WO2018142142A1 - Strings for musical instrument - Google Patents

Abstract

A set of strings for a musical instrument comprising: at least a first and second string, wherein the first string comprises a core and a winding and the second string comprises at least a core; wherein each string is configured to be held at a selected tension in a musical instrument to produce a tone of a respective different desired pitch; and wherein at least one property of the cores and windings of one or both of the first and second strings is selected such that a change in pitch of the first string matches a change in pitch of the second string for a variation of the string tensions away from the selected tensions.

Description

Strings for musical instrument

Introduction

The present invention relates to a set of strings for a musical instrument, for example a guitar.

Background

A tremolo mechanism allows a change of tension to be applied to all of the strings of an instrument simultaneously to create a pitch change. Known tremolo mechanisms increase or decrease the length of all strings belonging to a set of strings when the mechanism is actuated.

A standard guitar string is constructed to sound a specific pitch when arranged under a particular string tension. In a standard set of guitar strings, the strings have different values of string tension. Known guitar string sets also include "balanced tension" string sets. Standard strings and "balanced tension" strings suffer from the problem that different strings in the set bend through different pitch intervals when a player performs the same action on each string. For example, moving a tremolo arm through a distance causes notes sounded by each guitar string to drift through different pitch intervals thereby giving an out of tune sound for chords when the tremolo arm is away from rest position. Similarly, the force required for conventional string bends through the same intervals on different strings will be different within existing "balanced tension" string sets- Prior art in this field involves significantly modified tremolo system hardware that allows for bending chords in tune. This hardware has never been made available in a form compatible with the instruments of the most popular guitar manufacturers (such as the Fender Stratocaster and Floyd Rose hardware). Also, the only system made commercially available for any length of time (the Steinberger TransTrem system) is very complicated to set up and requires calibrated double ball-end strings, thus requiring both specific hardware and specific calibrated strings.

Similarly, US patent 6875911 describes a tremolo device for a stringed instrument comprising a base adapted to attach to an associated stringed instrument. The tremolo device requires additional hardware to be built into the guitar or existing guitar to be substantially modified. Summary

According to a first aspect of the present invention, there is provided a set of strings comprising at least a first and second string, wherein the first string comprises a core and a winding and the second string comprises at least a core, wherein each string is configured to be held at a selected tension in a musical instrument to produce a tone of a respective different desired pitch, and wherein at least one property of the cores and windings of one or both of the first and second strings is selected such that a change in pitch of the first string matches a change in pitch of the second string for a variation of the string tensions away from the selected tensions.

The selected tension of each string may be tension of the string in a rest configuration.

The musical instrument may comprise a guitar.

The variation of the string tensions may be caused by operation of a tremolo arm mechanism.

The change in pitch of the first string may vary from a change in pitch of the second string by less than 25 cents, preferably by 10 cents or less, for a variation of the string tensions away from the selected tensions that provides a mean change in pitch of the first and second strings of a minor third and/or substantially 300 cents.

The change in pitch of the first string may vary from a change in pitch of the second string by less than 10 percent, optionally by less than 5 percent for a variation of the string tensions away from the selected tensions.

The at least one property may comprise the diameters of the cores of the first and second string.

The at least one property may comprise a value of Young's modulus for each of the cores. The value of Young's modulus for each of the cores may be substantially the same.

The diameters of the cores of the first and second strings may be substantially equal. The diameters of the cores of the first and second strings may vary by less than 20%, optionally, the diameters of the cores may vary by less than 10%, further optionally the diameters of the cores may vary by less than 5%.

Substantially equal cores may be selected to provide substantially equal engineering strains for the first and second strings and substantially equal selected tensions.

Substantially equal cores may be selected to provide engineering strains for the first and second strings that vary by less than 15%, optionally by about 12% or less, optionally by less than 5% and optionally by about 2% or less.

The first and second strings may be configured such that the engineering strain of at least the first and second strings is substantially the same when the strings are held at said selected tensions to produce the desired pitches.

