, and for C5 the octave stretches range from less than to greater than the reference. The optimum neck 1 length parameter will, of course, depend on whether one is trying to match the reference headjoint or make a correction to it. In any event, the calculations of Fig. 15 make it clear that the length of neck 1 offers an additional way to fine-tune headjoint harmonicity.

This work examined the resonances in a variety of ergonomic flutes using a pressure-based method and model recently developed for determining the passive resonance structures of open input resonators (Ref. 10). It was found that resonance structures quite similar to those of the conventional flute could be realized in a wide variety of ergonomic flutes (end-blown or transverse, single or double elbow, etc.) with proper selection of stopper position and neck lengths. This suggests there is no fundamental reason why the ergonomic flutes cannot be made to sound as good or better than the best of the conventional ones.

This work also showed that the modeling technique, previously demonstrated for the relatively simple cases of a cylindrical tube and a conventional flute, could work in unusual Tee headjoint geometries such as those containing a double cork cavity. It also explained why notes in the flute's bottom octave have one peak intensity vs frequency behavior in the input admittance (|1/Z_in|) spectra (a monotonic decrease with increasing frequency), but a different behavior (an initial increase followed by a decrease) in the I_cork spectra (i.e., $Pcork++Pcork−2$ vs frequency, the ratio of the raw power spectrum at measured at the cork to the power spectrum of the sound source at the flute's input aperture).

In addition, this work provided insights on two issues commonly encountered with conventional flutes. First, it was suggested that plotting the harmonicity of the flute's passive resonances might be a more satisfactory way to visualize the effects of headjoint cork position (more typically explained in terms of the acoustic length L_emb of the embouchure hole/cork cavity combination); second, tuning ring inserts were shown to have an effect potentially large enough to be noticeable (at least to a trained flutist with a discriminating ear).

However, many questions remain. It is still hard to know whether the differences in tone quality and responsiveness of these flutes (better than the conventional flute in some respects, perhaps not as good in others) result from a changed alignment of the resonances, discontinuities in the bore diameter at connecting joints (surely unlikely to be of help), the details of the embouchure hole cut, or just the idiosyncrasies of the player.