The acoustics of mandolins

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1 PAPER The acoustics of mandolins David Cohen and Thomas D. Rossing Physics Department, Northern Illinois University, DeKalb, IL 60115, USA ( Received16 July 2001, Acceptedfor publication 16 May 2002 ) Abstract: Using electronic TV holography, we have studied the vibrational modes of four mandolins and a mandola. The lowest (0,0) modes may appear either as a triplet (as in a guitar) or as a doublet. The modal frequencies correlate well with the frequency response curves. Sound spectra indicate that sound radiation is quite uniform over the 0 5 khz range with some rolloff above 2.5 khz. Keywords: Mandolin, Normal modes, TV holography PACS number: Gh, Dx 1. INTRODUCTION The mandolin is a plucked string instrument whose origins appear to go back to the medieval gittern (also known as guittarra, chitarra, and guitaire in various European countries). The modern mandolin is descended from two instruments which developed during the 18th century in Italy. The first was the mandola or mandolino, which carried six courses of two strings tuned in 3rds and 4ths and is sometimes referred to as a Milanese mandolin. The second was the mandoline or Neapolitan mandolin, which had four courses of two strings tuned like a violin. The modern mandolin is tuned like the latter; the six-course instrument fell into disuse in the late 19th century. The history of the mandolin is discussed by Tyler and Sparks [1,2], by Gill and Campbell [3], and in an earlier paper by the authors [4]. The mandola might be called an alto mandolin. It has four double courses of strings tuned to C 3,G 3,D 4, and A 4 (as in a viola) compared to the G 3,D 4,A 4, and E 5 on the mandolin. The body parts are about 15% larger than those of the mandolin. Mandolins have become popular in many parts of the world. In the 1920s, a neo-baroque style of mandolin music developed in Germany, and the instrument also became popular in Japan. The mandolin was brought to America by Italian immigrants about the beginning of the 20th century, and it became especially popular in bluegrass music. In 1995 the Japan Mandolin Union had over 10,000 members and there were over 500 mandolin orchestras in Permanent address: J. Sargeant Reynolds Community College, Richmond, VA Germany [2]. In an earlier paper [4] we described the normal modes of two Gibson F-type mandolins, one with an elliptical sound hole and one with f-holes. In this paper we compare the normal modes of vibration and other acoustical properties of two mandolins and a mandola with c-holes with the previous results on the Gibson mandolins. 2. THE INSTRUMENTS The five instruments, all constructed by the first author, are shown in Fig. 1. Mandolin 1 has a carved redwood top and a maple back with a symmetrical body, mandolin 2 has a redwood top and a walnut back with a scroll-shaped cavity added to the body. Mandolins 3 and 4 are the Gibson F-type mandolins constructed from carved spruce plates, carved maple backs, and other parts supplied by the Gibson company. Bracing patterns in mandolins 1 and 2 are shown in Fig NORMAL MODES OF VIBRATION A normal mode of vibration represents the motion of a linear system at a normal frequency (eigenfrequency). It should be possible to excite a normal mode of vibration at any point in a musical instrument that is not a node and to observe motion at any other point that is not a node. A normal mode is a characteristic only of the structure itself, independent of the way it is excited or observed. Normal mode shapes are unique for a structure, whereas the deflection of a structure at a particular frequency, called an operating deflection shape (ODS) may result from the excitation of more than one normal mode. Normal mode testing has traditionally been done using sinusoidal 1

Roberts and Rossing [5]. Experimentally, all modal testing is done by measuring operating deflection shapes and then interpreting them in a specific manner to define mode shapes [6].

In practice, however, if the mode overlap is small, the single-frequency ODSs provide a good approximation to the normal modes.