The first and second strings may be configured such that the engineering strain of at least the first and second strings varies by less than 15%, optionally by about 12% or less, when the strings are held at said selected tensions to produce the desired pitches.

The first and second strings may be configured such that the engineering strain of at least the first and second strings varies by less than 5%, optionally by about 2% or less, when the strings are held at said selected tensions to produce the desired pitches.

The at least one property may be selected such that the selected tensions to produce the desired pitches are substantially equal for the first and second strings.

The at least one property may be selected such that the selected tensions to produce the desired pitches for the first and second strings vary by less than 12%, optionally, by less than 10%, optionally by less than 5%. The at least one property may comprises a ratio of string mass per unit length to core mass per unit length for the string. String mass is a sum of core mass and winding mass. The ratio of string mass per unit length to core mass per unit length for the strings may be selected so that the engineering strains of the first and second strings are substantially the same.

At least one of the cores of the first and second strings may have one of a circular, hexagonal, pentagonal cross-section or any other suitable shape, or may have a braided core.

The change in pitch may be characterised by the relative change in frequency between the tone produced at the selected tension and the tone produced away from the selected tension. The pitch of each string may be dependent on a ratio of tension to mass per unit length for the string. Each string may be configured such that the variation in tension caused by operation of the tremolo arm mechanism is primarily determined by properties of the core and the engineering strain applied in tuning the instrument. The mass of the windings per unit length of the string for the first and second strings may be selected to provide the desired different pitches for the different strings when under the selected tension.

The matching of the changes in pitch for the strings may be such that for each of the first and second strings the ratio of the change in fundamental frequency to the fundamental frequency in response to said variation of the string tensions is substantially the same

The matching of the changes in pitch may be approximate or exact. The changes in pitch may be represented by an interval. The interval may be relative to a reference pitch. Matching of changes in pitch may be such that the difference in interval for the first and second string is less than 10 % of the overall pitch change, preferably less than 8 %. The material of the core of the first string may be the same as the material of the core of the second string, and/or the material of the winding of the first string may be the same as the material of a winding of the second string. At least one of the cores may comprise steel. At least one of the windings may comprise nickel or nickel-plated steel.

The first and second strings may have a core diameter in a range 0.1 mm to 4 mm. The set of strings may comprise at least one unwound string having a string diameter that is matched to a core diameter of the first string and/or second string.

The string diameter of the at least one unwound string may be substantially the same as the core diameter of the first string and/or second string.

The set of strings may comprise at least a third string comprising a core and a winding and configured to be held at a selected tension in the musical instrument to produce a tone of a different desired pitch to that of the first and second strings. At least one property of the cores and windings of the first and/or second and/or third string may be selected such that a change in pitch of the third string matches the change in pitch of the first and second strings for a variation of the string tensions away from the selected tensions.

The third string may have properties and characteristics in common with the first and second strings. For example, the at least one property may comprise the diameters of the cores of the first, second and third string. The at least one property may comprise a value of Young's modulus for each of the cores. The value of Young's modulus for each of the cores of the first, second and third string may be substantially the same. The diameters of the cores of the first, second and third strings may be substantially equal. The first and second strings may be configured such that the engineering strain of at least the first, second and third strings is substantially the same when the strings are held at said selected tensions to produce the desired pitches. The at least one property may be selected such that the selected tensions to produce the desired pitches are substantially equal for the first, second and third strings.

The mass of the windings per unit length of the string for the first and second strings may be selected to provide the desired different pitches for the different strings when under the selected tension.

The matching of the changes in pitch for the strings may be such that for each of the first, second and third strings the ratio of the change in fundamental frequency to the fundamental frequency of the string at rest tension is substantially the same.

The material of the core of the first string may be the same as the material of the core of the second string and the same as the material of the core of the third string, and or the material of the winding of the first string may be the same as the material of a winding of the second string and the material of a winding of the third string.

