Resonance Modes of a Flute with One Open Tone Hole

Acoust. Sci. & Tech. 38, 1 (2017) #2017 The Acoustical Society of Japan PAPER

Resonance modes of a ﬂute with one open tone hole

Seiji Adachi Department of Acoustics, Fraunhofer Institute for Building Physics, Nobelstr. 12, 70569 Stuttgart, Germany (Received 1 September 2015, Accepted for publication 3 August 2016)

Abstract: A minimal model explaining intonation anomaly, or pitch sharpening, which can sometimes be found in baroque flutes, recorders, shakuhachis etc. played with cross-fingering, is presented. In this model, two bores above and below an open tone hole are coupled through the hole. This coupled system has two resonance frequencies !, which are respectively higher and lower than those of the upper and lower bores !U and !L excited independently. The ! differ even if !U ¼ !L. The normal effect of cross-fingering, i.e., pitch flattening, corresponds to excitation of the !-mode, which occurs when !L ’ !U and the admittance peak of the !-mode is higher than or as high as that of the !þ-mode. Excitation of the !þ-mode yields intonation anomaly. This occurs when !L / !U and the peak of the !þ-mode becomes sufficiently high. With an extended model having three degrees of freedom, pitch bending of the recorder played with cross-fingering in the second register has been reasonably explained.

Keywords: Cross ﬁngering, Pitch bending, Intonation anomaly, Avoided crossing PACS number: 43.75.Qr, 43.20.Ks, 43.75.Ef [doi:10.1250/ast.38.14]

electrical circuits etc. In Gough [6], a string coupled to a 1. INTRODUCTION soundboard by a bridge is discussed in detail as well as Cross-fingering is a technique of playing woodwind two strings coupled with each other. An important result of instruments in which one or more tone holes are closed these analyses is that two coupled oscillators have two below the first (closest to the input end) open hole. It different resonance modes; one of these frequencies is usually yields a pitch a semitone lower than that played higher and the other is lower than the frequencies of these with normal fingering. There is, however, an exception oscillating independently. Mathematically this corresponds resulting in pitch sharpening. This is called intonation to two degenerate modes of a linear system spearating by anomaly [1,2]. adding non-diagonal elements to the system matrix. Pitch flattening with cross-fingering has been tradition- Applying this theory to coupled piano strings, Weinreich ally understood using the lattice tone hole theory [3]. This [7,8] showed that different vibration modes yield charac- theory explains, for example, the fact that pitch is lowered teristic piano tone decaying in two steps. Speech spectra as more holes are closed and the reason why pitch have zeros in a high frequency region (4 to 5 kHz) caused flattening is greater in the second register than in the first. by a pair of branches at the bottom of the pharynx acting On the other hand, intonation anomaly or pitch sharpening as silencers. Frequencies of these zeros slightly deviate has been scarcely explained except for pointing out the from those of independent branches. This deviation is possibility for the open hole to act as a register hole [4]. explained by a model of coupled resonators [9]. The same This paper proposes understanding these pitch bending theory is applied to the flute played with cross-fingering in phenomena — both flattening and sharpening — in a uni- this paper. fied manner with a model of coupled mechanical oscil- Incidentally, Yoshikawa and Kajiwara [1,2] examined lators. pitch bending in a shakuhachi played with cross-fingering Coupled oscillating systems have been extensively experimentally in detail. Although this study has thrown studied in the past. One chapter of Fletcher and Rossing new light on this problem regarding cross-fingering, the [5] is devoted to analyzing coupled pendulums, masses, idea of ‘mode’ (or spectrum) switching presented there seems not successful in explaining intonation anomaly. e-mail: [email protected] Various resonance modes found in their experiment are