2 Roberts and Rossing [5]. Experimentally, all modal testing is done by measuring operating deflection shapes and then interpreting them in a specific manner to define mode shapes [6]. Strictly speaking, some type of curve-fitting program should be used to determine the normal modes from the observed ODSs, even when an instrument is excited at a single frequency. In practice, however, if the mode overlap is small, the single-frequency ODSs provide a good approximation to the normal modes. Ways to deal with overlapping modes include the use of multiple drivers and the use of phase modulation [7]. Fig. 2 Fig. 1 Clockwise from top left: mandolin 1 (redwood top, maple back and sides); mandolin 2 (redwood top, walnut back and sides); mandolin 3 (Gibson f-hole); mandolin 4 (Gibson oval hole); mandola (redwood top, walnut back and sides). Bracing patterns in mandolin 1 (left) and mandolin 2 (right). excitation, either mechanical or acoustical. Detection of motion may be accomplished by attaching small accelerometers, although optical methods are less obtrusive. For further discussion of normal modes of vibration, see 4. EXPERIMENTAL METHOD 4.1. Modal Analysis Using Holographic Interferometry Holographic interferometry offers by far the best spatial resolution of operating deflection shapes (and hence of normal modes) [8]. Recording holograms on film tends to be rather time consuming, however. Electronic TV holography, on the other hand, offers one the opportunity to observe vibrational motion in real time and a fast, convenient way to record operating deflection shapes and to determine the normal modes [9]. The TV holography system used in these experiments is similar to that previously described [3]. Reflected light from the mandolin reaches a CCD camera, along with the reference beam, to produce the holographic image. A phase-stepping mirror, driven by a piezoelectric crystal, varies the phase of the reference beam in four steps so that the holographic interferogram can be computed. The holographic image is displayed on a TV screen. The sinusoidal exciting force was applied by attaching a small NdFeB magnet to the bridge or the body of the mandolin and applying a sinusoidally varying magnetic field by means of a coil driven by an audio amplifier. ODSs were observed in the top and back plates both when a force was applied directly and when the force was applied to the other plate Frequency Response A small magnet was placed on the treble side of the bridge and a coil driven by an audio amplifier was placed next to it. Wideband noise of constant amplitude was fed to the amplifier so that a wideband force would be applied to the bridge. A small accelerometer was mounted on the mandolin next to the bridge, and the amplified signal from the accelerometer was fed to the fast-fourier-transform (FFT) analyzer. A graph of accelerance vs frequency was thus created Determination of Air Cavity Modes In order to determine the modes of the enclosed air 2

D. COHEN and T. D. ROSSING: ACOUSTICS OF MANDOLINS cavity, the instruments were imbedded in sand to immobilize the plates.

An electret microphone was guided by a stiff wire to probe the sound field inside the cavity. 4.

immobilized and the sound holes closed. This was done by placing each instrument top up and then back up in sand. The exposed plate was driven sinusoidally. 4.5.

3 D. COHEN and T. D. ROSSING: ACOUSTICS OF MANDOLINS cavity, the instruments were imbedded in sand to immobilize the plates. A horn compression driver was fitted with a stopper and a rubber tube, passing through the sound hole, transmitted the sinusoidally varying sound pressure to selected positions inside the air cavity. An electret microphone was guided by a stiff wire to probe the sound field inside the cavity Determining Plate Frequencies In order to determine plate coupling in the fundamental mode, it was necessary to determine the vibrational frequency of each plate when the other plate was immobilized and the sound holes closed. This was done by placing each instrument top up and then back up in sand. The exposed plate was driven sinusoidally Sound Spectra Sound spectra were recorded with an Ono Sokki 360 fast-fourier transform (FFT) analyzer and a microphone placed about 0.5 m from the instrument. Spectra were averaged by plucking a wide range of notes on all strings and also by exciting the instruments mechanically on the bridge and off the bridge with broadband noise [10]. 5. RESULTS 5.1. Normal Modes Operating deflection shapes (ODSs) were recorded at a large number of frequencies using electronic TV holography. Included were frequencies at which the mandolins appeared to radiate strongly and those at which the entire body or some particular part of it appeared to vibrate at large amplitude. From these we attempted to deduce the most important normal modes of vibration. We focused especially on normal modes that appeared to have symmetry since the mandolin is a nearly symmetric instrument. The lowest radiating mode in all the instruments is one in which the top and back plate move in opposite directions so that the instrument breathes through its sound holes. This so-called (0,0) a mode occurred at 234 Hz in the mandola and ranged in frequency from 209 to 313 Hz in the four mandolins. A second mode in which the two plates move outward together, which we call the (0,0) b mode occurred anywhere from 134 to 273 Hz higher. In this mode air moves out through the sound holes as the plates move outward. The (0,0) mode pair in mandolin 1 is shown in Fig. 3. Mandolins 2 and 4 had a third (0,0) mode in between these two modes in which the top and back plates moved in the same direction, similar to the (0,0) mode of middle frequency in a guitar [11,12]. The next mode in all the instruments was characterized by a longitudinal nodal line in the top plate, and we describe this as (1,0) motion in the top plate. In mandolins Fig. 3 (0,0) mode pair in mandolin 1. Top: top and back plates at 284 Hz; Bottom: top and back plates at 415 Hz. 3 and 4, which we studied earlier [4], the back plate vibrated in a similar way, but this was not the case in the other two mandolins or the mandola. The (1,0) mode was observed at 402 Hz in the mandola top plate and at 451 to 575 Hz in the mandolins. The (1,0) mode in mandolin 2 is shown in Fig. 4. Additional modes with 2 or 3 longitudinal nodal lines were observed in the top plates of most of the instruments. These are shown for the mandola in Fig. 5. The (0,1) mode is characterized by a transverse nodal line near the center. Although this is an important mode in Fig. 4 (1,0) mode in mandolin 2 at 474 Hz. Fig. 5 Modes in the mandola top plate. Left: (2,0) mode at 796 Hz; Right: (3,0) mode at 929 Hz. 3