The first, second and third strings may have a core diameter in a range 0.1 mm to 4 mm. The set of strings may comprise at least one unwound string having a string diameter that is matched to a core diameter of the first string and/or second string and/or third string.

The string diameter of at least one unwound string may be substantially the same as the core diameter of the first string and/or second string and/or third string.

The pitches of the first, second and third strings may be E₂, A₂, D₃.

The set of strings may further comprise at least three unwound strings. The at least three unwound strings may produce associated tones at their selected string tensions in the musical instrument, for example comprising pitches: G₃, B_3| E₄.

The variation of the string tensions may comprise a variation of at least 40% away from the selected tensions. Said variation of the string tensions may be substantially the same for at least the first and second string. The variation of the string tensions may be substantially the same for at least three wound strings and at least one unwound string. The variation of string tensions may be caused by lateral movement of the strings, for example, as part of a string bend. The set of strings may be configured to produce a matched change in pitch between the first and second string by moving the first string through a first lateral distance and moving the second string through a second lateral distance, wherein the first lateral distance is substantially equal to the second lateral distance.

A lateral distance of movement required to achieve a conventional pitch bend for the strings may be such that for each of the first and second strings for substantially the same ratio of pitch change to respective fundamental frequency to be obtained requires a lateral distance of movement to be substantially the same.

The set of strings may be configured to produce a matched change in pitch between the first and second string by applying a first lateral force to the first string and a second lateral force to the second string, wherein the first lateral force is substantially equal to the second lateral force.

A lateral force applied to a string required to achieve a conventional pitch bend for the strings may be such that for each of the first and second strings the same ratio of pitch change to respective fundamental frequency to be obtained requires a lateral force to be substantially the same.

The at least one property may comprise the cross-sectional area of the cores of the first and second string. The cross-sectional area of the first and second strings may be substantially equal.

The set of strings may further comprise at least one unwound string having a cross- sectional area that is matched to a cross-sectional area of the core of the first string and/or second string. In a further aspect there is provided a musical instrument including a set of strings as claimed or described herein. In another aspect of the invention there is provided a method of matching variations of pitch for a set of strings in a musical instrument comprising installing in the musical instrument a set of strings as claimed or described herein, wherein each string is installed to be at its selected tension to produce a tone of a respective different desired pitch, and the strings are such that for at least some of the strings changes in pitch for a variation of the string tensions away from the selected tensions are matched.

In another aspect of the invention, there is provided a method of designing a set of strings for a musical instrument, the set of strings comprising at least a first and second string, wherein the first string comprises a core and a winding and the second string comprises at least a core, wherein each string is configured to be held at a selected tension in a musical instrument to produce a tone of a respective different desired pitch, the method comprising:

inputting values for one or more input parameters;

determining values for at least one property of the cores and windings of one or both of the first and second strings using the inputted values for the one or more input parameters, such that a change in pitch of the first string matches a change in pitch of the second string for a variation of the string tensions away from the selected tensions. Determining values for the at least one property of the cores and windings may comprise calculating said values using the values of the one or more input parameters.

Determining values for the at least one property may comprise performing an iterative method. Determining values for the at least one property may comprise performing one or more approximations.

The values of the input parameters may be input by a user via a web interface. One or more outputs may be displayed to the user via the web interface. Theoretical values of one or more determined quantities may be displayed to the user. The determined quantities may include one or more of: tension, relative tuning for a tremolo arm move, a distance of lateral movement for a whole tone conventional pitch bend; lateral force required for a whole tone conventional pitch bend. The values of the determined quantities may be displayed together with values of said quantities for a standard string set.

The method may further comprise producing the strings in accordance with the determined values for the at least one property of the cores and windings. The method may further comprises producing the strings in accordance with the determined values for the at least one property of the cores and windings and at least one of the input parameters.

The input parameters may include at least one of: one or more sizes or dimensions of the instrument; one or more materials of the strings; number of strings; string pitches or tuning; scale length; stretchable length behind conventional nut; stretchable length between the bridge saddles and tailpiece; fingerboard or bridge radius; height of middle strings from the pivot plane; string core shape; wound or unwound string; core material; winding material.

