Resonance Modes of a Flute with One Open Tone Hole
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Acoust. Sci. & Tech. 38, 1 (2017) #2017 The Acoustical Society of Japan PAPER Resonance modes of a flute with one open tone hole Seiji Adachià 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 !Æ, which are respectively higher and lower than those of the upper and lower bores !U and !L excited independently. The !Æ differ even if !U ¼ !L. The normal effect of cross-fingering, i.e., pitch flattening, corresponds to excitation of the !À-mode, which occurs when !L ’ !U and the admittance peak of the !À-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. Ãe-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 (c) lower 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 harmon- ically 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.