Smulatng a gutar wth a conventonal sonometer Zly Bursten, Chrstna M. Gower and Gabrele U. Varesch, Loyola Marymount Unversty, Los Angeles, CA 90045 Muscal acoustcs s an nterestng sub-feld of physcs whch s usually able to engage students n a dual perspectve, by combnng scence and art together. The physcs prncples nvolved n most muscal nstruments 1 can be easly demonstrated wth standard laboratory equpment and can become part of lecture or lab actvtes. In partcular, we wll show n ths paper how to smulate a gutar usng a conventonal sonometer, n relaton to the problem of the nstrument ntonaton,.e., how to obtan correctly tuned notes on a gutar or smlar strng nstruments. Ths problem s more complex than what mght appear at frst; t obvously begns wth the correct tunng of the open strngs of the nstrument to the desred notes, whch can be easly accomplshed wth the help of a dgtal tuner. It s then related to the correct placement of the frets on the fngerboard whch enable the nstrument to produce all the dfferent notes, but t s further complcated by other subtle effects, whch are usually dealt wth n an emprcal way by luthers and gutar manufacturers. In a recently publshed paper 2 we descrbed mathematcal and physcal models to be used for a more scentfc approach to ths problem, resultng n a complex compensaton procedure whch s very effectve n mprovng the ntonaton of ths type of nstruments. In ths paper we present a smplfed approach to the problem, whch s more sutable to be used as an n-class demonstraton, or as a laboratory actvty, to smulate how muscal notes are produced on a gutar. The expermental apparatus and the ntonaton problem A conventonal (monochord) sonometer, such as the PASCO 3 WA-9613 or smlar, can be used to smulate a (sngle strng) gutar, wth the addton of a fngerboard whch can be easly obtaned from a local luther or through a musc shop. Fgure 1 shows our expermental apparatus composed of the PASCO sonometer to whch we added part of a classcal gutar fngerboard, complete wth twenty metallc frets, placed rght under the steel strng. It s mportant to know the scale length of the fngerboard (or fretboard) beng used,.e., the correct strng length for whch the fngerboard was desgned (usually between 640-660 mm for a classcal gutar) and to set the dstance between the two sonometer brdges accordngly. These two brdges wll smulate the nut and the saddle of a gutar, as shown n Fg. 1. A gutar outlne s also supermposed on ths fgure, to llustrate how our apparatus can smulate the functonalty of a real gutar. The frets are placed accordng to a precse mathematcal relaton: X = X 2 12 0 X (0.943874) 0, (1) where X 0 s the (open) strng length, or scale length, mentoned above and X s the poston of the -th fret, as measured from the saddle of the nstrument (see our prevous paper 2 for full detals). 1
The fretboard needs to be postoned rght under the sonometer strng so that the 12 th fret s placed exactly at half the strng length (as requred by Eq. (1) for = 12 ) and as close as possble to the strng as n a real gutar. After tunng the open strng to the desred note, one should test all the other notes by pressng down on each fret and pluckng the strng. Usually the sonometer has enough resonance to produce an audble sound, although not as strong as the one of a real nstrument. In ths setup process, care should be taken to adjust the acton of our smulated nstrument,.e., placng the fretboard close enough to the strng so that t s easy to press down on t at any poston along the fngerboard, but not too close n order to avod producng buzzng sounds when playng the notes on the fretboard. A good compromse can usually be acheved after tryng dfferent heghts of the fretboard above the base level of the sonometer. To press unformly on the strng at any poston we used a sprng-loaded devce (also shown n Fg. 1), but fnger pressure wll also work. The strng can be plucked by hand or wth a standard gutar pck. After ths ntal setup a chromatc scale can be played,.e., all the notes on the fretboard n successon, and the frequency of the produced sounds can be measured, to check f the smulated gutar s n tune. To measure these frequences one can use a mcrophone connected to a dgtal osclloscope or to a dgtal nterface lnked to a computer, usng one of the many software packages commercally avalable, or other devces. In our prevous work 2 we preferred to use a very precse dgtal tuner 4 whch could dscrmnate frequences at the level of ± 1 cent 5. The measured frequences should be checked aganst the theoretcal frequences whch are expressed n a way smlar to Eq. (1): ν 2 = ν (1.05946) 0 12 ν 0, (2) where ν 0 and ν are respectvely the frequency of the open strng note and of the -th note. Eqs. (1) and (2) are smply related, snce Mersenne s law 2 states that the frequency of a vbratng strng s nversely proportonal to ts length, and the vbratng length of a gutar strng s expressed by X n Eq. (1). At ths pont, a traned ear should perceve that the sonometer s not perfectly n-tune, even f the setup procedure outlned above was perfectly followed. Ths lack of ntonaton, notceable for most of the fretted notes,.e., those obtaned by pressng the strng onto the fretboard, s a well-known effect n gutar manufacturng. It s manly due to the mechancal acton of the player s fnger, whch presses the strng on the fngerboard, thus