A Theoretical and Experimental Analysis of the Acoustic Guitar. Eric Battenberg ME


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1 A Theoretical ad Experimetal Aalysis of the Acoustic Guitar Eric Batteberg ME
2 1 Itroductio ad Methods The acoustic guitar is a striged musical istrumet frequetly used i popular music. Because it is commo i dorm rooms across the coutry ad ca be aalyzed usig a simple liear model, we thought it a fittig istrumet for a theoretical ad experimetal aalysis. I this project, we will attempt to quatify certai aspects that affect the way soud is produced by the acoustic guitar. I the remaider of this sectio, we will summarize the theoretical model ad the sigal aalysis techiques we will use. I Sectio, we will cover the affects that pickig locatio has o the harmoic distributio of a guitar ote. I additio, we show that by comparig the results of the sigal aalysis with the theoretical model for pickig locatio, we ca approximate the pickig locatio used to create a recorded ote. I Sectio 3, we look ito the subtle implicatios of tuig systems used i wester music ad how they ifluece the guitar. I Sectio 4, we examie atural harmoics, which, i additio to frettig, are a way to create ew pitches o a guitar strig. 1.1 Theoretical Aalysis Sice the acoustic guitar is a striged istrumet, we pla to carry out theoretical aalysis usig the oe dimesioal wave equatio. This is a completely liear model of a vibratig strig, so the harmoic implicatios derived from it igores oliearities such as strig stiffess, dulless itroduced by ski oil ad agig, ad other material properties of the strigs beyod liear desity ad tesio. The 1D wave equatio is show below, where T represets the strig tesio, ρ represets its liear desity, ad c the wave speed alog the strig: y y =, x t The geeral complexform solutio to the above differetial equatio is show below, where the leadig costats are complexvalued: y( x, = Ae ~ i( ωt+ kx) + Be ~ i( ωt kx), ω k = c For a guitar strig that is fixed at both eds (x=0 ad x=), we have boudary coditios that restrict the solutio space further. Whe we plug i these boudary coditios ad take the real part of the possible solutios, we are left with a solutio that ca be writte as a liear combiatio of harmoicallyrelated siusoids. y( 0, = 0, y(, = 0 I this case, the fudametal frequecy of the strig is are multiples of this frequecy. c = 1 T = ρ ad all other possible frequecies We will use these implicatios of the 1D wave equatio as a startig poit for our theoretical aalysis. c y( x, = ( A cosωt+ B siω sik cπ ω 1= x π k = (1) () (3)
3 1. Experimetal Aalysis Our experimetal aalysis will cosist of frequecy domai sigal aalysis. To do this, we use Matlab s pwelch fuctio, which implemets Welch s method for estimatig power spectral desity, alog with a Ha widow of legth Welch s method is a method of spectral aalysis that reduces oise itroduced by timedepedet variatios at the expese of decreased frequecy resolutio. Welch s method breaks the sigal up ito overlappig frames of samples ad applies a optioal widow fuctio. The discrete Fourier trasform is the applied to each frame. The resultig magitude spectrum is averaged across frames to get the fial spectrum estimate. The Ha widow is a type of raised cosie widow that is very widely used i sigal aalysis. Compared to the flat rectagular widow, the Ha widow has a wider mai lobe ad much smaller side lobe height i its frequecy respose (see plots below). This meas that a Ha widow affords much greater suppressio of farreachig frequecy compoet iterferece while itroducig a bit of local distortio i the frequecy domai. Whe examiig a audio sigal to fid the magitude of its harmoic compoets, we felt that the Ha widow would work well. We use halfoverlappig frames of legth 3768 o 3 secod moaural audio clips sampled at 44.1kHz. This results i about 8 widowed frames ad a frequecy resolutio of 1.35Hz.
4 Aalysis of Pickig ocatio Ayoe who plays guitar or aother striged istrumet kows that pickig locatio iflueces the acoustic qualities of the resultig ote. I this sectio, we attempt to quatify this effect. I additio, we describe our attempt to use the theoretical ad experimetal aalyses to develop a method to determie pickig locatio based o oly recorded audio data..1 Pickig ocatio ad the 1D Wave Equatio Startig from where we left off i Sectio 1.1, we ca further restrict the solutio space of the modes of vibratio allowed o a guitar strig by imposig zero iitial velocity i the strig: y (x,0) = 0 dt B = 0 y( x, = = 1 A cosωt sik x The if the iitial positio of the strig is a fuctio f(x),, we ca solve for the coefficiets A. (4) y ( x,0) = f ( x) A = 0 f ( x) sik xdx (5) Note that the solutio for y(x, i eq. (4) is ot time limited ad would oscillate forever. I reality, air resistace ad iteral strig frictio would result i losses that would cause harmoic compoets to expoetially decay at differet rates. We hope that the short 3 secod audio sigals will help reduce the ifluece of these factors o the spectral aalysis. We use a geeral triagular shape for the iitial positio fuctio f(x), show below. Usig equatio (5), the amplitude of each harmoic resultig from f(x) above ca be writte as: A h = π pπ si p( p) (6)
5 . Determiig Pickig ocatio from a Audio Sample Sice we will already be comparig the theoretical ad experimetal aalyses of the effects of pickig locatio, we thought it fittig to attempt to automatically determie a best fit pickig locatio based oly o the data cotaied i a audio clip of a guitar ote beig played. The first step i this process is automatically determiig the frequecy locatio ad amplitude of the harmoic compoets of a sigal. To do this, we wrote a Matlab script that fids a best fit fudametal frequecy ad the locates the harmoic compoets at multiples of this frequecy. The steps we use are listed below. Fidig the fudametal (f0) 1. Welch s method with Ha widow to extract spectrum. Covert magitudes to decibels relative to maximum magitude 3. Fid all local maxima i the spectrum 4. Elimiate local maxima below a strig depedet frequecy threshold 5. Elimiate local maxima which are t the largest withi a 0Hz widow 6. f0 is set to be the locatio of the remaiig local maximum with the lowest frequecy Fidig the harmoics Steps 1,, ad 3 as above 4. Elimiate local maxima which are below f0 5. Elimiate local maxima which are t the largest withi 0.75xf0 Usig these procedures, we locate the first 14 harmoics of each guitar ote. The harmoics are deoted by gree circles overlaid o the correspodig magitude spectrum for three differet pickig locatios below 1 th fret (31.65cm from the bridge), 15cm from the bridge, ad 5cm from the bridge. As you ca see, the method summarized above does a good job of pickig out oly the harmoics while igorig all oisy spikes. Oce we have located the first 14 harmoics of a guitar ote, we ca try to match the actual harmoic distributio with the theoretical harmoic distributio for a give pickig distace. To do this, we iterate over the theoretical harmoic distributios for pickig locatios alog the eck of the guitar separated by 0.1mm. Each theoretical distributio is compared to the actual distributio usig a
6 weighted measquared error criterio that places more importace o lower harmoics. The weights are computed usig a Gaussia curve that decays to 0.5 by the time it reaches the 14 th harmoic. The harmoic distributio with the lowest weighted measquared error is chose as the matchig pickig locatio. For the three pickig locatios show i the previous plots, the theoretical harmoic distributios chose by our matchig algorithm are show below as red circles. The matchig pickig locatios are show i the table below. Though the matched distaces are ot exact, they do correlate well with the actual distaces used to produce the sigals. Differeces are most likely caused by oliearities missig from the model such as strig stiffess ad the rate of decay of harmoics. Actual Pickig ocatio Matched Pickig ocatio 31.65cm (1 th fre 3.31cm 15cm 13.79cm 5cm.75cm
7 3 Aalysis of Tuig Systems I music theory, a tuig system is the way i which the pitches of musical istrumets are related. The acoustic properties of harmoies created by multiple pitches soudig simultaeously are related to the itervals betwee the pitches. Just itoatio is a system i which all pitch itervals correspod to a ratio of whole umbers betwee the frequecies of each pitch. For example, the frequecy ratio for a perfect fifth, a iterval regarded as cosoat or pleasat soudig, is 3:. The problem with the just itoatio system of tuig is that because pitch frequecies are determied with respect to a sigle referece ote, a istrumet would have to be retued every time the musicia wats to chage keys or chords. To address this problem, wester music has widely adopted the equal temperamet system of tuig. Equal temperamet approximates the itervals defied by just itoatio usig a costat ratio betwee adjacet frequecies. This ratio is calculated such that there are 11 distict pitches, or semitoes, before the ext octave is reached (or before the origial pitch is doubled i frequecy). This ratio is the equal to. Usig equal temperamet, the ratio for a perfect fifth, a iterval of 7 semitoes, is as opposed to the 1.5 prescribed by just itoatio. This differece, while small, ca actually ifluece the timbre or spectral quality of a musical iterval eough that it is upleasat to a traied ear. A table comparig the two tuig systems for the first 8 musical itervals is show below. Iterval Name Exact value i Equal Temperamet Decimal value i Equal Temperamet Just itoatio iterval Uiso (C) ^(0/1) /1= Mior secod (C ) ^(1/1) /15= Major secod (D) ^(/1) /8= Mior third (D ) ^(3/1) /5= Major third (E) ^(4/1) /4= Perfect fourth (F) ^(5/1) /3= Augmeted fourth (F ) ^(6/1) /5= Perfect fifth (G) ^(7/1) /= Differece i Hz For a 400Hz root The right colum of the table shows the frequecy differece i Hz betwee the two tuig systems whe the root pitch is set to 400Hz. Differeces of a few Hertz are see for some itervals. These differeces have eve stroger implicatios whe harmoies cosistig of greater tha two otes are costructed. For example, a major triad, made up of a root, major third, ad perfect fifth, at 400Hz cotais pitches with fudametal frequecies 400Hz, 500Hz, ad 600Hz usig just itoatio. Usig equal temperamet, these frequecies are 400Hz, 504Hz, ad 599Hz. Below, sigals cotaiig these frequecies are plotted. The left plot cotais the just itoatio frequecies; the right plot, the equal temperamet frequecies. Strog beatig is see i the amplitude evelope which is caused by the frequecy deviatios i the equal temperamet itervals.
8 Eve though just itoatio preserves the ideals aspects of musical harmoy, it would make sese that acoustic guitars are costructed usig equal temperamet itervals betwee the frets. We examie the frequecy ratios usig the fudametal frequecy fidig method covered i Sectio.. The first eight otes o the G strig are aalyzed ad their fudametals are extracted. The frequecy ratios for these otes are listed i the table below alog with the correspodig ratio for the two tuig systems. Iterval (Note) Frequecy Ratio ET Ratio JI Ratio Closer Uiso (G) Mior d (G#) JI Major d (A) JI Mior 3 rd (A#) ET Major 3 rd (B) ET Perfect 4 th (C) JI Augmeted 4 th (C#) ET Perfect 5 th (D) ET The rightmost colum shows which tuig system explais the observed frequecy ratio best. While equal temperamet seems to barley edge out just itoatio, it is likely that oise i the measuremets combied with possibly lowquality costructio of the test guitar made the results closer tha they should ve bee. Usig equal temperamet tuig i a guitar makes much more sese from a practical stadpoit.
9 4 Aalysis of Natural Harmoics I additio to creatig ew pitches by frettig a guitar, thereby chagig the legth of the strig, ew otes ca be created by lightly touchig a strig at a specific locatio to prevet vibratios at that poit. New pitches created i this way are called atural harmoics. Doig this adds additioal coditios to those imposed by the boudary coditios covered i eq. (3) of Sectio 1.1. The boudary coditios have already limited the solutio space to the followig: π y( x, A cosω t sikx, k = (7) = = 1 By forcig the displacemet o the strig to zero at a specific distace from the bridge d, we force the amplitude of the th harmoic to zero uless is a whole umber, i.e. d y( d, = 0 A = 0 uless is a whole umber We approximate the fret locatios usig the just itoatio iterval ratios betwee the ope strig pitch ad the pitch at that fret locatio. These ratios the represet the ratio of the origial legth of the strig to the fretted legth. The chart below shows which of the harmoics of a strig are still allowed to vibrate whe a figer is placed at the 5 th, 7 th, ad 9 th frets. The just itoatio ratio for the 5 th fret (a perfect 4 th ) is 4/3. Therefore, the strig must be shorteed to ¾ the origial legth to get a perfect 4 th, ad. This ratio is oly equal to a whole umber whe multiplied by a multiple of 4, so we are left with harmoics 4, 8, 1, 16, etc. These harmoics correspod to those of the same pitch as the ope strig raised by two octaves. Therefore, the atural harmoic created by placig a figer at the 5 th fret gives us a iterval of octaves. ikewise, the atural harmoic created at the 7 th fret leaves us with harmoics 3, 6, 9, 1, etc. These harmoics correspod to a iterval of a octave (x) plus a perfect fifth (x1.5). A atural harmoic at the 9 th fret gives us harmoics at multiples of 5, correspodig to octaves (x4) plus a major third (x1.5). These harmoic distributios ca be visualized more easily with the aid of the table below. (8) Fret (1/JI ratio) 5 (3/4) 7 (/3) 9 (3/5)
10 As a example, we examied the audio sigal created by a atural harmoic at the 5 th fret o the G strig. The spectrum of the origial ope strig is show o the left with the harmoics labeled with gree circles. The 5 th fret atural harmoic spectrum appears o the right, ad we ca see that all harmoics except for the 4 th, 8 th, 1 th, ad 16 th have bee practically elimiated. 5 Discussio This project has attempted to quatify certai aspects of the acoustic guitar, amely, the ifluece of pickig locatio o harmoic distributio, the tradeoff betwee coveiece ad iterval quality preset i differet tuig systems, ad the theoretical ad acoustic properties of atural harmoics. We have show that the ifluece of pickig locatio ca be modeled ad used to recogize the pickig locatio used to play a particular ote. Eve though a simple liearized model is used, the accuracy is eough to provide useful iformatio. I order to use this method to aotate a automatic guitar music trascriptio system, the robustess must be icreased. This could be accomplished by itroducig additioal compoets ito the model to describe the decay of harmoics ad the ifluece of the material properties of differet strigs. I additio, usig some sort of supervised (traied) machie learig method to help make decisios about best matchig distaces would icrease the accuracy of the system ad make it less sesitive to oise.
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