The Acoustics of Mandolins

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Acoust. Sci. & Tech. 24, 1 (2003) 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: 43.75.Gh, 43.40.Dx

Germany [2]. 1. INTRODUCTION In an earlier paper [4] we described the normal modes The mandolin is a plucked string instrument whose of two Gibson F-type mandolins, one with an elliptical origins appear to go back to the medieval gittern (also sound hole and one with f-holes. In this paper we compare known as guittarra, chitarra, and guitaire in various the normal modes of vibration and other acoustical European countries). The modern mandolin is descended properties of two mandolins and a mandola with c-holes from two instruments which developed during the 18th with the previous results on the Gibson mandolins. century in Italy. The first was the mandola or mandolino, which carried six courses of two strings tuned in 3rds and 2. THE INSTRUMENTS 4ths and is sometimes referred to as a Milanese mandolin. The five instruments, all constructed by the first author, The second was the mandoline or Neapolitan mandolin, are shown in Fig. 1. Mandolin 1 has a carved redwood top which had four courses of two strings tuned like a violin. and a maple back with a symmetrical body, mandolin 2 has The modern mandolin is tuned like the latter; the six-course a redwood top and a walnut back with a scroll-shaped instrument fell into disuse in the late 19th century. The cavity added to the body. Mandolins 3 and 4 are the Gibson history of the mandolin is discussed by Tyler and Sparks F-type mandolins constructed from carved spruce plates, [1,2], by Gill and Campbell [3], and in an earlier paper by carved maple backs, and other parts supplied by the Gibson the authors [4]. company. Bracing patterns in mandolins 1 and 2 are shown The mandola might be called an alto mandolin. It has in Fig. 2. four double courses of strings tuned to C ,G,D, and A 3 3 4 4 3. NORMAL MODES OF VIBRATION (as in a viola) compared to the G3,D4,A4, and E5 on the mandolin. The body parts are about 15% larger than those A normal mode of vibration represents the motion of a of the mandolin. linear system at a normal frequency (eigenfrequency). It Mandolins have become popular in many parts of the should be possible to excite a normal mode of vibration at world. In the 1920s, a neo-Baroque style of mandolin any point in a musical instrument that is not a node and to music developed in Germany, and the instrument also observe motion at any other point that is not a node. A became popular in Japan. The mandolin was brought to normal mode is a characteristic only of the structure itself, America by Italian immigrants about the beginning of the independent of the way it is excited or observed. Normal 20th century, and it became especially popular in bluegrass mode shapes are unique for a structure, whereas the music. In 1995 the Japan Mandolin Union had over 10,000 deflection of a structure at a particular frequency, called an members and there were over 500 mandolin orchestras in operating deflection shape (ODS) may result from the excitation of more than one normal mode. Normal mode Permanent address: J. Sargeant Reynolds Community College, testing has traditionally been done using sinusoidal Richmond, VA 23261

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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 approx- imation to the normal modes. Ways to deal with over- lapping modes include the use of multiple drivers and the use of phase modulation [7]. 4. EXPERIMENTAL METHOD 4.1. Modal Analysis Using Holographic Interferome- try 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 Fig. 1 Clockwise from top left: mandolin 1 (redwood produce the holographic image. A phase-stepping mirror, top, maple back and sides); mandolin 2 (redwood top, driven by a piezoelectric crystal, varies the phase of the walnut back and sides); mandolin 3 (Gibson f-hole); reference beam in four steps so that the holographic mandolin 4 (Gibson oval hole); mandola (redwood top, walnut back and sides). 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.

4.2. 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 Fig. 2 Bracing patterns in mandolin 1 (left) and mandolin 2 (right). 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 excitation, either mechanical or acoustical. Detection of thus created. motion may be accomplished by attaching small accel- erometers, although optical methods are less obtrusive. For 4.3. Determination of Air Cavity Modes further discussion of normal modes of vibration, see 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. 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.

4.4. 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. Fig. 3 (0,0) mode pair in mandolin 1. Top: top and back 4.5. Sound Spectra plates at 284 Hz; Bottom: top and back plates at Sound spectra were recorded with an Ono Sokki 360 415 Hz. fast-Fourier transform (FFT) analyzer and a microphone placed about 0.5 m from the instrument. Spectra were 3 and 4, which we studied earlier [4], the back plate averaged by plucking a wide range of notes on all strings vibrated in a similar way, but this was not the case in the and also by exciting the instruments mechanically on the other two mandolins or the mandola. The (1,0) mode was bridge and oﬀ the bridge with broadband noise [10]. observed at 402 Hz in the mandola top plate and at 451 to 5. RESULTS 575 Hz in the mandolins. The (1,0) mode in mandolin 2 is shown in Fig. 4. 5.1. Normal Modes Additional modes with 2 or 3 longitudinal nodal lines Operating deﬂection shapes (ODSs) were recorded at a were observed in the top plates of most of the instruments. large number of frequencies using electronic TV hologra- These are shown for the mandola in Fig. 5. phy. Included were frequencies at which the mandolins The (0,1) mode is characterized by a transverse nodal appeared to radiate strongly and those at which the entire line near the center. Although this is an important mode in 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 Fig. 4 (1,0) mode in mandolin 2 at 474 Hz. 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 Fig. 5 Modes in the mandola top plate. Left: (2,0) mode describe this as (1,0) motion in the top plate. In mandolins at 796 Hz; Right: (3,0) mode at 929 Hz.