The input parameters may further include at least one of: which strings of the string set are to be matched; which matching condition to use.

The input parameters may be one of a standard set of parameters corresponding to parameters of a standard instrument setup. The standard set of parameters may be selected by selecting a standard instrument setup from a plurality of different standard instrument setups. The input parameters may be further varied from the values of the standard set of parameters.

According to a further aspect of the invention there is provided:

a system for designing a set of strings for a musical instrument, the set of strings comprising at least a first and second string, wherein the first string comprises a core and a winding and the second string comprises at least a core, wherein each string is configured to be held at a selected tension in a musical instrument to produce a tone of a respective different desired pitch, the system comprising:

a user interface, for example a web interface, for receiving input parameters from a user; a processing resource configured to use the input parameters to determine values for at least one property of the cores and windings of one or both of the first and second strings, such that a change in pitch of the first string matches a change in pitch of the second string for a variation of the string tensions away from the selected tensions.

User interface may include an input device and a graphical display.

According to a further aspect of the invention, there is provided a computer program product comprising computer-readable instructions that are executable to perform a method of designing a set of strings for a musical instrument, the set of strings comprising at least a first and second string, wherein the first string comprises a core and a winding and the second string comprises at least a core, wherein each string is configured to be held at a selected tension in a musical instrument to produce a tone of a respective different desired pitch, the method comprising:

receiving input value data for one or more input parameters; determining values for at least one property of the cores and windings of one or both of the first and second strings using the received input value data for the one or more parameters, such that a change in pitch of the first string matches a change in pitch of the second string for a variation of the string tensions away from the selected tensions.

Features in one aspect may be provided as features in any other aspect, in any appropriate combination.

Brief Description of the Drawings

Various aspects of the invention will now be described by way of example only, and with reference to the accompanying drawings, of which:

Figure 1 is a perspective view of part of a wound guitar string;

Figure 2 is a cross-sectional view of a guitar and a tremolo mechanism;

Figure 3 is a plot of theoretical pitch deviations for a first set of strings;

Figure 4 is a plot of theoretical pitch deviations for a second set of strings,

Figure 5 is a plot of theoretical and experimental pitch deviations for a full tremolo arm pull up; Figure 6 is a plot of theoretical and experimental pitch deviations after an ambient temperature increase;

Figure 7 is a cross-sectional view of a guitar showing a string bend, and

Figure 8 is a plot showing distance of movement required across the 12^th fret.

Detailed Description

A string of a string set for a guitar is designed to produce a sound having a fundamental frequency when arranged to be under a string tension. When secured to the guitar and tuned up to its tuned configuration, the string, when struck, produces a sound having a pitch characterised by a fundamental frequency. The fundamental frequency can be expressed in terms of v_f is the fundamental frequency, T_f is tension in the string, l_f is the length of the vibrating portion of string and μ is linear density or mass per unit length of the string as:

(1)

Each string of the string set when placed under a tension, will exhibit a change in pitch corresponding to a change in fundamental frequency when the tension of the string is changed. For example, the string may be placed under more tension or less tension by manual string bending or via operation of an attached tremolo arm mechanism. Changes in pitch can be characterised by a change in frequency relative to the fundamental frequency. Due to their physical dimensions, strings of standard string sets are arranged and tuned up to different tensions and respond differently to extra applied tension. For example, standard strings under a single tremolo arm movement produce different changes in pitch of each string. Therefore, a chord made up of tones from two or more strings will drift out of tune under tremolo arm movement.

The present invention relates to a set of guitar strings designed to provide control and match the sensitivity and feel of different guitar strings through an appropriate specification of core and winding diameters of the strings. These strings allow a chord of up to four notes to be made to bend in tune through use of the tremolo arm, through the appropriate specification of core and winding diameters in the three, and optionally fourth, lowest pitch strings. The present invention allows in-tune chords to be played during tremolo arm use on guitars of the most popular designs without any modification of the hardware and therefore at a very low cost.

