Senior Honours Project

Physics 4

The Vibration of Piano Strings

December 2011

Abstract

The inharmonicity of piano and guitar strings was examined using Fourier trans-forms. Typical deviations from Fletcher’s theory of stiff strings could be attributedto the nonuniformity of wound strings and soundboard perturbations by extendingthe analysis to a rigid monochord. Additionally, the decay rates showed a beatingbehaviour, originating from the two polarisations of string motion and a series ofextra peaks in the spectrum was identified, known as even phantoms.

Contents

1 Introduction 1

2 Theory 22.1 Inharmonicity of vibrating strings . . . . . . . . . . . . . . . . . . . . . . 22.2 Phantom partials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Interactions among strings and soundboard . . . . . . . . . . . . . . . . . 42.4 Nonuniform overwound strings . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Method 7

4 Results & Discussion 94.1 Inharmonicity of the Piano . . . . . . . . . . . . . . . . . . . . . . . . . . 94.2 Inharmonicity of different guitars . . . . . . . . . . . . . . . . . . . . . . 124.3 Decay rates and polarisation . . . . . . . . . . . . . . . . . . . . . . . . . 134.4 Phantoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.5 The Monochord and soundboard perturbations . . . . . . . . . . . . . . . 154.6 Nonuniform overwound strings . . . . . . . . . . . . . . . . . . . . . . . . 17

5 Conclusion 22

A Theory and Experiment - Complete data 25

B Maple code for nonuniformity simulations 26

1 Introduction

The prime objective of the project was to study the inharmonicity of steel strings. This isthe phenomenon of sharp harmonics (frequencies are higher than expected) with respectto the fundamental mode, originating from the stiffness of real strings. To study thiseffect, guitar and piano strings were recorded and their spectra subsequently analysedusing Fourier transforms. The data was then fitted to the prevalent theory derived byFletcher [1] to determine the inharmonicity of each string, such that comparisons betweentheory and experiment could be made.

In order to examine the effect of the piano soundboard, one string was isolated on amonochord (essentially a rigid steel bar). Contrary to expectations, the lower harmonicsshowed a stronger deviation from the theory than the piano strings. Although resonancesfrom the soundboard and monochord are a contributing factor, the primary cause of thisphenomenon could be identified as the nonuniformity of wound strings: Each wound stringon a piano (or monochord) has a small portion of bare string (core of the string withoutwinding) at the tuning end, resulting in a nonuniform mass density. A simple experimentwith a wound guitar string, adjusted such that the amount of plain string could be varied,confirmed this theory, along with computer simulations, based on theoretical work byChumnantas [2].

Another aspect that was analysed were the decay rates of the harmonics, by usingFourier transforms for short time intervals of the recorded strings. Plotting the amplitude

1

as a function of frequency and time, a trend of exponential decay was found, with manycurves exhibiting a superimposed sinusoidal behaviour. This peculiarity has been widelyobserved in literature [3] for different stringed instruments and is believed to be causedby the two polarisations for transverse vibrations (the string is not restricted to a singleplane).

In addition to the expected peaks in the spectrum, a number of frequency-doubledrelatives were seen, forming a series known as even phantoms [4]. This series also exhibitsinharmonicity, such that comparisons of the relevant coefficient with the original valueswere made.

2 Theory

2.1 Inharmonicity of vibrating strings

For a vibrating string, solving the wave equation and applying boundary conditions leadsto the well known expression [5]:

fn =n

2L

√T

ρS(1)

where fn are the allowed frequencies, known as partials, defined through the integern > 0, L the length of the string, T its tension, ρ the density and S the cross sectionof the string. Thus the partials are integer multiples of the fundamental frequency andform a harmonic series, i.e. fn = nf1.

However, this analysis is only valid for a perfectly flexible string, due to the stiffness ofreal strings, the normal modes depart from this relation. First noted by Lord Rayleigh inthe 19th century [6], this problem was also solved by Nobel laureate Richard Feynman in aprivate letter to his piano tuner [7] (despite him not being able to play the piano), in 1961.An accurate treatment was published by Fletcher in 1964 [1], by adding an restoring forcedue to elastic stiffness ( d4

dx4dependence) in the equation of motion. Again, solving the

equation and applying boundary condition leads to an expression of the normal modes:

fn = nf0√

1 +Bn2 (2)

where:

B =π3Qd4

64L2T(3)

where Q is Young’s modulus, d the diameter of the string, all other symbols aredefined as in equation 1. As the allowed frequencies fn are not harmonic anymore, theyare customary called “partials”.

B is the inharmonicity coefficient, the correction to the original equation, Bn2, growswith n2, which is understandable: As the stiffness adds restoring force, the frequencies ofthe partials must increase, the higher n, the more nodal points or “arcs” will be on thestring, thus increasing the force. The constant f0 can be understood as the first partial ifthere were no inharmonicity present, as setting B = 0 recovers the ideal string behaviour,but it is not a real physical quantity.

2

The acoustical implications of inharmonicity are believed to be characteristic of pi-ano strings, computer simulations (sound synthesis) without inharmonicity have beenattributed as “unnatural” [8]. Perhaps more importantly, it affects the tuning of pianos.If n is small, the correction due to the stiffness is negligible, as B is typically of order10−5, but due to the n2 dependence it quickly becomes crucial: If a piano were tunedby adjusting the first mode (f1) of each key to the common twelve-tone scale, which isessentially the way to tune a guitar and most other stringed instruments, beats would beaudible when playing high and low keys simultaneously. As the inharmonicity increasesthe frequencies of the upper partials on the lower strings, they appear sharp relative tothe lower partials on upper strings (lower partials: small n → Bn2 ≈ 0). To avoid this,piano makers give their instruments a stretched tuning, resulting in first modes higher infrequency for treble keys and lower on the bass side, to balance the effect of inharmonic-ity. Figure (1), taken from [9], shows a typical tuning of a piano (aural, red data), alongwith averaged values over many pianos, known as the Railsback stretch (green data).

Figure 1: Graph (taken from [9]) showing stretched tuning of piano, due to inharmonicity.The red curve displays typical aural tuning, the green one is an average over many tunedpianos.

It should also be noted that inharmonicity affects the perceived pitch of a string[10]. As it is widely known in psychoacoustics, the human ear responds poorly to verylow frequencies. This implies that most of what is heard from a low string will be thehigher partials, which are heavily affected by inharmonicity, therefore raising the stringsapparent pitch. Despite not being a surprising result, it provides further explanation ofthe stretched tuning, but essentially beating and sharp pitch due to inharmonicity in thepiano are two coins of the same side.

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2.2 Phantom partials

Additionally, another series was discovered in the spectra of piano strings, with an inhar-monicity of about a quarter of the original partials [11]. Conklin was first to investigatethese unexpected peaks and named them phantoms [4]. He discovered that phantomsoccur at frequencies harmonic to inharmonic partials (at 2fn) and at frequencies equal tothe sum of partials (e.g. at fn−1 + fn+2). As a consequence, the frequency doubled peaksform a series known as even phantoms:

fp ≈ 2fn = f0p

√1 +

1

4Bp2 (4)

where fp are the frequencies of the even phantoms, p the integer counting the phantoms(set to p = 2n), f0 and B are as previously defined. It is interesting to note that onlythrough inharmonicity phantoms are distinguishable from the regular partials. The reasonfor the ≈ sign is that more recent research by Bank [12] has shown that even phantomsdo not originate from frequency doubling of, for example, partial 13, but rather from thesums of partials 12 and 14. In this case, the phantoms will have slightly higher frequenciesthan predicted by equation 4.

By recording the output of a string with both a microphone and pickup, Conklinshowed that the pickup, which only responds to transverse string motion, had relativelylow amplitude phantom partials. This provided evidence that the additional peaks involvelongitudinal vibrations, which are excited by the original transverse motion.

A successful theory was derived by Naganuma [13] and Takasawa [14] in 2000, bytaking into account the variation of tension in the string. This means that the tension isnot constant when solving the transverse wave equation, as it depends on the position, x.The equation may be solved by decomposing the tension into a constant and a fluctuationterm depending on the local elongation (longitudinal waves). Using this substitution, theoriginal ideal string equation has two extra terms: One for the aforementioned stiffnessand the other generated by longitudinal elongation of the string. The latter term explainsthe generation of phantoms in the spectrum.

This analysis is based on the assumption that the longitudinal motion is excited bythe initial string displacement, more recent research by Bank in 2005 [12] examined thetransverse to longitudinal coupling and found that longitudinal motion is continuouslygenerated by transverse waves. A full analysis has the tendency to be fairly involved, asit would not only include energy transfer between the two transverse modes, transverseto longitudinal coupling, but also longitudinal to transverse coupling and the impedanceof the bridge and is still an active area of research.

The acoustical implications of phantoms have been studied by Bank in 2010 [15] byresynthesizing real piano tones with and without phantoms. Listeners have describedthe tone with the phantoms as more realistic, “livelier”, but also as “out of tune” and“distorted”. In comparison, the tone without the phantoms was found to be softer andduller, but it needs to remembered that all of these effects are rather subtle.

2.3 Interactions among strings and soundboard

Fletcher’s equation was derived by assuming perfectly rigid supports, i.e. setting thedisplacement of the string to zero at boundaries 0 and L. Obviously, in a real instrument,

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the strings are coupled to a mobile soundboard, which not only affects the timbre andtone duration but also the frequencies of the partials [16, 17]. As shown by Kobayashi[18] in 2010, by introducing a simple spring mass system at one end of the string (inthis case the bridge of a guitar) instead of the rigid boundary, the influence of bodyresonances can be examined. The simulations showed that this results in partials beingbend towards the soundboard resonance, such that around this frequency perturbationsin the inharmonicity are visible. However, as this approach is still novel, theory andexperiment do not yet match, but it does show that resonances can affect single partialsin the low frequency regime. These soundboard perturbations will therefore manifestas random errors when fitting Fletcher’s equation, and are not expected to affect themeasured B coefficient greatly, provided a large number n of partials is given for theanalysis.

Apart from modifying the eigenfrequencies fn of the string, coupling to the soundboardis of particular interest for piano designers, as it affects both the loudness and sustain(tone duration) of the instrument. A commonly observed characteristic of pianos is thecompound decay of strings [19]. As the name suggests, strings decay with two distinctrates, resulting in a large initial decay followed by a much slower final decay rate. Thisphenomenon originates from the two modes of polarisation, see Figure (2).

Figure 2: Vertical and horizontal polarisations of a string (showing first mode, f1, forboth).

Note that the polarisations do not necessarily have to be vertical or horizontal tothe soundboard, but are a set of orthogonal eigenstates. When the hammer strikes thestring, the initial vibration is going to be vertical to the soundboard. Over time, thestring will transfer energy to the horizontal polarisation, parallel to the soundboard.In this direction, less energy is being transferred (the soundboard is stiffer), resultingin a slower decay. Loosely speaking, the vertical mode ensure loudness, whereas thehorizontal mode provides sustain, a compromise of both is usually desired, explaining theimportance of the soundboard when designing pianos. As an aside, there have been pianosdesigned that only allow their strings to vibrate vertically. In the Stuart Piano [20], this is

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achieved by a bridge agraffe, which bends the strings vertically (akin to a guitar bridge, asopposed to the usual horizontal bridge construction), however, apart from manufacturer’sspecifications, little evidence is available for this ensuring only vertical coupling.

This exchange of energy between modes of polarisation is continuous for the durationof the note. It is interesting to note that the two modes are often of slightly differentlength: On guitar strings, research by Woodhouse [3] resulted in additional peaks inthe spectrum that were to some extent dependent on the plucking angle of the string.When plotted on a sonogram, a plot that shows the variation of the spectrum with time,the amplitudes of the partials exhibited a superimposed sinusoidal behaviour. Throughexamining the geometry of the frets it became clear that this originated from the differentlengths of the two modes of polarisation.

2.4 Nonuniform overwound strings

When looking at strings, may it be on a guitar or a piano, two types are encountered:Plain strings, essentially steel wires, are used for the treble, high frequency range, whereaswrapped strings are used in the bass region. The winding is needed to add the requiredmass for low frequencies without making the strings overly stiff.

While for plain strings, Fletcher’s equation has been shown to accurately predict theinharmonicity [16], he concluded that for wrapped strings, the stiffness is mostly due tothe core, with wrapping primarily adding mass. Experimental data confirms that thisassumption is roughly valid, manifesting in a discontinuity in the inharmonicity whengoing from plain to wrapped strings. A number of correction terms for the B coefficientin equation (2) were proposed, methods by Roberts and Sanderson did not prove to beaccurate [16]. However, these theories leave out a crucial aspect of piano strings, namelyits nonuniformity. On a piano, at the very end of each wound string there is a smallportion of plain string, defined as a1 in Figure 3, which not just affects the B parameter,but results in a behaviour that is not explicable by Fletcher’s equation.

Figure 3: Diagram of a wound piano string. The amount of plain string is defined as a1,the length of the wound part is a2.

An accurate theory was derived by Chumnantas [2], by taking into account the non-linear mass density. This increases the complexity of the problem, once a number ofparameters is known, the partials fn can be calculated by solving the following transcen-dental equation (5):

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(µ211

µ222

+ 1

)(µ212

µ221

+ 1

)(µ11 tanh(µ21a2)

µ21 tanh(µ11a1)+ 1

)(µ12 tan(µ22a2)

µ22 tan(µ12a1)+ 1

)−(µ211

µ221

− 1

)(µ212

µ222

− 1

)(µ11 tan(µ22a2)

µ22 tanh(µ11a1)+ 1

)(µ12 tanh(µ21a2)

µ12 tan(µ12a1)+ 1

)= 0 (5)

where:

µij =

√( Ti2(QSk2)i

)2

+ (2πfn)2ρi

(Qk2)i+ (−1)j

Ti2(Qk2)i

1/2

(6)

where i, j = 1, 2.where a1 and a2 are the lengths of plain and wound portions of the string (Figure 3),

all terms with subscript 1,2 correspond to the respective part of the string, k is the radiusof gyration, all other symbols have been previously defined.

Clearly, equation (5) does not include a simple inharmonicity coefficient, due to itsnature, it is difficult to understand the effect of nonuniformity. Computer simulations willbe given in the results section of this report, but the important parameter is the ratio a1

a2(plain over wound portion of string): This value determines the departure from Fletcher’sequation (at a1

a2= 0 the original behaviour is recovered). As opposed to soundboard

perturbations, nonuniformity will therefore manifest as a systematic error in the fit.This phenomenon has been repeatedly observed in literature (but rarely identified), forexample Anderson writes in 2005 [10], “[each fit has] a slight tendency to underestimatethe lower partials and overestimate the higher partials”, which is precisely the effect ofnonuniformity.

3 Method

The method for acquiring the raw data was simple: The strings of the piano or guitar wereplucked at a position close to the end while the sound was recorded using a microphone.The instruments used were a 19th century Broadwood grand piano, a Fender AmericanSeries Stratocaster (electric guitar) and a Martin 00-15M acoustic guitar.

A 1 mm plectrum was used to displace the string, as the piano hammer did not providesignificant amplitude for the upper mode frequencies.1 On a more practical note, onlywith a plectrum it is possible to excite a single string for the keys where the hammerwould hit three strings at once2, which is necessary for a sound analysis.

The reason for plucking the string close to the end is to minimise the amount ofmissing or low amplitude partials, i.e. to avoid the plucking position from coincidingwith a nodal point of a partial. For example, on a guitar, the plucking position as afraction of the total length 30/650 will approximately result in a nodal point of partial 22(given by the inverse of the fraction of the string), therefore the amplitudes are expectedto drop in this region.

1This can be easily justified by listening to the different methods: Using the plectrum results in amuch “brighter” sound.

2One might assume that the three strings are tuned to the same pitch, but this is not universallytrue, moreover, the piano used was far from being in-tune.

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For recording the sound, an Audio Technica ATM 33a microphone was used, placednear the centre of the string, as close as possible to the source to minimise the signalto noise ratio (this was to ensure that the partials can be correctly identified and arenot “swallowed” by background noise). The error from the microphone can be neglected(dynamic range of 20 - 20kHz), as it only manifests in the way the microphone responds todifferent frequencies, i.e. the amplitudes are to some extent dependent on the microphoneused, which is irrelevant for the experiment, as only the positions of the partials andrelative changes of amplitudes are of interest.

In order to analyse the spectrum of the recorded sound, it first needs to be digitalised,an Edirol UA-1EX USB interface converted the signal at 24 bits/ 96 kHz, such that thesampling error from conversion is negligible. The Fourier program [21] in conjunction withMATLAB plotted the spectrum for each respective string, this was also the main sourceof errors: The resolution of the FFT (Fast Fourier Transform) is inversely proportionalto the length of the recorded sound, e.g. a 10s sample leads to a resolution of 0.1 Hz,where each sample has an error of 0.05 Hz. However, in practice, even higher resolutionsdid not result in a lower error in each measurement, as the peaks did not resembleideal delta functions, but have finite width, which made it difficult to determine themaximum/position of the partial. The error was therefore set to 0.1Hz. For the guitarstrings, the spectra were analysed using Audacity, which resulted in a larger error (2.5Hz),due to the unavailability of the MATLAB software in the early stages of the project (Assensible results were obtained, the data was not reanalysed). An unsuccessful attempthas been made to write a JAVA code for identifying the partials in the spectrum.

To confirm the origins of different kinds of peaks in the spectrum the same string wasrecorded both with a microphone and an electrostatic pickup. Due to the nature of thepickup, it is only sensitive to transverse vibrations, but its response is far from linear,such that no precise analysis was possible.

In order to compare the experiment to the theory, the length and diameter of thestrings were measured. Vernier scales allowed a precision of 0.02 mm in the diameter,due to the construction of the piano bridge, the uncertainty in the length was estimatedto be 1 cm. For Young’s modulus and the density literature values were taken. Thetension was found by using equation (1), assuming a negligible inharmonicity of the firstmode. A complete table of all values used is given in Appendix A.

In order to study the effect of soundboard resonances a monochord was used, whichis a rigid steel bar with a single piano string that can be adjusted to a variety of tuningsand lengths. Also, the mobility of the monochord and piano soundboard were measuredby hitting it with a hammer: The resulting output was not normalised, as for this itwould have been necessary to measure the impact force of the hammer, but neverthelesscould be used to find resonances of the soundboard.

As it was realised that the plain portion of an overwound string is significant, anexperiment with a guitar string and a variety of different fractions of bare string wasperformed. For comparing the experiment to the theory, numerical calculations of Chum-nantas’ equation (5) with Maple provided the eigenfrequencies fn, the full code can befound in the Appendix.

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4 Results & Discussion

4.1 Inharmonicity of the Piano

In order to provide a sensible graphical representation of the measured frequencies of thepartials, Fletcher’s equation (2) was linearised by plotting f 2

n/n2 versus n2. The gradient

is then equal to Bf 20 , with an intersection of f 2

0 . A weighted least squares fit using IgorPro (scientific graphing and data analysis program) was used to find the parameters. Thegraphs were plotted using Microsoft Excel.

From the fitted parameters, B and f0 were determined, the error of f0 was given by:

σf0 =2

f0σf20 (7)

And the error in B was found using:

σB = B

√σ2Bf20

(Bf 20 )2

+σ2f20

(f 40 )

(8)

Figure 4 shows a typical set of data, displaying partials 1-42 of string number 15(overwound), where string number 1 corresponds to the lowest key on the piano. Theobtained value for number 15 was B = 7.92(1) ∗ 10−5.

Figure 4: Typical set of data, taken from string number 15, linearised. A least squaresfit gives the values of the intersection f 2

o and gradient f 20B.

9

The plot of the residuals (Figure 5) provides a quantitative account on the qualityof the fit. In general, amongst overwound strings, the lower partials appear consistentlylower than predicted by the fit, while the higher partials are overestimated. This is causedby the nonuniformity of the string (more information will be provided subsequently).

Figure 5: The residuals of the fit from string number 15 show that the uncertainties forthe upper partials appear underestimated.

Once the fitted parameters have been found, a number of strings were compared tothe theory. For the plain strings, excellent agreements have been found, a prime exampleis string number 18, which has a measured B coefficient of 1.082(2)∗10−4 and a calculatedvalue of 1.081 ∗ 10−4. Other results agree within 1% or 3σ.

For the wound strings, it was found that the winding contributes stiffness as well asmass. Theoretical predictions using only the stiffness of the core string resulted in valuesthat were 8-12% lower than the measured inharmonicity.

It should also be noted that the calculated inharmonicity had an uncertainty orders ofmagnitude larger than the error in the measured value (See appendix A). This originatesfrom the measured diameter of the strings, Vernier scales only allowed a precision of 0.02mm. A solution would be to acquire the precise manufacturer’s specifications, a typicalproducer claims a wire diameter tolerance of ±0.1µm (this would lead to a 20 fold increasein precision).

By extending the analysis across the whole spectrum of the piano, the variation ofthe B coefficient was studied. This is important, as the inharmonicity of a piano shouldvary smoothly, otherwise the instrument is difficult to tune [16]. Figure 6 shows theinharmonicity against string number (essentially key) for strings 1-33.

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Wound strings Plain strings

Single Doubled Tripled

Figure 6: Variation of B parameter for piano strings 1-33. The error bars are smallerthan the symbols used.

A striking pattern is visible: Every time there is a discontinuity in the inharmonicitycoefficient B, there is also a discontinuity in the scale of the piano (not displayed). Thishappens at the transition from the eight single strings (one string per key) to the threedoubled strings (two strings per key), from the doubled strings to the tripled strings,and at the transition from wound strings to plain strings. This is in agreement withliterature values of other pianos. However, despite there being a discontinuity in the scale(higher key but longer scale) at these points, this is not the reason for the increase ininharmonicity. In fact, increased length should lead to a smaller B coefficient, appealingto equation (3): B = π3Qd4

64L2T.

To understand this phenomenon, it is easiest to consider Fletcher’s original assump-tion, stating that the stiffness of wound strings is primarily due to the core, with thewinding mostly adding mass. At the transition from wound to plain strings, measure-ments showed a much larger diameter for the plain string than for the core of the adjacentwound string, therefore explaining the increased inharmonicity.

For the discontinuity in the value of B between the single and doubled wound strings,a close examination of the strings revealed that the core wire remained almost constant(0.06(2)mm thinner) while the diameter of the whole string (including winding), changeddrastically (0.82(2)mm thinner). As mentioned before, the theoretical value of the inhar-monicity of wound strings is determined by using the diameter of the core wire, but thethinner winding does matter: Equation (1), relating fundamental frequency to tensionand length, is dependent on a linear mass density term (cross section ∗ density). Inthis case, the reduced cross sectional area leads to a reduced tension, which then in turn

11

requires the B coefficient to rise. The same behaviour can be observed when going fromthe doubled to tripled strings.

To avoid this, an easy solution would be to ensure smoothly varying string gauge inthe bass range. However, this would entail a very low inharmonicity for the last woundstring, resulting in a large discontinuity between wound and plain, which might not bebeneficial for the tunability of the instrument. With the current setup, the inharmonicityis kept within a certain level for the wound strings.

4.2 Inharmonicity of different guitars

By extending the analysis to guitar strings, the difference between plain and overwoundstrings becomes clear. Figure 7 shows the values of B obtained from the acoustic andelectric guitar.

Figure 7: Variation of B parameter for acoustic (circles) and electric (triangles) guitar.

While for most strings the two guitars exhibited similar inharmonicity, at string num-ber 4 (G3) a large difference was observed. This originates from the strings used: It iscustomary to have four overwound strings on acoustic guitars, as opposed to just threeon an electric guitar, which shows the lower inharmonicity of overwound strings.

Again, from the parameters given by the manufacturer, comparisons between theoryand experiment showed a good agreement with the plain steel strings, limited by theprecision of the method (Audacity). As an example, for the B3 (fifth) string on theelectric guitar, the experimental value was found to be B = 4.4(4) ∗ 10−5, while thetheory predicted B = 3.93 ∗ 10−5, such that agreement within 1.1 sigma or 10 percentis given (Note that the values for Youngs’ modulus and density were taken for standard

12

steel, as opposed to the nickel plated steel used for guitar strings). A complete descriptionof all values used for obtaining the theoretical inharmonicity is given in appendix A.

It is interesting to note that despite the inharmonicity in guitars being of the sameorder of magnitude as for pianos, stretched tuning are generally not encountered. Thismay simply be because of the much smaller range available on six string guitars, makingthe inharmonicity a more subtle effect. However, a number of professional guitaristsprefer to tune by ear using harmonics [22], which may result in a slightly stretchedtuning - further work in this area is necessary to verify these assumptions.

4.3 Decay rates and polarisation

Figure 8 shows the decay rates of the first three partials of string number 57. This plotwas created by using 0.1s intervals of the FFT. The first mode is at about 760Hz, apartfrom the partials, the plot also displays soundboard vibrations.

0

0.5

1

1.5

2

2.5

0

500

1000

1500

2000

2500

0

10

20

30

40

50

Time (s)

Decay rates of first 3 partials of string #57

Frequency (Hz)

Am

plit

ud

e (

dB

)

Figure 8: Sonogram of string number 57, using FFT of 0.1s

As visible, only the third partial shows a linear decay, f1 and f2 exhibit a superimposedsinusoidal behaviour. Especially the first mode is a convincing example of the compounddecay described in the theory. A large initial decay rate is seen up to about 0.3s, indicatinga vertical polarisation. After 0.3s energy is being transferred to the horizontal mode,parallel to the soundboard, resulting in a much slower final decay rate. To further examinethis effect, a plot of a short FFT from 0-0.5s superimposed with the spectrum using thefull length of the recording is conclusive (Figure 9).

13

44

46

48

50

52

54

56

58

60

62

756 757 758 759 760 761 762 763 764 765 766

Ampl

itude

(db)

Frequency (Hz)

9 seconds FFT0.5 seconds FFT

Figure 9: Graph showing the first partial in detail for 0.5s FFT and the full 9s FFT

By increasing the resolution of the Fourier transform, the single peak from Figure 8splits into two maxima, about 1 Hz apart. This shows that the two modes of polarisationare of different length, explaining the observed “beating” behaviour in the decay.

4.4 Phantoms

Here the analysis was confined to even phantoms, as they are expected to form a similarseries to the partials, with a factor of 1

4B instead of B. This made it possible to compare

the fit of the phantoms with the fit of the partials. In Figure 10 the phantoms areplotted along with a fitted function and a calculated line using the B parameter from thetransverse vibrations.

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Figure 10: Example data for even phantoms. Error bars are smaller than the symbolsused.

Four piano strings were used for this analysis, all of the fits for even phantoms resultedin a larger value than the expected 1

4B, ranging from 1

3.9(6)B to 1

3.8(3)B. Although within

3σ, consistently larger values support the more recent theory by Bank rather than Con-klin’s original doubled frequency approach. As outlined in the background section, theparents of the even phantoms are e.g. f12+f14 rather than 2f13, such that the predictionswith equation 4 (1

4B) will be underestimated. This resulted in a larger parameter, and

therefore shows that the doubled frequency approach is only an approximation.

4.5 The Monochord and soundboard perturbations

In order to quantify the effect of the piano soundboard, experiments with the Monochordwere performed. Contrary to expectations, the monochord showed strong deviationsfrom Flechter’s equation in the low partial regime. Figure 11 shows the partials fn/(nf0)against the partial number n, this method allows to examine the quality of the fit. Asvisible, the first few partials are consistently lower than the fit predicts, a similar, butless pronounced deviation was found on the equivalent piano string.

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Figure 11: Lower partials of the monochord string show deviation from the fitted function.

The most ostensible difference between the monochord and the piano string is thesoundboard/steel block, the mobility of the monochord (i.e. response to different fre-quencies) is given in Figure 12, along with partials 1-18.

Figure 12: Monochord mobility with the measured partials (circles) showing a clearmaximum 240.2(1) Hz, coinciding with the 7th partial.

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This graph shows a remarkable maximum at 240.2(1)Hz, almost exactly coincidingwith the 7th partial of the previous measurement. Note that due to the nature of themethod (hitting it with a hammer), the amplitudes of the frequencies are not normalised.To test the hypothesis of this soundboard perturbation, the monochord was tuned to ahigher frequency, such that the 7th partial was shifted by 9.6(1)Hz to avoid the depictedresonance. Figure 13 shows how the higher tuning affects the quality of the fit.

Figure 13: Monochord tuned to a higher frequency, lower partials are still overestimatedby fit.

Comparing the two tunings, the deviations from Fletcher appear unchanged, withinerror bars, the first 8 partials are again consistently lower than predicted by the fit.Even the 7th partial, which coincided with a resonance, did not show a notable change.While resonances certainly affect the frequencies of the partials, they were not primarilyresponsible for this deviation. Therefore, instead of looking for external causes, theexplanation lies in the string itself.

4.6 Nonuniform overwound strings

As outlined in the theory, for the case of the nonuniform string, Fletcher’s equation isonly an approximation (Note that usually all overwound piano string are nonuniform!).Nonuniformity refers to the plain portion of the wound string, and the fraction a1/a2(plain/wound portion) is the crucial parameter. Figure 14 shows the results obtained forthe guitar string with a1/a2 = 5.83(3)%, along with a fit of Fletcher’s equation and thevalues obtained by numerical methods using Chumnantas’ theory.

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Figure 14: “Linearised” data from guitar string with high nonuniformity ratio, along withfitted function (Fletcher, solid line) and theory by Chumnantas (dotted line).

The graph shows an excellent fit of Chumnantas’ theory and sheds light on the limitsof Fletcher’s equation. For further analysis, Figures 15 and 16 show the residuals of boththeories.

Figure 15: Residuals of the previous graph, showing the characteristic parabolic shapewhen Fletcher is fitted to a nonuniform string.

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Figure 16: Residuals of the previous graph, Chumnantas’ theory provides an excellentfit.

Figure 15 shows the characteristic ‘parabolic’ shape of the residuals, i.e. the lowerpartials are underestimated and upper partials are overestimated by the fit. This trendhas been seen throughout the analysis of the wound piano strings, though it was not asconvincing due to the smaller fraction of bare string (Compare with Figure 5, residualsfrom piano string 15). Thus, nonuniformity provides the systematic error that has beenseen in the monochord and wound piano strings.

Figure 16 supports Chumnantas’ theory, an excellent fit to the data has been achieved,rendering the uncertainties overestimated. This high accuracy would therefore permit adetailed study of the effects of the soundboard.

To further quantify the effect of nonuniformity, computer simulations using Maplewere extended to a variety of different a1/a2 ratios from 0-8%, the same parameters wereused (guitar string). Figure 17 displays the results.

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Figure 17: Maple simulations for nonuniformity a1/a2 ratios from 0-8%, showing a sig-nificant departure from Fletcher’s straight line.

If the ratio a1/a2 is zero, Fletcher’s equation is recovered and a straight line is ob-tained. As the fraction of bare string increases, so does the inharmonicity of the string(larger gradient), until at very high portions of plain string, a straight line fit wouldactually result in a lower value for B. However, these large non uniformities are far fromreality, the highest a1/a2 ratio encountered on the Broadwood piano was about 1.5%.3

Looking at Figure 17, it also becomes immediately clear why the fit overestimated thelowest partials on the monochord (Figure 13). At around 2% nonuniformity, which is ap-proximately the a1/a2 ratio found on the monochord, the upper partials form a straightline, whereas at the low end they appear to be bend away and lie lower than expected.

To quantify how the B coefficient varies with nonuniformity, Fletcher’s equation wasfitted to both the calculated and measured partials, shown in Figure 18. It is not entirelysensible to fit a straight line to the data, which results in high uncertainties for theobtained inharmonicity. For clarity, error bars were only included in the observed data.

3On a subjective note, it may be said that a high nonuniformity (> 5%) results in a very unpleasantsound.

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Figure 18: Inharmonicity against nonuniformity parameter a1/a2. White circles are the-oretical values, full circles the experiment.

As mentioned, this analysis is somewhat unsound, but what it does show is hownonuniformity affects the standard inharmonicity measurement. A realistic a1/a2 ratioof 1.5% (as seen on the piano) can increase the measured inharmonicity by about 1*10−5.For comparison, the discontinuity between plain and wound strings in the B value wasof order 4*10−5.

A better way to analyse the effect of nonuniformity is to look at the frequency of asingle partial. Figure 19 shows the inharmonicity in cents for partial 16 against the a1/a2ratios. As 100 cents are one semitone, a high nonuniformity can therefore increase thepitch of a partial by about half a semitone. A quick fit to this data showed that theinharmonicity in cents is roughly proportional to the square of the a1/a2 ratio.

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Figure 19: Inharmonicity in cents of 16th partial.

As every problem is an opportunity in disguise, this could be used to ensure a smoothlyvarying inharmonicity for a better tunability: The obvious candidate would be to grad-ually reduce the amount of bare string on the bass strings leading to a1/a2 = 0 on thevery lowest string, as this could ‘even out’ the ubiquitous inharmonicity rise. This mightbe very efficient on an upright piano, which, due to the shorter scale, will have a com-paratively high a1/a2 nonuniformity ratio (Piano string winders tend to increase the barelength with each string replacement to have a “safety margin”, which can be as large as4 cm [23]. This would result in a a1/a2 ratio of about 4%, drastically increasing inhar-monicity). Another obstacle for tuners is the inharmonicity discontinuity between plainand wound strings, gradually increasing the amount of bare string of a few strings up tothe transition may result in a more uniform sound.

5 Conclusion

The inharmonicity of various guitar and piano strings was found, Fletcher’s equationproved to be accurate for plain strings. In the case of wound strings, nonuniformitymanifests as a systematical error, increasing the inharmonicity with the fraction of barewire. Observations from decay rates and phantom partials were in agreement with theprevalent theories. The effect of soundboard resonances could not clearly be identified,future work with Chumnantas’ equation and high precision values for the core and windingdiameter could shed light into this area.

22

References

[1] H. Fletcher, “Normal vibration frequencies of a stiff piano string,” J. Acoust. Soc.Am., vol. 36, pp. 203–209, 1964.

[2] P. Chumnantas, C. Greated, and R. Parks, “Inharmonicity of nonuniform overwoundstrings,” Journal de physique, vol. 4, pp. 649–652, 1994.

[3] J. Woodhouse, “Plucked guitar transients: Comparison of measurements and syn-thesis,” Acust. Acta Acust., vol. 90, pp. 945–965, 2004.

[4] H. A. Conklin, “Generation of partials due to nonlinear mixing in a stringed instru-ment,” J. Acoust. Soc. Am., vol. 105, pp. 536–545, 1999.

[5] L. E. Kinsler, A. R. Frey, A. B. Coppens, and J. V. Sanders, Fundamentals ofAcoustics. John Wiley & Sons, Inc., 4th ed., 2000.

[6] J. S. L. Rayleigh, The Theory of Sound. Dover Publications, 2nd ed., 1954.

[7] J. C. Bryner, “Stiff-string theory: Richard feynman on piano tuning,” Physics Today,vol. 12, pp. 47–49, 2009.

[8] A. Reinholdt, E. Jansson, and A. Askenfelt, “Analysis and synthesis of piano tones,”J. Acoust. Soc. Am., vol. 81, no. S1, p. S61, 1987.

[9] N. H. Fletcher and T. D. Rossing, The Physics of Musical Instruments. SpringerVerlag, 1998.

[10] B. E. Anderson and W. J. Strong, “The effect of inharmonic partials on the pitch ofpiano tones,” J. Acoust. Soc. Am., vol. 117, pp. 3268–3272, 2005.

[11] R. J. Alfredson, “Fourier transform methods for analysing the sound of a piano,”Acustica., vol. 39, pp. 130–132, 1964.

[12] B. Bank and L. Sujbert, “Generation of longitudinal vibrations in piano strings:From physics to sound synthesis,” J. Acoust. Soc. Am., vol. 117, pp. 2268–2278,2005.

[13] D. Naganuma and I. Nakamura, “Numerical simulation of a piano string vibrationregarding transversal wave velocity,” Proc. Acoust. Soc. Jpn., pp. 575–576, 2000.

[14] Y. Takasawa and I. Tokuhiro, “On the inharmonicity in string vibration,” Acoust.Soc. Jpn., vol. 19, no. 3, pp. 27–32, 2000.

[15] B. Bank and H. Lehtonen, “Perception of longitudinal components in piano stringvibrations,” J. Acoust. Soc. Am., vol. 128, pp. 3268–3272, 2010.

[16] H. A. Conklin, “Design and tone in the mechanoacoustic piano. part iii. piano stringsand scale design,” J. Acoust. Soc. Am., vol. 100, pp. 1286–1298, 1996.

[17] K. Ege and A. Chaigne, “End conditions of piano strings,” International Symposiumon Musical Acoustics, Barcelona., 2007.

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[18] T. Kobayashi, N. Wakatsuki, and K. Mizutani, “Inharmonicity of guitar string vi-bration influenced by body resonance and fingering position,” Proceedings of theInternational Symposium on Music Acoustics, Sydney, p. 43.40.CW, 2010.

[19] G. Weinreich, “Coupled piano strings,” J. Acoust. Soc. Am., vol. 62, no. 6, pp. 1474–1484, 1977.

[20] W. Stuart, Stuart and Sons. http://www.stuartandsons.com/, Nov. 2011.

[21] D. Skulina, The Fourier program. Edinburgh University, 2011.

[22] S. Ryan, “Alternate guitar tunings: an introduction,” Guitar Techniques Magazine,vol. 8, p. 34, 2009.

[23] D. Sanderson, Sanderson Piano Services. http://www.sandersonpiano.com/, Nov.2011.

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A Theory and Experiment - Complete data

E4 B3 G3 D3 A2 E2

Length (cm) 65 65 65 65 65 65

diameter (cm) 0,03 0,03302 0,043 0,076 0,094 0,122

core (cm) 0,04 0,042 0,05

Young's (g/(cm*s2)) 2,00E+12 2,00E+12 2,00E+12 2,00E+12 2,00E+12 2,00E+12

density (g/cm3) 7,85 7,85 7,85 7,85 7,85 7,85

crosssection (cm2) 0,000707 0,000856 0,001452 0,004536 0,0069398 0,0116899

Tension(g*cm)/s2

10212137 6987209 7401083 13182472 11140055 10427822

first mode (Hz) 330 248 196 148 110 82

B (calc) 1,82E-05 3,90E-05 1,06E-04 5,80E-04 1,61E-03 4,87E-03

B (core) 1,82E-05 3,90E-05 1,06E-04 4,45E-05 6,41E-05 1,37E-04

B (measured) 2,10E-05 4,40E-05 1,27E-04 5,50E-05 8,70E-05 1,60E-04

Uncertainty: 1,00E-06 4,00E-06 7,00E-06 4,00E-06 4,00E-06 4,00E-06

Difference % 13,38 11,32 16,58 19,02 26,37 14,09

Within # sigma: 2,81 1,25 3,01 2,62 5,74 5,64

#5 #8 #9 #11 #12 #18 #23

Length (cm) 178,5 175,1 173,782 172,552 173,01 180,58 139,91

diameter (cm) 0,384 0,346 0,264 0,239 0,184 0,122 0,119

core (cm) 0,136 0,126 0,12 0,119 0,114 0,122 0,119

Young's (g/(cm*s2)) 2,E+12 2,E+12 2,E+12 2,E+12 2,E+12 2,E+12 2,E+12

density (g/cm3) 7,85 7,85 7,85 7,85 7,85 7,85 7,85

crosssection (cm2) 0,115812 0,094025 0,054739 0,044863 0,0265904 0,0116899 0,011122

Tension(g*cm)/s2

1,34E+08 1,43E+08 9,16E+07 9,42E+07 6,15E+07 6,08E+07 6,08E+07

first mode (Hz) 34 39,8 42 47,4 49,6 71,3 94,3

B (calc) 4,94E-03 3,16E-03 1,70E-03 1,13E-03 6,03E-04 1,082E-04 1,633E-04

B (core) 7,77E-05 5,56E-05 7,27E-05 6,93E-05 8,89E-05 1,082E-04 1,633E-04

Uncertainty: 4,86E-06 3,74E-06 5,19E-06 5,02E-06 6,87E-06 8,58E-06 1,34E-05

B (measured) 8,45E-05 6,29E-05 8,43E-05 7,78E-05 1,01E-04 1,081E-04 1,623E-04

Uncertainty: 2,00E-07 2,00E-07 3,00E-07 3,00E-07 2,00E-07 2,00E-07 3,00E-07

Difference % 8,08 11,68 13,81 10,99 11,96 0,07 0,61

Within # sigma: 34,14 36,74 38,81 28,49 60,38 0,39 3,30

# Theory sigma: 1,40 1,96 2,24 1,70 1,76 0,01 0,07

Wound Plain

Plain Wound

Electric Guitar

Piano

The error in the theoretical B value was given by:(σBB

)2=

(4σdd

)2

+ 8(σLL

)2+

(8σddπ

)2

+

(2σf1f1

)2

(9)

where σd = 0.0002 cm, σL = 1 cm and σf1 = 0.1 Hz.

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B Maple code for nonuniformity simulations

> >

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Define equation to solver e s t a r t ;

e q n : = ( u 1 1 ( f ) ^ 2 + u 2 2 ( f ) ^ 2 ) * ( u 1 2 ( f ) ^ 2 + u 2 1 ( f ) ^ 2 ) * ( u 1 1 ( f ) * t a n h ( u 2 1 ( f ) *a 2 ) + u 2 1 ( f ) * t a n h ( u 1 1 ( f ) * a 1 ) ) *( u 1 2 ( f ) * t a n ( u 2 2 ( f ) * a 2 ) + u 2 2 ( f ) * t a n ( u 1 2 ( f ) * a 1 ) ) - ( u 1 1 ( f ) ^ 2 - u 2 1 ( f )^ 2 ) * ( u 1 2 ( f ) ^ 2 - u 2 2 ( f ) ^ 2 ) *( u 1 1 ( f ) * t a n ( u 2 2 ( f ) * a 2 ) + u 2 2 ( f ) * t a n h ( u 1 1 ( f ) * a 1 ) ) * ( u 1 2 ( f ) * t a n h ( u 2 1 ( f ) *a 2 ) + u 2 1 ( f ) * t a n ( u 1 2 ( f ) * a 1 ) ) ;

Define constants

p1, d1, a1: Density, diameter and length of plain portion (p2,d2,a2 for wound)p 1 : = 7 . 8 5 :

a 1 : = 0 :

a 2 : = 6 5 :

d 1 : = 0 . 0 5 2 :

S 1 : = P i * d 1 ^ 2 / 4 :

Q1 := 2*10^12:

p 2 : = 8 . 9 3 :

d 2 : = 0 . 1 3 6 :

S 2 : = P i * d 2 ^ 2 / 4 :

k : = d 1 / 4 :

d : = ( d 2 - d 1 ) / 2 :

d d : = d + d 1 :

mass1 := Pi*d1^2*p1/4:

mass2 := mass1 + Pi^2*d*dd*p2/4:

f 0 : = 7 7 :

T := 4*(a1+a2)^2*f0^2*mass2:

n : = T :

P1 := mass1:

m := Q1*S1*k^2:

P2 := mass2:

u 1 1 : = ( f ) - > ( ( n ^ 2 / ( ( 2 * m ) ^ 2 ) + 4 * P i ^ 2 * f ^ 2 * P 1 / m ) ^ ( 1 / 2 ) + n / ( 2 * m ) ) ^ ( 1 / 2 ) ;

u 1 2 : = ( f ) - > ( ( n ^ 2 / ( ( 2 * m ) ^ 2 ) + 4 * P i ^ 2 * f ^ 2 * P 1 / m ) ^ ( 1 / 2 ) - n / ( 2 * m ) ) ^ ( 1 / 2 ) ;

u 2 1 : = ( f ) - > ( ( n ^ 2 / ( ( 2 * m ) ^ 2 ) + 4 * P i ^ 2 * f ^ 2 * P 2 / m ) ^ ( 1 / 2 ) + n / ( 2 * m ) ) ^ ( 1 / 2 ) ;

u 2 2 : = ( f ) - > ( ( n ^ 2 / ( ( 2 * m ) ^ 2 ) + 4 * P i ^ 2 * f ^ 2 * P 2 / m ) ^ ( 1 / 2 ) - n / ( 2 * m ) ) ^ ( 1 / 2 ) ;

26

> >

> >

> >

> >

> >

Find solutions

Define increment for solving equationi n c : = 6 0 :

Define number of solutions & create arrayi t e r : = 1 0 0 :

s o l n s : = A r r a y ( 1 . . i t e r ) :

Start loop, solve equation and store into arrayf o r j f r o m 1 t o i t e r d o

t h i s _ s o l n : = f s o l v e ( e q n = 0 , f = ( 1 / 2 ) * i n c * j . . ( 3 / 2 ) * i n c * j ) :

s o l n s [ j ] : = t h i s _ s o l n :

end do;

Get rid of double entries via Maple's set method.S 3 : = { s e q ( s o l n s [ j ] , j = 1 . . 1 0 0 ) } ;

27