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Air and Structural Modes of a Harpsichord Williamr Air and structural modes of a harpsichord WilliamR. Savage Departmentof Physics and Astronomy, The University of lowa, Iowa City,Iowa 52242 Edward L. Kottick Schoolof Music,The Universityof Iowa,Iowa City,Iowa 52242 Thomas J. Hendrickson Departmentof Physics,Gettysburg College, Gettysburg, Pennsylvania 17325 Kenneth D. Marshall TheUniroyal Goodrich Tire Company, Research and Development Center, Brecksville, Ohio 44141 (Received17 May 1991;accepted for publication30 December1991 ) The acousticalbehavior of a harpsichordmodeled after 17th-centuryFlemish prototypes was studiedusing both experimentaland analyticaltechniques. The vibrationalmodes of its enclosedair volumewere measuredand foundto correspondclosely to thosepredicted by the Joand J0 solutionsto the Besselequation for a wedgeshaped space. A modalanalysis of the completeharpsichord revealed that the soundboardhas 36 vibrationalmodes over a frequency rangeof 0 to 600 Hz, and that thereare numerousmodes where the instrument'scase has a significantamplitude of motion.Additional information is reportedshowing that theacoustic outputof theharpsichord is reasonablyfiat overa frequencyrange of 50-2000Hz. It is concluded that the resonance behavior of both the soundboard and the enclosed air are importantto thetone quality of theharpsichord, and that itsgenerally uniform acoustic output resultsfrom the excitation of a largenumber of woodand air modesby the stringpartials. PACS numbers:43.75.Mn, 43.75.Gh LISTOF SYMBOLS eigenfunction(mode shapecoefficient), soundboard o circularfrequency, radians effectiveangle of the air cavity,20 deg • viscousdamping factor h heightof air cavity, 18 cm c velocityofsound, m/s L effectivelength of the air cavity, 170cm 2[ wavelength,m m number of radial nodal surfaces inside the air Ca eigenfunction(mode shape coefficient), air cavity INTRODUCTION tury Flemishexamples. • It wasbuilt in 1976by thesecond author,from parts and plans supplied by ZuckermannHarp- The harpsichordis a pluckedstring instrument whose sichords(see Fig. 1). It hasone keyboard with 52 keys,and tone-producingmechanism is activated from one or two key- oneset each of 8' and4' stringsand jacks. Its 8' rangeis from boards.Harpsichords normally haveone, two, or three sets G t (49 Hz) to D6 ( 1176Hz). 2Depressing a key raises both (sometimescalled choirs or ranks) of strings:if one set, it setsof jacks;however, a setcan be turned"on" or "off' by will be at normal (8') pitch;if two, eitherboth will be at 8' movingits registerslides in or out, causingthe plectra either pitch, or one will be at 8' pitch and the secondan octave to engageor to miss the strings.The instrumentcan be higher(4' pitch); if three,two will beat 8' pitchand the third playedwith the 8' stringsalone, the 4' alone,or bothtogeth- at 4'. Somevery largeharpsichords may havea 16' choir as er. In normal useit sitson a standwith its lid proppedup well. Eachset of stringshas a setof jacks.These sit on the approximately45 deg.We call this instrumentour "acous- distal endsof the keys,and their plectra (traditionally bird tics harpsichord."We have marked it, photographedit, re- quill,but nowusually Delrin plastic)pluck the appropriate cordedit, wired it, shakenit, hammeredit, drilled it, blasted stringswhen the keysare depressed. The 8' stringstransmit it with sound,filled it with sand,and dismantledit. We have theirvibrations to thesoundboard through the 8' bridge;a 4' evenplayed it. setof strings,if present,requires a separate,shorter bridge. Developedsome 500 yearsago, the harpsichordwas in Harpsichordbuilding generally followed either North- continualuse throughthe end of the 18th century;but its em or SouthernEuropean constructional practices, al- rigid levels&loud and softsounds did not suitthe require- though the productsof some regional schoolsshow in- mentsof the classicalperiod for flexibility.By the endof the fluencesof both.The instrumentused for the investigations 18thcentury it wassuperseded by the moreexpressive piano. describedin thisreport is Northern, modeledafter 17th-cen- It hasenjoyed a revivalin this century,and is now accepted 2180 J. Acoust. Sec. Am. 91 (4), Pt. 1, April 1992 0001-4966/92/042180-10500.80 ¸ 1992 AcousticalSociety of America 2180 Previousstudies on the acousticsof harpsichordscan be found in papers by Kellner (1976), Fletcher (1977), SOUND Spencer( 1981), and Kottick ( 1985); and of its relative,the BENTSIDE clavichord,by Thwaites ( 1981) and Thwaitesand Fletcher 4' HITCHPIN-... ( 1981). Theseare valuablearticles, but they are limited ei- RAIL ther to a few aspectsof the acousticalbehavior of harpsi- chords,or to purely theoreticalconsiderations without ex- 4' 81RIDGE .... ---- J•3TTOM perimentalverification. Therefore, since 1976, the first two