changng slghtly ts length and tenson and therefore alterng the frequency of the sound beng produced. As a result, a fretted strng nstrument, such as a gutar, mandoln, or smlar nstrument, s never perfectly n tune, snce the pressure of the player s fnger s unavodable. On the contrary, non-fretted strng nstruments, such as those of the voln famly, do not have ths complcaton, as long as a sklled player can compensate for ths effect by pressng on the fngerboard at a poston whch naturally elmnates the problem. Fgure 2 shows a detal of our apparatus, llustratng how the sprng-loaded devce s pressng at a partcular poston (the ffth fret n ths case), thus alterng the orgnal length and tenson of the strng, n the same way the player s fnger acts on a real gutar fngerboard. The dgtal tuner, placed near the sonometer, can accurately measure the frequency of the produced sound, when the strng s plucked. 2
In the next secton we wll outlne how a partcular compensaton procedure can mnmze ths lack of ntonaton of a fretted nstrument and show how t can be practcally mplemented on our monochord sonometer. The compensaton procedure In order to correct the ntonaton problem descrbed above the compensaton procedure usually nvolves movng slghtly both saddle and nut, from ther ntal postons. In partcular, the saddle s usually moved a few mllmeters away from the nut ( setback of the saddle, ncreasng the open strng length), whle the nut s usually moved forward toward the saddle by a smaller amount ( set-forth of the nut, decreasng the open strng length). These movements are performed wthout changng the postons of all the frets and they can be easly performed, n our expermental apparatus, by adjustng the postons of the two movable brdges of the sonometer whch smulate the saddle and the nut. In our prevous work 2 on the subject, we analyzed a complex mathematcal procedure whch maxmzed the compensaton effects, and we computed the best possble saddlenut postons. In ths paper we descrbe a smplfed procedure, smlar to the one used by gutar manufacturers, whch can be effectvely used as a lab actvty for students n a muscal acoustcs class. The procedure we used nvolves the followng steps: 1. Set the nut and saddle at ther ntal postons for the strng length beng used and tune the strng to the requred ptch (open strng note). 2. Usng a sprng-loaded devce or a fnger, press at all fret postons and record the frequences of all notes wthout compensaton (at least three measurements for each fret, n order to compute average and standard devaton for each frequency). 3. Move the nut to a new set-forth poston (by small fxed ncrements, of one mllmeter or less) and retune the open strng to ts proper value. 4. Compare the fretted note at the 5 th fret wth the theoretcal frequency expected for that note (obtaned usng Eq. (2)). 5. Return to pont 3 and terate the procedure untl the best poston for the nut s found (the one for whch the fretted 5 th note s more n-tune). Record the fnal nut set-forth. 6. Leavng the nut at ths new poston, proceed to adjust the saddle to a new setback poston (by small fxed ncrements, of one mllmeter or less for example) and then retune the open strng to ts proper value. 7. Compare the fretted note at the 12 th fret wth the theoretcal frequency (obtaned usng Eq. (2)). 8. Return to pont 6 and terate the procedure untl the best poston for the saddle s found (the one for whch the fretted 12 th note and the correspondng theoretcal frequency are best matched). Record the fnal saddle setback. 9. Leave the saddle and nut at these new compensated postons and retune once agan the open strng to the proper frequency. 3
10. Fnally, press at all fret postons and record the frequences of all notes wth compensaton (three measurements for each fret, average and standard devaton for each frequency). Expermental results We appled the procedure outlned above to two steel strngs ncluded n the PASCO sonometer and also used n our past analyss 6 (see Table I n our prevous paper 2 for the physcal propertes of these strngs). The frst strng was tuned at an open strng frequency ν 0 = 130.813 Hz, correspondng to a C 3 note. Usng Eq. (2), t s straghtforward to compute the theoretcal frequences of all the notes correspondng to the twenty frets of our smulated gutar. We then used the procedure descrbed above to measure all the expermental frequences of the notes wth and wthout compensaton, and compared these values wth the theoretcal ones. The results are more effectvely presented n terms of the frequency devaton (n cents 5 ) between the expermental and the theoretcal values. A zero frequency devaton for a partcular note means perfect tunng, whle a frequency devaton of about ± 10 cents (ptch dscrmnaton range) would stll be consdered acceptable, snce the human ear cannot dfferentate two sound frequences wthn ths range (see Ref. 2 for more detals). Therefore, n Fg. 3 the frequency devatons for the frst strng are presented as a functon of the fret number of our smulated gutar. The perfect ntonaton level corresponds to zero frequency devaton (black dotted lne) and t s obvously acheved only by the open strng (zero fret number). Red crcles represent results wthout compensaton, whle blue trangles are obtaned wth our compensaton procedure (wth a nut set-forth of 5.0 mm and a saddle setback of 5.0 