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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, 415 237, 267, 400 313, 464 209, 398, 482 234, 368 1,0451 (top) 474 570 575 (595) 402(top) 0,1 650 620 824 775 558? 666? 2,0907(top) 837 (top) 886 918 796 3,0 1,062 (top) 1,076 1,117 1,149 929 1,1 862 821 728 A0 293 283 272 214 218 A1 866 865 895 804 556 A2 1,085 1,082 816 905 792 ft 400 404 412 466 350 fb 580569 618 633 490 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. A0 is the Helmholtz mode in which air vibrates in and out of the sound holes. In the A1 mode, air vibrates longitudinally with a pressure node across the waist. In the A2 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 ft and fb when the opposite plate is immobilized in sand.

5.2. Frequency Response Function Accelerance level (log of acceleration at constant force Fig. 6 Frequency response (accelerance at treble side of amplitude) is plotted vs frequency in Fig. 6. Peak values of the bridge). (a) mandolin 1; (b) mandolin 2; (c) accelerance occur at most of the modal frequencies in mandolin 3; (d) mandolin 4; (e) mandola. Vertical Table 1. In addition most instruments show a broad scale is 20dB/division. Peaks corresponding to the (0,0), (1,0), and (3,0) modes are indicated. maximum around 3,500 Hz.

5.3. 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 D. COHEN and T. D. ROSSING: ACOUSTICS OF MANDOLINS quite similar to those of the mandolins. The treble rolloff Mandolin makers have used a wide variety of brace begins at a lower frequency (about 1.3 kHz) than in the patterns. Gibson mandolins generally feature longitudinal mandolins and it is not quite as great. braces, whereas such makers as Gilchrist and Smart prefer crossed braces. A systematic study of the effect of different 6. DISCUSSION bracing patterns on the acoustics of mandolins would be Normal modes of vibration in the mandolin and desirable. Also desirable would be theoretical studies of mandola in the low-frequency range are somewhat akin mandolin body vibrations, perhaps by the application of to those observed in guitars, as might be expected. In finite element methods, as has been done in instruments guitars, (0,0) modes of the top and back plates coupling such as the guitar [12]. strongly to the Helmholtz resonance lead to three normal Although we have only sampled a few different styles modes within the span of about an octave from 100 to of instruments, the acoustical observations that we have 200 Hz [13]. In mandolins 1 and 3 and the mandola, only made should be of interest to mandolin makers, who are two (0,0) modes were observed and these most nearly constantly experimenting with methods to improve their resemble the lowest and the highest (0,0) modes in guitars. instruments. The coupling is weaker than in guitars, and the A0 cavity mode (Helmholtz resonance) is often lower in frequency 7. CONCLUSION than the (0,0) modes in the top and back plates. Also fb is The normal modes of vibration in mandolins are similar considerably larger than ft in all the instruments, indicating to those observed in guitars. The coupling between top and that the back plate is much stiffer than the top plate (in back plates appears to be less than in guitars. The lowest most guitars ft and fb are fairly close to each other). modes may appear as either a doublet or a triplet. Modal Because the stiff back plate vibrates at relatively low frequencies observed with holographic interferometry amplitude at the (0,0) mode frequencies, one might expect correlate well with peaks in the frequency response curves. the two mass model to describe the coupling [14] As we suggested in our earlier paper [4], the historical reasonably well. According to this model, and current interest in the mandolin as a musical instrument 2 2 2 2 f1 þ f2 ¼ ft þ fA0 . In mandolins 1 and 4, these sums would suggest the desirability of further research on are within 5% of each other; in mandolin 2 they differ by mandolin acoustics. The effect on tone of stronger plate 11% and in mandolin 3, they differ by 22%. In the coupling could profitably be studied, as well as the effect of mandola, they differ by 11%. In all cases different bracing patterns. The authors thank the Gibson 2 2 2 2 (f1 þ f2 Þ > ðft þ fA0 ), which is to be expected since Company for furnishing materials for two of the mandolins the compliance of the back plate will lower the frequency studied. of the A air cavity mode. 0 REFERENCES If stronger plate coupling in the fundamental mode were desired, it would be well to raise the A air cavity [1] J. Tyler and P. Sparks, The Early Mandolin (Clarendon Press, 0 Oxford, 1989). mode (Helmholtz resonance) frequency by enlarging the [2] J. Tyler and P. Sparks, ‘‘Mandolin,’’ in The New Grove sound hole(s) and to lower fb by making the back plate less Dictionary of Music and Musicians, 2nd ed., S. Sadie, Ed. stiff. This should produce an instrument with a more bass (Macmillan Publishers Ltd., London, 2001). sound and greater support for the lowest string, a desirable [3] D. Gill and R. 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[10] T. D. Rossing, ‘‘Sound radiation from guitars,’’ Am. Lutherie, Chap. 9. 16, 40–49 (1988). [13] T. D. Rossing and G. Eban, ‘‘Normal modes of a radially- [11] T. D. Rossing, ‘‘Physics of guitars: An introduction,’’ J. Guitar braced guitar determined by electronic TV holography,’’ J. Acoust., 4, 45–67 (1981). Acoust. Soc. Am., 106, 2991–2996 (1999). [12] N. H. Fletcher and T. D. Rossing, The Physics of Musical [14] G. Caldersmith, ‘‘Guitar as a reﬂex enclosure,’’ J. Acoust. Soc. Instruments, 2nd ed. (Springer-Verlag, New York, 1998), Am., 63, 1566–1575 (1978).