14 S. ADACHI: FLUTE WITH ONE TONE HOLE

(a) model of a ﬂute

(b) upper bore

Fig. 1 (a) A simplified model of a flute with one open tone hole, definitions of (b) the upper bore and of (c) the lower bore. classified in upper-, lower- and whole-bore ‘modes’. These definitions are, however, not clearly given. In their theory, Fig. 2 Admittance spectra of a flute with one open tone two ‘modes’ are sometimes switched with each other as the hole are plotted in blue thick lines as the lower bore instrument is perturbed by fingering, by changing the length lL is increased from 100 to 375 mm while the length of the instrument or by shifting tone holes. If the upper bore length lU is fixed to 379 mm. Spectra of the ‘modes’ are switched, resonance frequency of each ‘mode’ upper bore and of the lower bore are also plotted in green and red thin lines, respectively, to help under- should inevitably jump. This departs from the basic standing how resonance of this flute is generated. The property of the conventional resonance mode whose plots are shifted vertically for better visibility. frequency changes continuously during perturbation. ‘Mode’ switching may be confused with transition of the state among the resonance modes in which the instrument Table 1 Resonance frequencies fn of the simplified sounds. flute below 2,000 Hz when the lower bore length lL is changed from 100 to 375 mm. 2. ADMITTANCE CALCULATION BY NEDERVEEN lL [mm] f1 f2 f3 f4 f5 f6 f7 f8 [Hz] 100 436 865 1,246 1,464 1,798 By following Nederveen [10], pitch bending with 125 435 858 1,136 1,361 1,762 cross-fingering is first demonstrated by calculating input 150 434 843 1,014 1,330 1,729 admittance of a simplified flute with one open tone hole. 175 434 808 931 1,313 1,664 1,862 This flute is depicted in Fig. 1(a). The input end of this 200 433 746 896 1,297 1,549 1,792 225 432 682 883 1,271 1,437 1,760 instrument is on the left. The bore from the input end to the 250 431 627 876 1,217 1,365 1,729 1,939 open hole is called upper bore and that from the hole to the 275 430 580 871 1,142 1,334 1,676 1,839 output end is called lower bore. The upper bore length lU is 300 428 539 865 1,069 1,317 1,591 1,787 325 425 504 858 1,005 1,303 1,502 1,758 fixed to 379 mm, while the lower bore length lL is increased 350 419 476 846 951 1,283 1,425 1,729 1,899 from 100 to 375 mm with 25 mm step. The bore radius is 375 408 457 825 913 1,250 1,369 1,683 1,825 assumed to be 9.5 mm, the tone hole radius to be 9.5 mm and its acoustical length te to be 22 mm. Mouth impedance at the input end is disregarded here after Nederveen. The blue thick lines in Fig. 2 show calculated input embouchure hole induced by an air-jet is proportional to admittance of the flute. Each peak of the admittance input admittance of the instrument. For the flute to sustain represents a resonance mode of the entire flute. This figure sound by overcoming radiation and other energy dissipa- is essentially the same as Fig. A6.3 in [10], but lL is more tion, this volume flow should be supported by large elongated in our calculation. In the figure, admittances of admittance. The flute therefore operates at admittance the upper and lower bore are additionally shown in green peaks, or sound is generated at a frequency close to one of and red thin lines. These are calculated for the parts of the peak frequencies. Although sound frequency can be varied bore shown in Figs. 1(b) and 1(c). The entire bore is cut in a certain degree by jet velocity, length and embouchure and closed just below and above the open hole to calculate control, we approximate it by an admittance peak fre- upper and lower bore resonances. The calculation model is quency, i.e., resonance frequency, for simplicity in this summarized in Appendix A. paper. According to a sound production theory (e.g., chap- By keeping this theory in mind, let us see the ters 16.3 and 16.4 of [5]), acoustic volume flow through the admittance spectra in detail by referring to the resonance

15 Acoust. Sci. & Tech. 38, 1 (2017)

(a) lL = 150 mm (b) lL =175mm (c) lL = 200 mm

Fig. 3 Standing-wave pressure patterns of the lowest three modes.