authorsof this paper,with considerableassistance from the CUTOFF BAR ß secondtwo, haveattempted to betterunderstand the myster- LOWER ies of this instrument.We have gatheredexperimental data on soundboardand air resonancesand their interactionby meansof responsecurves and Chladni patterns.We have LOWER testedour acousticsharpsichord with and without strings, BEL• RAIL with and without the bottom, and with the soundboardboth in and out of the case.The propertiesof the air cavityhave beenmeasured with the soundboardrendered immobile, and LEGS with the bellyrailslot both openand closed. Our understandingof the harpsichord'sbehavior was ME •KEYFRAME enhancedby field work performedin 1980,when response curvesand Chladni patternswere obtainedfor 39 harpsi- FIG. 1. Schematicview of the Flemish (acoustics)harpsichord. chords,both new and antique(Kottick, 1985). Until 1986, the informationwe gatheredon the vibrationalbehavior of the instrumentwas limited to the studyof the forceand mo- tion at one location at a time. Since then, the use of modal analysistechniques to study the harpsichordas an input/ as the appropriateinstrument for keyboardmusic written outputsystem has resulted in a moreglobal description, and beforeca. 1750.The harpsichordseems destined to remain a greaterunderstanding of its dynamicbehavior. with usas long as we continueto enjoythe sounds of "acous- tic" instruments. The work describedin thispaper, therefore, represents a Becauseof itsshape the harpischord would seem to have blendingof the experimentaland mathematicalprocesses muchin commonwith thepiano, particularly since the earli- necessaryto study the generalacoustical behavior of the estexamples of the latterwere essentially harpsichords with harpsichord.As such,it suggestsa framework for thefuture actionsthat struck,rather than pluckedthe strings.But the studyof otherharpsichords, and for instrumentswith simi- resemblanceis specious.The modern piano has a thick lar physicalproperties such as the virginal, the bentside spin- soundboard,massive sides, no bottom, and doesnot enclose et and the early piano. a volumeof air, while the harpsichordhas a thin sound- I. AIR MODES board, somewhat flexible sidesand bottom, and an enclosed air mass.In theseways, the harpsichordmore closely resem- Our understandingof the air cavity'sresonances and blesthe guitarthan the piano--aninstrument with whichit their contributionto our acousticsharpsichord's properties evenseems to sharethe presenceof "soundholes." Like the wasincreased by both theoreticaland experimentalstudies. relationshipbetween the harpsichordand the piano,how- An approximatemathematical model provided us with valu- ever,these apparent similarities are outweighedby sharpdif- able cluesto the interpretationof the experimentalresults ferences.The guitar is symmetricalin shape,but the harpsi- obtained for the instrument itself. The tests on the acoustics chord is decidedly asymmetrical,both in shape and in harpsichordwere carried out at the Universityof Iowa, ei- barring(ribbing). While the guitaris internallyunobstruct- ther in the "soundroom"--a resonantroom with a relatively ed, the interior of the harpsichordis heavilybraced. The flat response--inthe acoustics laboratory in theDepartment ratio of the area of the rose hole to the enclosed volume of air of Physicsand Astronomy,or in the anechoicchamber is far largerin the guitar than in the harpsichord.Further- housedin the Departmentof Speechand Hearing. more,Northern harpischordsusually have a sizableopening The air cavityof the acousticsharpsichord is bounded just behindthe keyboard (the bellyrail slot) whosearea is on top by a flexiblesoundboard about 2.5 mm thick,and on muchgreater than that of its rosehole. The guitar is a com- the bottomand sidesby boardsabout 12 mm thick to which pact instrument,and its vibrationalmodes can be excited areglued several braces (these and subsequent relationships overa widerange of frequenciesthrough its relativelysmall are madeclear by Fig. 1). The wallsof thiscavity consist of a bridge. This cannot happen with a harpsichord,whose spineabout 152cm long,a cheekabout 57 cm long,a bent- bridgemay be from 1.5-2.0 m in length.Finally, the sheer sideapproximately parabolic in shape,an angledtail about sizeof the harpsichordhas discouraged experimental study 27 cm long, and upper (12 mm thick) and lower (15 mm of its vibrationalbehavior. It is not an easyinstrument to thick) bellyrailsabout 77 cm long.The depthof the inside, suspend,mount, shake, blast with sound,or subjectto holo- from bottom to soundboard,is 18 cm. At the keyboard end graphicinterferometry. of this asymmetricalbox is a horizontalopening, the belly- 2181 J. Acoust.Soc. Am., VoL91, No. 4, Pt. 1, April1992 Savageeta/.: Air and structuralmodes of a harpsichord 2181 rail slot,
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