mm). Results wthn the ptch dscrmnaton range (green dashed lnes) should be consdered practcally n-tune, n vew of the prevous consderatons. We can see that the compensaton procedure was effectve snce the results wth compensaton are n fact wthn the requred range. The nut adjustment corrected the ntonaton at lower fret numbers, whle the saddle adjustment took care of the hgher frequency notes. Fg. 4 presents smlar results for a second strng, whch was tuned an octave hgher than the frst one (open strng frequency ν 0 = 261.626 Hz, correspondng to a C 4 note). The meanng of the symbols n ths fgure s the same as n Fg. 3, and agan we can see that the compensaton procedure was effectve (wth a nut set-forth of 3.0 mm and a saddle setback of 2.0 mm for ths strng). Conclusons We presented a smple way to convert a standard sonometer nto a smulated fret nstrument, by addng a gutar fngerboard. Ths modfed apparatus can be useful to ntroduce students to strng vbratons, Mersenne s law and ts applcatons to fretted strng nstruments. The subtle effects of ntonaton and compensaton can also be studed n a practcal way, wthout usng complex mathematcal models. The experments 4
presented n ths work can easly become a more structured laboratory actvty, whch can be used n general physcs courses or more specalzed acoustcs classes. Acknowledgments Ths research was supported by a grant from the Frank R. Seaver College of Scence and Engneerng, Loyola Marymount Unversty. The authors would lke to acknowledge luthers Greg Byers and Hugh Greenwood for useful dscussons and advce. References 1. N. H. Fletcher and T. D. Rossng, The Physcs of Muscal Instruments, 2 nd ed. (Sprnger-Verlag, New York, 1998). 2. G. U. Varesch and C. M. Gower, Intonaton and compensaton of fretted strng nstruments, Am. J. Phys. 78, 47-55 (January 2010). 3. PASCO, WA-9613 Sonometer Instructon Manual (1988). 4. TurboTuner, Model ST-122 True Strobe Tuner, www.turbo-tuner.com. 5. The cent s a logarthmc unt of measure used for muscal ntervals. The muscal octave s dvded nto 12 semtones, each of whch s subdvded n 100 cents. Snce an octave corresponds to a frequency rato of 2:1, one cent s equvalent to an nterval of 2 1/1200. Gven two dfferent notes of frequences a and b, the number n of cents between the notes s n = 1200 log2( a / b). 6. The results n ths paper are slghtly dfferent from those reported n our prevous work 2, although we used the same strngs. The compensaton procedure used here was dfferent; we reduced the dstance between fretboard and strng, and also used the sprng-loaded devce n a dfferent manner, n order to make our apparatus more smlar to a real gutar. PACS codes: 43.75.+a; 43.75.Bc; 43.75.Gh; 01.50.Pa Keywords: acoustcs, muscal acoustcs, gutar, sonometer, ntonaton, compensaton. Zly Bursten s a sophomore honors student at Loyola Marymount Unversty, majorng n physcs and Englsh. She s nterested n explorng many areas of physcs durng her remanng three years at LMU, before attendng graduate school. Department of Physcs, Loyola Marymount Unversty, 1 LMU Drve, Los Angeles, CA 90045; zburste@lon.lmu.edu Chrstna M. Gower s a senor physcs and mathematcs major at Loyola Marymount Unversty. Durng her four years at LMU Chrstna has conducted research n dfferent areas of physcs ncludng space physcs, condensed matter, acoustcs, and cosmology. Chrstna s plannng to attend graduate school n astrophyscs n the fall of 2011. Department of Physcs, Loyola Marymount Unversty, 1 LMU Drve, Los Angeles, CA 90045; Chrstna.Gower@gmal.com 5
Gabrele U. Varesch s an assocate professor of physcs at Loyola Marymount Unversty. He receved hs Ph.D. n theoretcal partcle physcs from the Unversty of Calforna at Los Angeles. Hs research nterests are n the areas of astro-partcle physcs, cosmology and general physcs. In hs free tme he enjoys playng classcal gutar and renassance lute musc. Department of Physcs, Loyola Marymount Unversty, 1 LMU Drve, Los Angeles, CA 90045; gvaresch@lmu.edu FIGURES: Fg. 1. Our tabletop apparatus composed of a sonometer and a gutar fngerboard, wth twenty metallc frets. Also shown are the two movable brdges, whch smulate the nut and saddle of a gutar, the sprng-loaded devce used to press on the strng at dfferent postons, and the dgtal tuner used to measure sound frequences. A gutar outlne s supermposed on the pcture, n order to show how the apparatus smulates the functonalty of a real gutar. 6
Fg. 2. A detal of our expermental apparatus, showng the sprng-loaded devce pressng at the ffth fret poston. The resultng strng deformaton and the slght change n the strng tenson wll cause the lack n ntonaton of the note beng produced. The dgtal tuner postoned near the sonometer can accurately measure the sound frequency. 7
Fg. 3. Frequency devaton from perfect ntonaton level for notes obtaned wth our frst strng. Red crcles ndcate results wthout compensaton, whle blue trangles denote results wth compensaton. The regon between the green dashed lnes s the approxmate ptch dscrmnaton range for frequences related to ths strng. 8
Fg. 4. Frequency devaton from perfect ntonaton level for notes obtaned wth our second strng. Red crcles ndcate results wthout compensaton, whle blue trangles denote results wth compensaton. The regon between the green dashed lnes s the approxmate ptch dscrmnaton range for frequences related to ths strng. 9