frequencies listed in Table 1. When lL ¼ 100 mm, the first two resonance frequencies of the flute are 436 and 865 Hz, which align almost harmonically and agree with those of the upper bore. These are pitches that can be played in the first and second registers. In the third register, or in a frequency region three times higher than f1, no harmonically aligned peaks exist. Instead, two peaks with smaller Fig. 4 A part of a flute with one open tone hole modeled with springs and masses. magnitude appear. On the other hand, the third resonance peak of the upper bore (green) and the first peak of the lower bore (red) stay there. As these two frequencies are 225 mm corresponding pitch flattening. At lL ¼ 250 mm, close, it is conjectured that a coupling is made between the the fourth peak becomes smaller than the fifth, which upper and lower bores, which may cause the two small would be excited more likely and intonation anomaly admittance peaks of the flute. Such two peaks appear in all would occur. When lL finally reaches 375 mm that is nearly the registers, e.g., at lL ¼ 175 mm in the second register, the same as lU, four pairs of double peaks are observed in and at lL ¼ 375 mm in all the registers, when a lower bore all the registers. resonance peak crosses an upper bore resonance peak. In the three cases of lL ¼ 150, 175 and 200 mm, Apparently the first resonance frequency f1 does not standing-wave pressure patterns of the three lowest modes change very much at first when lL is increased from are drawn in Fig. 3. The f1-mode shows a typical pattern of 100 mm. As lL is more increased, the second resonance the lowest mode of a cylindrical resonator with one open frequency f2 approaches to f1 with increasing magnitude. hole in these cases. In the f2-mode, pressure oscillates in- The f1 is then a little bit lowered as if these two peaks repel phase on both sides of the open tone hole at x ¼ 379 mm, each other. These two frequencies thus make a pair. In the whereas in the f3-mode, a pressure node appears in the second register, when lL is increased from 125 to 175 mm, vicinity of the tone hole and pressure oscillates anti-phase the third peak is shifted downward with increasing on both sides of the node. This difference in the standing- magnitude and f2 is gradually lowered. The f2 and f3 thus wave patterns characterizes the f2- and f3-modes making a make a repelling pair in this case. As lL is more increased, pair when pitch bending occurs. The same characteristic the second peak then suddenly becomes smaller and f2 is difference can be seen in all the pairs generated in all the rapidly lowered, while the third peak becomes larger and registers. stays in the second register. It is therefore estimated that 3. MECHANICAL MODEL AND RESONANCE the second peak would be excited (i.e., the flute could be FREQUENCIES played at this frequency.) for lL ¼ 100, 125 and 150 mm and that the third peak would be excited for lL ¼ 200 mm. To investigate a mode pair concerning pitch bending, a For lL ¼ 175 mm, both the second and third peaks may be mechanical model composed of the upper and lower bores excited. Namely, the sound frequency in the second interacting with each other through an open tone hole is register is gradually decreased in order of 865, 858, 843 considered (Fig. 4). We examine how the resonance modes and 808 Hz as lL is increased. These correspond to pitch of this coupled system vary as the resonance frequency of flattening or the normal effect of cross-fingering. When lL the lower bore is changed while that of the upper bore is becomes 200 mm, the sound frequency is suddenly increas- fixed. ed to 896 Hz. This corresponds to intonation anomaly. In the model, we assume that one resonance mode Pitch flattening can also occur when lL is changed from 325 exists in the upper bore and another in the lower one. Each to 375 mm. In this case, the third and fourth peaks make a pair. The same observation can be made in the third Strictly speaking, these are not standing-wave patterns but those register. The f4 is lowered as lL is increased from 200 to forced by a sinusoidal volume flow at the input.

16 S. ADACHI: FLUTE WITH ONE TONE HOLE mode is modeled with a lumped spring-mass system as in [11]. Loss is disregarded here for simplicity. As the f2 and f3 modes in Fig. 3 show, pressure amplitude is smaller near the open hole; a local minimum appears at the hole in the f2-mode and a node appears near the hole in the f3-mode. This rationalizes treating the neighbor of the open hole where the air is less compressed as mass and the other part as spring. In Fig. 4, one spring-mass system replicates a quarter wave length part of the standing wave closest to the open hole either in the upper (left) or lower (right) bore. Note here that both ends of this mechanical model do not represent the input and output ends of the flute model in Fig. 1. Instead, they are velocity nodes closest to the open hole on both sides. The spring constants of these bores 2 are denoted by kU ¼ k and kL ¼ r k. The masses m are assumed to be the same. An open tone hole can be modeled with another mass M. This corresponds to considering only shunt impedance and neglecting series impedance in a matrix tone hole model. For simplicity, the radius of the hole is assumed to be the same as that of the bore. [If the hole radius differs, this can be taken by redefining mass M as we will see in Eq. (1).] It is also assumed that the junction surrounded by the three masses is filled with an incompressible fluid with a small density. Displacements xU, xL and x from their rest positions are defined as shown in Fig. 4. Fig. 5 Two frequencies ! at which the coupled system The equation of motion of this coupled system is can oscillate as functions of a ratio r ¼ !L=!U (upper) mx€UðtÞ¼kUxUðtÞSp pðtÞ; and amplitude ratios xU=xL at two different frequencies ! as functions of a ratio r ¼ !L=!U (lower). mx€LðtÞ¼kLxLðtÞSp pðtÞ; ð1Þ

Mx€ðtÞ¼Sp pðtÞ; where pðtÞ is pressure in the fluid and Sp is the cross- higher and lower than !U, respectively. The amplitude sectional area of the bore. Because of fluid incompressi- ratio is then xU=xL ¼1. This implies that the upper and bility x ¼ xU þ xL, xðtÞ can be eliminated from Eq. (1). lower bores oscillate anti-phase at !þ, and that these When the system oscillates sinusoidally at frequency !, the oscillate in-phase at !. In other words, the tone hole mass eigenvalue problem of this system becomes M does not move at !þ, while two masses of the upper and 2 lower bores show push-pull motion. As a result, both the ! 1 xU 10 xU ¼ ð2Þ spring-mass systems in the upper and lower bores feel mass !2 1 x 0 r2 x U L L m smaller than m þ M and therefore oscillate faster. At ! , pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with !U ¼ kU=ðm þ MÞ and ¼ M=ðm þ MÞ. The !U two masses in the upper and lower bores show symmetric and !L ¼ r!U are frequencies at which the air columns in motion. Velocity of M then becomes twice as large as these the upper and lower bores oscillate independently. The masses. This means that the lumped air in the tone hole takes a role of a coupling parameter. By solving Eq. (2), works as 2M effectively and the system oscillates slower. we can find two eigen frequencies and oscillation ampli- Figure 5 shows how ! and xU=xL vary when r ¼ tude ratio for each frequency: !L=!U is changed from 0.6 to 1.4. The is set to 0.25 here qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi as an example. As is increased, the deviations of ! from 2 2 2 2 2 2 2 2 !U þ !L ð!U !LÞ þ 4 !U!L ! and ! become greater. Conversely as approaches !2 ¼ ; ð3Þ U L 2 2ð1 Þ zero, ! are converged to !U and !L. We first consi- 2 der the case where !L is much larger than !U. The xU ! ¼ 2 2 : ð4Þ coupledpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi system has two resonance frequencies !þ xL ! !U 2 2 2 2 ð!L þ !UÞ=ð1 Þ and ! !U as found in the In a special case of ! ¼ ! , the two eigen frequencies frequency plot. The amplitude plot shows that the upper pffiffiffiffiffiffiffiffi L U pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi become !þ ¼ k=m and ! ¼ k=ðm þ 2MÞ, which are and lower bores oscillate anti-phase at !þ and that the

17 Acoust. Sci. & Tech. 38, 1 (2017) upper bore amplitude xU is smaller than the lower bore amplitude xL. In actual playing, the !þ-mode can not be excited. For self-excitation, a positive feedback between aerodynamic motion of the jet and sound in the embouchure hole is indispensable. In the !þ-mode, oscillation amplitude in the upper bore and thus that in the embouchure hole are too small to excite air reed oscillation. Instead, the !-mode is excited because the amplitude of xU is much larger than that of xL. This Fig. 6 A mechanical model of the recorder with one excitation corresponds to sounding with normal fingering. open tone hole that explains pitch bending (both Namely, the coupled system oscillates at a frequency very flattening and sharpening) in the second register. The upper bore has two resonance frequencies !U1 and !U2 close to !U without much influence of the lower bore where !U1 ¼ r1!U2 is assumed. Frequency of the resonance. lower bore is !L ¼ r!U2 continuously changed with a When !L approaches !U, the system has !þ that is ratio r. higher than !L and ! that is lower than !U. Due to the resonance in the lower bore, ! is considerably lower than !U. This corresponds to the normal effect of cross- fingering or pitch flattening. When !L becomes much closer to !U, excitation in the !þ-mode would become possible because amplitudes both in the upper and lower bores are comparable. This excitation corresponds to intonation anomaly leading to pitch sharpening. The amplitude ratio xU=xL is then negative and the two bores oscillate anti-phase. A large volume flow would be observed within the junction between the upper and lower bores, not through the tone hole. Namely, a pressure node occurs near the open tone hole. When !L is lower than !U, jxU=xLj becomes smaller than one in the !-mode. This means that excitation in this mode becomes difficult. Excitation in the !þ-mode yield- ing intonation anomaly then becomes easy. 4. RESONANCE FREQUENCIES OF RECORDER CROSS-FINGERINGS Although pitch bending on a flute with one open hole has been qualitatively explained with the model presented in the previous section, a more elaborated model having two degrees of freedom in the upper bore (and thus three in total) is needed to quantitatively predict resonance frequencies of the recorder (Fig. 6). The upper bore has two resonance frequencies !U1 and !U2. The ratio r1 ¼ Fig. 7 Three eigen frequencies !1, !2 and !3 (upper) !U1=!U2 is a model parameter. and amplitude ratios of xU1=xL and xU2=xL in eigenm- The eigenvalue problem of this model is odes 1, 2, and 3 (lower) for ¼ 0:25 and r1 ¼ 0:55.In 0 10 1 0 10 1 the upper figure, calculated resonance frequencies of 1 x r2 00 x 2 U1 1 U1 an alt recorder (alt-a) played with a normal fingering ! B CB C B CB C and three cross-fingerings @ 1 A@ xU2 A ¼ @ 010A@ xU2 A; ð5Þ !2 , and are also U2 2 1 xL 00r xL shown with circles, squares and triangles. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where !U2 ¼ k=ðm þ MÞ. Calculated eigen frequencies and oscillation amplitudes are drawn in Fig. 7 as functions of r ¼ !L=!U2, where r1 ¼ 0:55 implying the upper bore eigen frequencies are matched to the resonance frequencies has slightly inharmonic frequencies and ¼ 0:25 are of an actual recorder. Eigen frequency crossing is avoided assumed. These values have been optimized so that the twice near r ¼ 0:55 and r ¼ 1 as seen in the frequency