4 Table 1 Modal frequencies in mandola and four mandolins. Mode Man.1 (603) Man.2 (402) Man.3 (f-hole) Man.4 (oval) Mandola 0,0 284, , 267, , , 398, , 368 1,0451 (top) (595) 402 (top) 0, ? 666? 2,0907 (top) 837 (top) ,0 1,062 (top) 1,076 1,117 1, , A A A 2 1,085 1, f t f b guitars it is not seen so clearly in mandolins. This mode appeared at 558 Hz in the mandola and at 650to 824 Hz in the mandolins. In mandolin 2 at 821 Hz, the top vibrated in a (1,1) configuration while the back motion was (2,0), characterized by two longitudinal nodes. The (1,1) mode with one longitudinal and one transverse nodal line was observed in the back plates of several instruments in the 606 to 760 Hz frequency range but never seen in the top plates. Several modal frequencies are compared in the five instruments in Table 1. Also given are frequencies of the first three air cavity modes in each instrument. A 0 is the Helmholtz mode in which air vibrates in and out of the sound holes. In the A 1 mode, air vibrates longitudinally with a pressure node across the waist. In the A 2 mode air vibrates transversely with a pressure node at the plane of symmetry. Also included in Table 1 are the top and back plate (0,0) mode frequencies f t and f b when the opposite plate is immobilized in sand Frequency Response Function Accelerance level (log of acceleration at constant force amplitude) is plotted vs frequency in Fig. 6. Peak values of accelerance occur at most of the modal frequencies in Table 1. In addition most instruments show a broad maximum around 3,500 Hz. Fig. 6 Frequency response (accelerance at treble side of the bridge). (a) mandolin 1; (b) mandolin 2; (c) mandolin 3; (d) mandolin 4; (e) mandola. Vertical scale is 20dB/division. Peaks corresponding to the (0,0), (1,0), and (3,0) modes are indicated Sound Spectra Sound spectra of mandolin 4 driven with a random noise force applied to the treble and bass sides of the bridge and directly to the body about 5 cm below the treble side of the bridge are shown in Fig. 7(a). The microphone was placed approximately 0.5 m in front of the instrument. Although peaks associated with the (0,0) modes, the (1,0) mode, and other modes in Table 1 are apparent, the sound radiation is fairly well distributed over the entire range of 0 5 khz with some rolloff above 2.5 khz. Sound spectra from the other three mandolins were quite similar. Sound spectra of the mandola, shown in Fig. 7(b), are Fig. 7 Sound spectra (20dB/division). (a) mandolin 4; (b) mandola. 4

