Question

If a pluck on a guitar is restricted and you hear a click rather than a steady long tone, how is that a wider frequency bandwidth?

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The guitar and the piano are two of the most popular instruments which allow multiple notes to be played simultaneously.

The acoustic quality of the two instruments are, of course, different, but that is not the focus of this question.

The design of the two instruments affects what and how music are played. Here are some differences I have in mind:

• All semitones are treated equally on the guitar. Guitar chords (other than the open chords) can be moved up and down the fretboard without changing the finger shape. This also allows the player to shift the key easily. On the piano, the same chord takes different shapes in different roots due to the presence of black and white keys.

• Regarding playing multiple notes at the same time, the piano is more flexible than the guitar.

  1. The pianist use ten fingers to play the notes, while the guitarist uses only four (or five if you use the thumb), and at most six notes can be played at the same time (using a bar)

  2. Since you use two hands on the piano, it's easier to play rhythm and melody at the same time.

  3. On a guitar you can only play one note on a string, so the possible voicing of chords is restricted. On the piano you can pretty much play any voicing.

• Strings on the guitar can be bended, for example, to create a vibrating sound. I'm not aware of an equivalent technique on the piano.

• Abstract

  Electric guitar playing is ubiquitous in practically all modern music genres. In the hands of an experienced player, electric guitars can sound as expressive and distinct as a human voice. Unlike other more quantised instruments where pitch is a discrete function, guitarists can incorporate micro-tonality and, as a result, vibrato and sting-bending are idiosyncratic hallmarks of a player. Similarly, a wide variety of techniques unique to the electric guitar have emerged. While the mechano-acoustics of stringed instruments and vibrating strings are well studied, there has been comparatively little work dedicated to the underlying physics of unique electric guitar techniques and strings, nor the mechanical factors influencing vibrato, string-bending, fretting force and whammy-bar dynamics. In this work, models for these processes are derived and the implications for guitar and string design discussed. The string-bending model is experimentally validated using a variety of strings and vibrato dynamics are simulated. The implications of these findings on the configuration and design of guitars is also discussed.

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  Introduction

  The ability to explore micro-tonality and segue between pitches in a continuous manner is one element of guitar playing that sets it apart from other popular instruments where pitch is discrete. Radial pitch shifting or string-bending, the application of force to bend a string from its equilibrium position, and has become an integral part of electric guitar lead playing. Coupled with the huge array of amplification, effects and distortion options, the electric guitar can yield a vocal-like quality in lead playing, allusions to which are often made in popular culture; in Dire Strait's 1979 debut single “Sultans of swing”, songwriter Mark Knopfler refers to a jazz guitarist as being “strictly rhythm, he doesn't want to make it cry or sing”. Eric Clapton's thick guitar tone and use of vibrato is referred to by guitarists as the “woman tone”, which he famously contributed to the Beatles's classic “While my guitar gently weeps”. These are but some examples - An accomplished guitarist's tone and vibrato can be so intrinsic to that player that their idiosyncratic sound is as distinctive as a vocalist's to a trained ear.

  From a physical perspective, the basic acoustics of string instruments have been well studied for centuries. There has been considerable analysis of the physics of violins, practically all of which is equally applicable to guitar. Much of this work is concerns the mechano-acoustic properties of vibrating plates and applications of Chladni's Law [1]. Considerable investigation has been undertaken on the mechanics of acoustic guitar construction [2] and resonances [3], [4], as well as analysis into the tonal effects of removable back-plates [5]. Similarly, other research [6] has examined the acoustics of classical guitars and the effect of bridge design on top plate vibrations. Such analysis is understandably critical for acoustic instruments, but perhaps less for electric guitars which rely on suitable valve or solid-state amplifiers to project their sound. As a consequence, some research pertaining explicitly to the electric guitar has concentrated on the mechanical-electrical properties of electric guitar pick-ups [7]. Other research has examined the challenge of digitally simulating the tonal quality of electric guitars in syntheised instruments [8]–[10]. As guitars provide a tangible example of the many important physical principles, some literature on the subject is dedicated to the pedagogical advantages of using guitars as demonstrations of crucial acoustic principles such as standing waves [11].