18 S. ADACHI: FLUTE WITH ONE TONE HOLE plot. These are caused by the lower bore interacting with the first and second resonance modes of the upper bore, respectively. The plot of !2 near r ¼ 1 is a little bit different from that of ! in the simple two-degrees-of- freedom model. When r > 1:09, !2 becomes higher than !U2. As seen in the amplitude plot, xU1 in the second mode oscillating in anti-phase with xL has amplitude comparable with or larger than that of xL. Because of this effect, mass M oscillates with a smaller amplitude and then !2 tends higher. When r < 0:92, !2 becomes higher than !L. This implies that in this frequency range, xU1 starts interacting with xL. From the amplitude plot, we can roughly estimate which eigenmode is excited depending on r ¼ !L=!U2. Fig. 8 Three eigen frequencies !1, !2 and !3 (upper) When r > 1, the amplitudes of xU1 in the first mode and for ¼ 0:25 and r1 ¼ 0:50. Sound frequencies of a xU2 in the second mode are large. Both the first and second recorder [Aulos 509B(E)] played with a normal finger- modes are therefore likely to oscillate. When r decreases ing and three cross-fingerings , and are also from 1 and approaches to 0.55, the first mode is still likely shown with circles, squares and triangles. to oscillate because of large jxU1j in the first mode. The amplitude of xU2 in the second mode becomes smaller than that of xL. Instead, jxU2j in the third mode then becomes retrieved in a computer. The sound frequencies have been larger than jxLj. This implies that the second mode calculated using FFT. The results are again compared with excitation becomes harder and the third mode excitation the eigen frequency curves in Fig. 8. The frequencies are becomes easier. In the case of r < 0:55, jxU1j in the second plotted in circles, squares and triangles with note names. mode and jxU2j in the third mode are large. This means that The names not listed in the usual fingering chart are put in the second and third modes are likely to oscillate. parentheses. In these plots, all the reference frequencies By applying the transmission-line matrix model pre- !U1, !U2 and !L are borrowed from those calculated for sented in Appendix A to an actual geometry of an alt the alt-a recorder geometry as that of this played recorder is recorder named alt-a in [12], we have calculated resonance not known. Choice of ¼ 0:25 and r1 ¼ 0:50 provides a frequencies (i.e., peak admittance frequencies) for a normal reasonable fit to the sound frequencies. In this experiment, fingering and three cross-fingerings mode 3 could not be excited with fingering . , and where indicates F6 with fingering can only be played just the thumb hole is ‘pinched’ for playing in the second after playing E6 with . B5 with register. They are normally for notes D6, C#6, E6 and D#6 can be continuously played when the amount of in order. The first, second and third resonance frequencies air is a little bit decreased from that needed for playing E6. of these fingerings are marked with circles, squares and A5 with can also be played with moderate triangles in Fig. 7. In these plots, !U1 and !U2 have been amount of air smaller than for playing D#6, which is much calculated for the upper bore with fingering closed lower than the pitch the model predicts. at the position just below the open hole. The ratio r ¼ 1 5. CONCLUSIONS !U1=!U2 has been confirmed to be 0.55 for the geometry of this recorder. The !L=!U2 has been calculated to be 0.98, The mechanism of pitch flattening and sharpening 0.86, 0.72 and 0.60 for the lower bore closed at the position caused by cross-fingering has been clarified. When the just above the first open hole and fingered with , instrument is played with cross-fingering, a coupled system , and , respectively. The calcu- of the upper and lower bores interacting through the open lated resonance frequencies are in good agreement with the tone hole is formed. This system has two resonance eigen frequencies given in our mechanical model. For frequencies !: The !þ is higher and the other ! is lower completeness, calculated admittance spectra are shown in than resonance frequencies !U and !L of the upper and Appendix B. lower bores excited independently. Pitch flattening is Finally, another alt recorder [Aulos 509B(E)] has caused by excitation of the !-mode. This happens when been played in an ordinary room by the author with !L ’ !U and the admittance peak of the !-mode is these fingerings. Long tones have been recorded with a higher than or as high as that of the !þ-mode. Pitch Behringer C-2 microphone. The sound data have been sharpening occurs if !þ-mode is excited. This happens sampled at 48 kHz rate with a UMC202 audio interface and when !L / !U and the peak of the !þ-mode becomes

19 Acoust. Sci. & Tech. 38, 1 (2017) sufficiently high. By using a more elaborated model with [23] G. B. Thurston and J. K. Wood, ‘‘End corrections for a circular three degrees of freedom, pitch flattening and sharpening of tube,’’ J. Acoust. Soc. Am., 25, 861–863 (1953). the recorder played with cross-fingering in the second register have been reasonably explained. APPENDIX A: TRANSMISSION-LINE MATRIX MODEL OF A WOODWIND REFERENCES INSTRUMENT [1] S. Yoshikawa and K. Kajiwara, ‘‘Acoustics of cross fingerings in the shakuhachi,’’ Proc. Forum Acusticum Krako´w 2014 A.1. Sound Propagation in the Bore (2014). A bore with an arbitrary shape can be approximated [2] S. Yoshikawa and K. Kajiwara, ‘‘Cross fingerings and associated intonation anomaly in the shakuhachi,’’ Acoust. with a series of truncated cones. In a transmission-line Sci. & Tech., 36, 314–325 (2015). matrix model [13], pressure and volume flow ðp; UÞ at one [3] A. H. Benade, Fundamentals of Musical Acoustics (Oxford end of a truncated cone can be calculated by multiplying University Press, New York, 1976), Chap. 21.4D. ðp; UÞ at the other end by a 2 2 matrix. Sound [4] J. Wolfe and J. Smith, ‘‘Cutoff frequencies and cross fingerings in baroque, classical and modern flutes,’’ J. Acoust. Soc. Am., propagation loss due to friction and to heat exchange on 114, 2263–2272 (2003). the wall is considered. The parameters of this model are air [5] N. H. Fletcher and T. D. Rossing, The Physics of Musical constants such as sound velocity, density, viscosity, ther- Instruments, 2nd ed. (Springer-Verlag, New York, 1998). mal conductivity, specific heat at constant pressure and [6] C. E. Gough, ‘‘The theory of string resonances on musical specific heat ratio. These are set in this paper to c ¼ instruments,’’ Acustica, 49, 124–141 (1981). 3 [7] G. Weinreich, ‘‘Coupled piano strings,’’ J. Acoust. Soc. Am., 346:0 m/s (for temperature of 25 C), ¼ 1:2 kg/m , ¼ 5 3 62, 1474–1484 (1977). 1:8 10 kg/(ms), ¼ 6:2 10 cal/(msK), Cp ¼ [8] G. Weinreich, ‘‘The coupled motions of piano strings,’’ Sci. 240 cal/(kgK) and ¼ 1:4. Am., 240, 118–127 (1979). We assume that radiation impedance Z at the output [9] H. Takemoto, S. Adachi, P. Mokhtari and T. Kitamura, r ‘‘Acoustic interaction between the right and left piriform fossae end is that of a baffled circular piston of the same diameter in generating spectral dips,’’ J. Acoust. Soc. Am., 134, 2955– [14]. Let the instrument bore driven by a piston vibrating at 2964 (2013). the input end. Pressure and volume flow ðpout; pout=ZrÞ are [10] C. J. Nederveen, Acoustical Aspects of Woodwind Instruments, then generated at the output. The ðp; UÞ at any sections in 2nd ed. (Northern Illinois University Press, DeKalb, IL, 1998). [11] A. Barjau and V. Gibiat, ‘‘Study of woodwind-like systems the instrument can successively be calculated until through nonlinear differential equations. Part II. Real geom- ðpin; UinÞ at the input end. Input impedance of this etry,’’ J. Acoust. Soc. Am., 102, 3032–3037 (1997). instrument is then Zin ¼ pin=Uin. If mouth impedance Zm [12] J. Bouterse, ‘‘Alto recorders by Bressan: Bore profiles,’’ can not be disregarded, pressure exerted on the other side FoMRHI Commun. Q., 118, Comm 1929 (2011). of the piston, which amounts to Z U , should also be [13] R. Causse´, J. Kergomard and X. Lurton, ‘‘Input impedance of m in brass musical instruments — Comparison between experiment considered. Impedance felt by the piston is therefore and numerical models,’’ J. Acoust. Soc. Am., 75, 241–254 Zin þ Zm. In the low frequency limit, Zm becomes (1984). i!L=Sp where L is the end correctionffiffiffiffiffiffi and Sp is the [14] P. M. Morse, Vibration and Sound, 2nd ed. Sect. 28 (Acous- p pipe area. In [15], L 0:73Sp= Sm is given, where tical Society of America, New York, reprinted 1976). 2 [15] F. Ingerslev and W. Frobenius, ‘‘Some measurements of the mouth area Sm ¼ 12 4 mm is assumed for an alt end-corrections and acoustic spectra of cylindrical open flue recorder in this paper. organ pipes,’’ Trans. Dan. Acad. Tech. Sci., 1, 1–44 (1947). [16] A. Lefebvre and G. P. Scavone, ‘‘Characterization of wood- A.2. Two-port Matrix Model of a Tone Hole wind instrument toneholes with the finite element method,’’ J. Tone hole dimensions are shown in Fig. A.1. Lefebvre Acoust. Soc. Am., 131, 3153–3163 (2012). [17] C. Nederveen and D. W. van Wulfften Palthe, ‘‘Resonance and Scavone [16] present a matrix of a tone hole relating frequency of a gas in a tube with a short closed side-tube,’’ ðp; UÞ before and after passing a tone hole, Acustica, 13, 65–70 (1963). [18] D. H. Keefe, ‘‘Theory of the single woodwind tone hole,’’ J. Acoust. Soc. Am., 72, 676–687 (1982). [19] D. H. Keefe, ‘‘Experiments on the single woodwind tone hole,’’ J. Acoust. Soc. Am., 72, 688–699 (1982). [20] D. H. Keefe, ‘‘Woodwind air column models,’’ J. Acoust. Soc. Am., 88, 35–51 (1990). [21] J.-P. Dalmont, C. J. Nederveen, V. Dubos, S. Olivier, V. Me´serette and E. te Sligte, ‘‘Experimental determination of the equivalent circuit of an open side hole: Linear and non linear (a) An open tone hole (b) A tone hole ‘pinched’ by behaviour,’’ Acta Acust. united Ac., 88, 567–575 (2002). the thumb. [22] D. H. Lyons, ‘‘Resonance frequencies of the recorder (English flute),’’ J. Acoust. Soc. Am., 70, 1239–1247 (1981). Fig. A.1 Dimensions of the tone hole.