5 D. COHEN and T. D. ROSSING: ACOUSTICS OF MANDOLINS quite similar to those of the mandolins. The treble rolloff begins at a lower frequency (about 1.3 khz) than in the mandolins and it is not quite as great. 6. DISCUSSION Normal modes of vibration in the mandolin and mandola in the low-frequency range are somewhat akin to those observed in guitars, as might be expected. In guitars, (0,0) modes of the top and back plates coupling strongly to the Helmholtz resonance lead to three normal modes within the span of about an octave from 100 to 200 Hz [13]. In mandolins 1 and 3 and the mandola, only two (0,0) modes were observed and these most nearly resemble the lowest and the highest (0,0) modes in guitars. The coupling is weaker than in guitars, and the A 0 cavity mode (Helmholtz resonance) is often lower in frequency than the (0,0) modes in the top and back plates. Also f b is considerably larger than f t in all the instruments, indicating that the back plate is much stiffer than the top plate (in most guitars f t and f b are fairly close to each other). Because the stiff back plate vibrates at relatively low amplitude at the (0,0) mode frequencies, one might expect the two mass model to describe the coupling [14] reasonably well. According to this model, f 1 2 þ f 2 2 ¼ f t 2 þ f A0 2. In mandolins 1 and 4, these sums are within 5% of each other; in mandolin 2 they differ by 11% and in mandolin 3, they differ by 22%. In the mandola, they differ by 11%. In all cases (f 1 2 þ f 2 2 Þ > ðf t 2 þ f A0 2 ), which is to be expected since the compliance of the back plate will lower the frequency of the A 0 air cavity mode. If stronger plate coupling in the fundamental mode were desired, it would be well to raise the A 0 air cavity mode (Helmholtz resonance) frequency by enlarging the sound hole(s) and to lower f b by making the back plate less stiff. This should produce an instrument with a more bass sound and greater support for the lowest string, a desirable feature for performers and composers who wish to extend the compass of music played by this instrument. The frequency response (accelerance) curves in Fig. 6 show peaks corresponding to most of the modal frequencies in Table 1. Unlike guitars, the (0,1) modes do not appear to contribute noticeably to the frequency response. The significance of the broad maxima around 3,500 Hz is not understood. The sound spectra showed only small peaks at the modal frequencies. Sound radiation is quite even over the entire 0 5 khz range with some rolloff above 2.5 khz. The reason why a (0,0) mode triplet (similar to what is observed in guitars) appeared in mandolins 2 and 4, whereas the others showed a (0,0) mode doublet is not apparent. Mandolins 2 and 4 had two transverse braces (see Fig. 2) which added a small amount of transverse stiffness. Mandolin makers have used a wide variety of brace patterns. Gibson mandolins generally feature longitudinal braces, whereas such makers as Gilchrist and Smart prefer crossed braces. A systematic study of the effect of different bracing patterns on the acoustics of mandolins would be desirable. Also desirable would be theoretical studies of mandolin body vibrations, perhaps by the application of finite element methods, as has been done in instruments such as the guitar [12]. Although we have only sampled a few different styles of instruments, the acoustical observations that we have made should be of interest to mandolin makers, who are constantly experimenting with methods to improve their instruments. 7. CONCLUSION The normal modes of vibration in mandolins are similar to those observed in guitars. The coupling between top and back plates appears to be less than in guitars. The lowest modes may appear as either a doublet or a triplet. Modal frequencies observed with holographic interferometry correlate well with peaks in the frequency response curves. As we suggested in our earlier paper [4], the historical and current interest in the mandolin as a musical instrument would suggest the desirability of further research on mandolin acoustics. The effect on tone of stronger plate coupling could profitably be studied, as well as the effect of different bracing patterns. The authors thank the Gibson Company for furnishing materials for two of the mandolins studied. REFERENCES [1] J. Tyler and P. Sparks, The Early Mandolin (Clarendon Press, Oxford, 1989). [2] J. Tyler and P. Sparks, Mandolin, in The New Grove Dictionary of Music and Musicians, 2nd ed., S. Sadie, Ed. (Macmillan Publishers Ltd., London, 2001). [3] D. Gill and R. Campbell, Mandolin, in The New Grove Dictionary of Musical Instruments, S. Sadie, Ed. (Macmillan Publishers Ltd., London, 1984). [4] D. Cohen and T. D. Rossing, Normal modes of vibration in two mandolins, Catgut Acoust. Soc. J., 4(2), (2000). [5] M. Roberts and T. D. Rossing, Normal modes of vibration in violins, Catgut Acoust. Soc. J., 3(5), 3 9 (1998). [6] M. H. Richardson, Is it a mode shape, or an operating deflection shape? Sound Vib., 31(1), (1997). [7] F. Engström and T. D. Rossing, Using TV holography with phase modulation to determine the deflection phase in a baritone guitar, Proc. ISMA 98, D. Keefe, T. Rossing and C. Schmid, Eds. (Acoust. Soc. Am., Woodbury, NY, 1998), pp [8] R. L. Powell and K. A. Stetson, Interferometric vibration analysis by wavefront reconstruction, J. Opt. Soc. Am., 55, (1965). [9] E. V. Jansson, N.-E. Molin and H. O. Saldner, On eigenmodes of the violin Electronic holography and admittance measurements, J. Acoust. Soc. Am., 95, (1994). 5

6 [10] T. D. Rossing, Sound radiation from guitars, Am. Lutherie, 16, (1988). [11] T. D. Rossing, Physics of guitars: An introduction, J. Guitar Acoust., 4, (1981). [12] N. H. Fletcher and T. D. Rossing, The Physics of Musical Instruments, 2nd ed. (Springer-Verlag, New York, 1998), Chap. 9. [13] T. D. Rossing and G. Eban, Normal modes of a radiallybraced guitar determined by electronic TV holography, J. Acoust. Soc. Am., 106, (1999). [14] G. Caldersmith, Guitar as a reflex enclosure, J. Acoust. Soc. Am., 63, (1978). 6

Quarterly Progress and Status Report. On the body resonance C3 and its relation to top and back plate stiffness

Dept. for Speech, Music and Hearing Quarterly Progress and Status Report On the body resonance C3 and its relation to top and back plate stiffness Jansson, E. V. and Niewczyk, B. K. and Frydén, L. journal: