Acoust. Sci. & Tech. 24, 1 (2003) PAPER The acoustics of mandolins David Cohenà and Thomas D. Rossingà 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 ÃPermanent address: J. Sargeant Reynolds Community College, testing has traditionally been done using sinusoidal Richmond, VA 23261 1 Acoust. Sci. & Tech. 24, 1 (2003) 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.