20 S. ADACHI: FLUTE WITH ONE TONE HOLE ! ! ! ! ! 0:8 p1 1 Za=2 101 Za=2 p2 tr ¼½0:822 0:47fb=ða þ tÞg b ðA:6Þ ¼ U1 01 1=Zh 1 01 U2 using radiation impedance of a partially flanged tube in the ! ! low frequency limit. In the case of a ‘pinched’ hole shown 1 Za p2 ðA:1Þ in Fig. A.1(b), Eq. (A·6) can be modified to 1=Zh 1 U2 0:8 2 tr ¼½0:822 0:47fb^=ða þ tÞg b =b^; ðA:7Þ where Zh and Za are shunt and series impedances in a circuit analog. Zh is nothing but input impedance of the because the hole radius is narrowed to b^ but another factor 2 2 tone hole as seen from the instrument bore. The Za of b =b^ comes due to volume flow conservation. Accord- represents an effect of the stream lines in the bore ing to Lyons [22], the ‘pinched’ hole is modeled as penetrating toward the hole. As a result, the bore length covering the hole with tape that has a hole of 2.3 mm is a little bit shorten [17] and ta in Eq. (A·15) is thus diameter drilled in it. Therefore b^ ¼ 1:15 mm is assumed negative. This effect is, however, very small. As jZa=Zhj here. 1 is generally held, the approximation in Eq. (A·1) is taken. In [19], the loss factor has three terms corresponding This equation is used for calculating admittance in this to losses of radiation, passage through the hole and paper. viscosity at both ends of the hole: For further simplicity, the effect of series impedance Za 2 ¼ 0:25ðkbÞ þ t þ kdv lnð2b=rcÞ=4; ðA:8Þ can be incorporated into a transmission matrix of the main bore by reducing the length of the main bore by an amount where 2 sffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffi of ða=bÞ ta. In this case, Za can be regarded as zero in 2 ! 2 2 Eq. (A·1). For an open tone hole, Zh becomes i!te=ðb Þ ¼ fdv þð 1Þdtg; dv ¼ ; dt ¼ : 2bc ! Cp! in the low frequency limit where te is the acoustical length of the hole. This implies that an open tone hole can be Here, dv and dt are viscous and thermal boundary thick- 2 modeled with mass M ¼ teb . This is the basis of the ness, respectively. rc is the radius of curvature at either end mechanical models presented in this paper. of the hole, which is assumed to be 0.3 mm in this paper. In the case of the ‘pinched’ hole, the third term of Eq. (A·8) A.3. Shunt Impedance should be increased, because streamlines are considerably The is defined as the ratio of b=a. Let the actual hole bent there. According to Thurston and Wood [23], the height be t. According to Keefe [18–20], we write acoustic impedance of a thin orifice with a radius of r sufficiently larger than d is c v Z ¼ ðikt þ Þ; ðA:2Þ pffiffiffiffiffiffiffiffiffiffiffiffi h 2 e 3 2 b Z ¼ 2!ðto þ fRÞ=ðr Þþi!ðto þ fLÞ=ðr Þ; ðA:9Þ where k is wave number. The imaginary and real parts where to is the thickness of the orifice assumed to be zero in correspond to the end correction and to loss due to the hole, our case, and where fR is the end correction which is the respectively. order of r,orfR r. (See Eqs. (2) and (4) of [23].) The Dalmont et al. [21] express te with the sum of t and real part of Eq. (A·9) can be rewritten as three different corrections: c b2 ReðZÞ¼ kd ; ðA:10Þ t ¼ t þ t þ t þ t : ð : Þ 2 2 v e m i r A 3 b b^