  Yet despite the significant volume of research dedicated to guitar acoustics, there appears to be a paucity of research concerning the physics of electric guitar playing and the underlying physical principles of the distinct techniques which influence the instrument's unmistakeable tonality. As the electric guitar has become ubiquetous in rock, pop, metal, jazz and blues, it has developed a range of techniques which further distinguish it from its musical forebears. String-bending and vibrato add much to a guitarist's palette, and as these techniques are heavily influenced by the physical constraints of the guitar and strings used, the underlying mechanics are worthy of analysis. Guitarists also use a wide variety of legatotechniques to articulate their playing; these include hammer-ons, where a fretted string is picked and another one sounded by coming down sharply on it with the fretting hand, resulting in a smoother sound that would result from merely picking both notes. The opposite technique is a pull-off, where a picked note is released and a lower one sounded. A fusion of both these techniques practically unique to the guitar is tapping, where both fretting and picking hands are used to ‘articulate’ a flurry of notes. This is a staple of modern lead guitar playing, popularised by guitar virtuoso Edward Van Halen in the late 1970s. For these techniques, the fretting force required to ‘sound’ a note with or without picking becomes an important limiting factor and influences a player's choice of string and guitar set-up. Unlike more traditional instruments, there is also wide scope for modification and extension of tonal range by using external hardware. One example of this is the vibrato system or “whammy-bar” which many guitars opt for; these are mechanical systems for adding extra vibrato and come in a variety of designs. Rather misleadingly, they are sometimes referred to as tremolo bars, which is an unfortunate misnomer as tremolo is modulation of volume rather than pitch. Such units dramatically increase the sonic scope of the instrument and have been employed by famous players such as Dave Gilmour, Brian May, Steve Vai, John Petrucci, Frank Zappa and Joe Satriani amongst others.

  As the electric guitar has such a wide range of techniques and modifications, it is worthwhile to analyse these factors from first principles, and examine what implications these elements have on guitar design and playing. This work will concentrate on some of the unique principles of electric guitar playing, such as string-bending, vibrato, fretting force and whammy-bar effects. String design is discussed, and string-bending factors experimentally examined. Vibrato and related effects are simulated and discussed.

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  Materials and Methods

  0.1 String bending

  For a stretched string, the fundamental frequency may be derived from the one dimensional wave equation, and is given by

  

  (1)

  where is the length of the vibrating element, is the string tension and the linear density or mass per unit length of the string. The technique of string-bending is the application of a another force perpendicular to the fretboard - an illustration of this in shown in figure 1 where a bending force, , is applied to a string. This applied force causes a slight increase in the vibrating length, which would act to lower the pitch if not for the net increase in string tension due to the extension force acting along it, . This acts to raise the pitch. Assuming the distance from the string to the fretboard is small so that length increases in this plane are negligible, the fundamental frequency of a bent string in terms of the extension force and the bend angle is then

  

  (2)

  

  Figure 1

  Force diagram for a bent string.

  The bend angle is not independent of the extending force, and the relationship between the twain can be explicitly calculated; the string is elongated along its vibrating length when bent, and by Hooke's law the force and extension are related by

  

  (3)

  Hooke's stiffness can be related to the Young's modulus () and cross-sectional area of the string , and (3) can be rewritten in these terms as

  

  (4)

  Thus, the frequency of a bent string is described by first approximation to

  

  (5)

  Strictly speaking, (5) holds when is constant. This is defined as mass () per unit length; Hooke's law states that the string stretches in proportion to the applied force. Defining as the mass per unit length with no force applied, then the complete expression the bent frequency is

  

  (6)

  In practice, bend angles that can be employed by most players with a standard configuration are unlikely to exceed even . Guitar strings tend to be composed of steel or steel alloys and as such can have a Young's Modulus of up to 200 GPa [12], and so equation 6 predicts that even these relatively small bend angles can have a substantial effect on pitch. It is important to note that guitarists can bend either “up” or “down”, yet the net effect is always to raise pitch relative to an unbent string and it is the magnitude rather than the direction of the bend that is relevant. The reason for this is explored in the discussion section. Examples of string-bending are included in the supplementary material for reference (Video S1).

  0.2 Fretting-Force and Bending force

  On a fretted guitar, the string is depressed until it makes contact with the fret marker. The distance between the fingerboard and string is called the “action” of a guitar. Generally, low action is considered preferable as lower action guitars are easier to fret, but if the action is too low the string may buzz due to collisions with fret-markers or the fretboard. The force required to fret is not only important for physically sounding notes, but is also vital for some prominent guitar techniques such as hammer-ons and tapping, examples of which are included in the supplementary material (Video S2). If we denote the action as the distance from the string to the fret-marker and the angle the depressed string makes with the bridge as as depicted in figure 2, then the effective tension on the string can be resolved to yield an expression for fretting force of

  

  (7)

  

  Figure 2

  Fretting force diagram.

  Similarly, the force required to bend a string to the angle as in equation 6 is given by

  

  (8)

  0.3 Vibrato

  Vibrato is a regular pulsating change of pitch, and adds expression to a note. It is remarkably important in guitar music, and can be so utterly unique to different players it can be used in some cases to distinguish them by ear. In electric guitar playing in pop, rock and metal, vibrato is often produced by the player modulating the angle of the string, essentially bending it up and down to vary the pitch. This can be modelled by modifying equation 6 with the bend angle as a function of time, denoted . In this case, the rate of vibrato is given by the first derivative of frequency with respect to time by

  

  (9)

  In classical guitar playing, most vibrato is produced by “axial” vibrato techniques, where the tension is altered on a fretted note by modulating the tension applied so that the bend angle is kept at zero. In this case the variable is tension and the process can be modelled by modifying equation 1 with a time-varying tension for . The vibrato rate is then given by

  

  (10)

  In practice, guitarists will have vastly differing approaches to vibrato - it is not uncommon to hear guitarists describe to a particular vibrato as “wide” or “narrow”, in reference to how much angular or tensional displacement the player is applying. Similarly, other vibratos may be described as “fast” or “slow”, depending on the rate at which the player modulates the vibrato. Like bending, vibrato will not lower the pitch from the resting pitch. Examples of both vibrato types are included in the supplementary material (Video S3).

  0.4 Whammy-bar dynamics

  Whammy-bars come in a variety of designs, and famous models include the Bigsy Vibrato Tailpiece, the Fender Strat Tremolo and the Floyd Rose locking tremolo. These work by changing the string tension with a controlling lever at the bridge. In this case, the vibrato rate can be described very simply by modifying equation 1 to yield

  

  (11)

  where is the force applied to the vibrato arm. It is important to note that on many models tension can be either decreased or increased, with an increase in tension acting to increase pitch and vice versa, encapsulated in the term in equation 11 for an increase and decrease in pitch respectively.

  0.5 Experimental method for investigation of string-bending

  If the expression for the frequency of a bent string given by as derived in (6) describes string bending well, then the expression for fretting force and rate of change of pitch follow from this. To investigate this experimentally, a fixed bridge Gibson Epiphone special was stripped down (Gibson, Nashville, TN) and modified by the addition of of two holding pins at the 12th fret position, approximately 328 mm from the bridge of the instrument. The guitar output was connected to a chromatic tuner and then to an undistorted amplifier. A microphone was placed at the amplifier speaker and output frequency was measured using the Waves software package (Cohortor.org). On the modified guitar, taut strings in the 1st and 6th positions were used as supporting wires to prevent the nut from slipping when the test string was replaced. The test string was loaded in the 5th string position for each test, as depicted in figure 3. The holding pins were placed at 7.5 mm and 11.1 mm respectively above the equilibrium position of the test string as depicted in figure 4, yielding bend angles of 0°, 1.31° and 1.94° approximately. These positions were chosen to give a range of realistic bend angles below the threshold which strings may snap.

  

  Figure 3

  Modified experimental guitar with two bending pegs inserted at 12th fret (a) String in equilibrium position, string unbent.

  (b) String affixed above first fretting peg, a vertical displacement of approximately 7.5 mm from resting position.

  

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  Figure 4

  Results for test string on experimental guitar rig.

  The highest three strings depicted in (a), (b) and (c) had approximately the same Young's modulus, close to that of steel. For composite wound strings depicted in (d), (e) and (f), this was markedly lower and decreasing. Best fits are shown for all strings with the solid line, the the dotted and dashed lines depict best fits if bend displacement is plus or minus 0.5 mm from measured value.

  Ernie Ball Regular Slinky gauge strings (Ernie Ball, California) were then individually loaded in the test rig and tuned to their respective concert pitchs using a chromatic tuner. The pitch of the individual test strings were recorded unfretted, fretted at the 12th fret position and then respectively fretted on the first and second holding pin for each string in the set. The linear density of the strings was determined by cutting the strings into 30cm sections and then weighing them on a calibrated weighing scales. String diameter and area was measured using a digital micrometer screw gauge. The Slinky regular gauge strings were tempered for standard concert pitch for a six string guitar with gauges, with gauges of 46, 36, 26, 17, 13 and 10, corresponding to low E string, A string, D string, G string, B string and high E string respectively. Of these, the 17, 13 and 10 gauge strings were plain high carbon steel wire plated with tin. The 46, 36 and 26 gauge strings had a high carbon steel core and were wound with nickel. Measured properties are shown in table 1. When loaded in the test rig, the tension estimate for each string was obtained by manipulating equation 1. The action distance from test string to fret marker was mm for all string positions along fretboard, yielding small values of so effects in this plane were expected to be negligible.

• The piano has a wider pitch range than the guitar.

Do you agree? What other important differences can you think of? What are the implications of these differences, if any?

17

All of your points are good; one of the reasons guitar became so popular in western culture in the early days, is because its portable.

So it's sort of a travelling minstrel's piano. During the late 19th/early20th centuries in America(post emancipation act), there were lots of travelling musicians who couldn't afford a piano; and even if they could, they couldn't have carried one since most of their travelling was done on foot, they would literally play for a couple of meals and a night under a roof. These are the guys that brought us the earliest forms of blues, like the Delta blues; which were basically the old slave work songs Blues'd up (call and response etc.).

If you scout around you can find old photos of these guys with the most battered looking old instruments you can imagine.

However you want to look at it (poor man's piano/portable piano), the portability and relative cheapness (guys would knock them together out of bits of random wood), is a major difference which helped start the wheel of popular music/culture rolling.