The tm is a length correction accounted for a small volume which is proportional to kdv similarly to the third term of between the main bore and the tone hole, which is Eq. (A·8). Interpreting this to a correction to , we ﬁnd calculated [10] as b2 kd ðA:11Þ ¼ ð þ 3Þ ð : Þ 2 v tm b 1 0:207 =8: A 4 b^ One of the main results of a recent FEM analysis [16] is an should be added to Eq. (A·8). In addition, when b^ ¼ b, estimation of the inner end correction toward the main bore should be equal to Eq. (A·8), we therefore decided to ti, which is rewrite the third term of Eq. (A·8) to be 2 3 ti ¼ð0:822 0:095 1:566 þ 2:138 b2 : 2 kdv lnð2b=rcÞ=4 ðA 12Þ 1:6404 þ 0:5025Þb: ðA:5Þ b^

In [16], the radiation end correction toward the ambient tr by assuming lnð2 3mm=0:3mmÞ=4 ¼ 0:75 is the same is also modeled as order of 1.0. In short, instead of Eq. (A·8),

21 Acoust. Sci. & Tech. 38, 1 (2017)

b2 2 : ¼ 0:25ðkbÞ þ t þ 2 kdv lnð2b=rcÞ=4 ðA 13Þ b^ is used for a ‘pinched’ hole.

(a) fingering . (b) fingering . A.4. Series Impedance fn : 700, 1158, 1381, 1744 Hz.fn : 684, 1065, 1376, 1684 Hz. By following [16], series impedance is just c Z ¼ i kt ; ðA:14Þ a b2 a where the effective length ta is 2 : (c) fingering . (d) fingering . ta ¼b f0:35 0:06 tanhð2:7t=bÞg: ðA 15Þ fn : 664, 958, 1347, 1666 Hz.fn : 634, 857, 1301, 1659 Hz.

APPENDIX B: CALCULATED Fig. B.1 Admittance spectra of alt-a played with a RESONANCE FREQUENCIES OF normal and three cross-fingerings. ALT-A RECORDER Admittance spectra of alt-a recorder calculated with the method presented in Appendix A for various fingerings are plotted in Fig. B.1. Those of the upper and lower bores are plotted in Figs. B.2 and B.3. The resonance frequencies fn are denoted in the captions. Fig. B.2 Admittance spectrum of the upper bore of alt-a fingered with . fn : 674, 1,231, 1,920 Hz.

(a) ﬁngering . (b) ﬁngering . fn : 1212, 1646 Hz.fn : 1058, 1570 Hz.

Fig. B.3 Admittance spectra of the lower bore of alt-a with various ﬁngerings.