A set of strings is provided having three lower pitched strings (D, A and E) that are wound and three higher pitched strings (E, G, B) that are not wound. An example of a wound string is shown in Figure 1. The wound string 40 has wire 42 helically wound round a core 44. The wire 42 is cylindrical and has a winding diameter d„ that is equal to a cross-sectional diameter, as indicated in Figure 1. The core 44 is circular in cross section and has a core diameter denoted by d^. The wound string 40 is an example of a roundwound string with round wire wrapped around a round core.

The set of strings are configured to be secured to a guitar and tuned up to produce their desired pitches. The three wound strings have cores and windings selected such that, in response to a bending action, for example a tremolo arm movement, the pitches produced by the first and second strings are changed by a substantially equal amount relative to their original pitches. A bending action, for example a tremolo arm movement, induces a change in string linear density, a change in tension in the string and a change in vibrating length. As can be seen from Equation 1 a change in pitch of a string is dependent on any induced change in string linear density, tension in the string and change in vibrating length.

Relative frequency changes are dependent on engineering strain which is a measure of the amount of stretch of a secured string along its axis when tuned to its fundamental frequency. As will be described in detail elsewhere, the value of engineering strain on the strings depends on, amongst other parameters, a ratio of the total string mass (mass of string and mass of winding) to the mass of the core of the string. Controlling this ratio for at least two strings, provides an equalization of engineering strain across the wound strings, thereby leading to equal relative change in pitch of at least two strings. The wound strings may be matched to an unwound string also, such that the engineering strain across the wound strings and the unwound string are substantially equal.

Physical dimensions of an example set of strings made in accordance with the present invention are set out in Table 1. The three lowest (in pitch) strings are roundwound with nickel plated steel windings and cylindrical steel cores. The strings are manufactured by Newtone Strings and are secured to a Fender Stratocaster.

Table 1: String design parameters for a set of strings.

Several parameters are shown in Table 1. For ease of reference, all strings of string sets are labelled by an index, n, chosen such that strings 1 , 2, 3, 4, 5 and 6 correspond to strings (high to low) E, B, G, D, A and E respectively. The strings have sounding pitches E_4| B_3| G_3| D_3| A₂ and E₂. Shape factor is equal to π for all strings. Shape

factor for circular cross-section core and for a hexagonal cross-

section core. Strings 3, 4, 5 and 6, hereby referred to as matched strings, have a ratio of string mass to winding mass selected to give an equal change in pitch for these strings. Ratio of string mass to core mass is dependent on ratio of winding diameter to core diameter.

The three lowest pitched strings (D_3| A₂ and E₂) have core diameters, d_core,

selected to match the diameter of the unwound third string G₃. In this case, the core diameters are equal to 0.016 inches. Through the choice of matched core diameters between the 3^rd, 4^th, 5^th and 6^th string, the string tensions of the first and second strings are substantially equal. Having selected the core diameter to match that of the G string, the corresponding winding diameters of the windings for these strings are selected to be the nearest available diameter of wire to give a ratio of string mass to core mass that provides an equal engineering strain and hence change in pitch on bending.

The ratio of string mass to core mass is also dependent on the ratio of winding material density to core material density and core shape factor. However both of these parameters are substantially constant for these strings due to same materials and shape being used. Windings are made from Nickel plated steel. The nickel adds a small correction to the overall density of the winding in relation to the core. In alternative embodiments, the materials and therefore densities and/or Young's moduli of the strings could be selected to be different from one another. Furthermore, guitar strings may be manufactured to have different shape factors for example, the strings could be flatwound around a circular, hexagonal or braided core, wherein the wrapped wire has a circular, rounded square cross-section or of another type of guitar string, for example, half-wound, ground-wound, multiple or double-wound, stranded and pressure-wound. Alternatively, the strings could have a core of another shape, for example, a pentagonal core.

Suitable materials for string cores include synthetic or gut cores. Also alloys other than steel, coated or plated and any other metals, for example, brass, phosphor bronze and nickel-bronze may be used. Synthetic or gut cores are particularly well-suited for acoustic guitars. Amorphous metal, also known as metallic glass or glassy metal may also be suitable material for a string. Amorphous metal could be used to give a metal string with different Young's Modulus for the core that would allow for matching the engineering strain of the unwound strings too. In summary, any other metals, amorphous metal, nylon, gut, silk, natural threads, perlon, vinylon, dynel, polythene, polyethylene, orlon, fortisan, trevira and terylene could be used.

A variety of coatings or plating can be applied for the purpose of corrosion resistance or cosmetics. A stranded string, also known as braided or corded strings, may be used for unwound strings. A stranded string, also known as braided or corded string, could serve as a core design with a winding over it. In this implementation, the tension holding core is one stranded wire. For example, a 7 strand steel wire may be provided with a single core that holds the tension and six other wires wrapped around with a variable diameter or winding angle to control winding mass.

Suitable winding materials include nickel plated steel, pure nickel, stainless steel winding, particularly suitable for electric guitar strings. Other suitable winding materials include, for example, phosphor bronze, in particular for acoustic guitar strings and aluminium-copper. Windings on top of windings could be used to provide high overall mass on a narrow core, for example, double-wound strings. Silk layers may also be used for windings, for example for a violin string. Stainless steel advantageously provides robust thin windings for D strings. Figure 2 is a schematic diagram of the set of strings, as described above, strung to an electric guitar with a typical tremolo system 10. The tremolo system is based on a design of Leo Fender. Some sizes are emphasised in Figure 2 in relation to others for illustrative purposes.

The guitar has a guitar body 12 and extending from the guitar body is a guitar neck 14 incorporating a fretboard. At a distal or head end of the guitar neck 14 is a headstock having a tuner for each string. In Figure 2 only one tuner 16 is shown for clarity. The tuner 16 has an aperture, through which one end of a string can be threaded, and a tuning peg that can be turned to secure the string at the head of the guitar. Further turning of the tuning peg adjusts the tension in the secured string. At an upper end of the neck, sitting below the headstock is a nut 18. At a lower end of the neck 4 is a bridge having a bridge plate 20. The bridge plate 20 is pivotally mounted at a first proximal end to the body 12 at a pivot point 22. As shown in Figure 2, the bridge plate 20 is attached at an angle to the upper surface of the guitar body. A saddle 24 is positioned on an upper surface of the bridge plate 20. The tremolo mechanism has a tremolo arm 26 connected at a distal end of the bridge plate 20. The tremolo arm 26 has a first portion extended out from the bridge plate, substantially normal to the upper surface of the guitar body 12 and a second portion that extends substantially parallel to the upper surface of the guitar body 12. At the end of the second portion is an angled tip. The tremolo arm 26 can be moved towards or away from the guitar body 12 thus causing an angular displacement of the bridge plate 20. Depending from the bridge plate 20 is a tremolo block 28 to which a second end of a string can be secured.

On a lower surface of the guitar body 12 is a recess in which a set of tremolo springs and a corresponding set of tremolo claws are positioned. Only one tremolo spring 30 and one tremolo claw 32 is shown in Figure 1. There are a number of springs attached to multiple hooks on a single tremolo claw 32, typically between 2 and 5 springs. The claw 32 is attached to an end of the recess with two screws that can be adjusted, for example, when setting up the instrument, to give a desired rest position for the bridge plate 20. The spring 30 is hooked onto the claw 32 at a first end and secured to the tremolo block 28 at a second end.

Figure 2 also shows a string 34 of the above set of strings secured to the guitar. A first end of the string 34 is secured to the tuner 16 and a second end of the string 34 is secured to the tremolo block 28. Between the first and second ends, the string 34 passes over the nut 18, the neck 14 and the saddle 24. Also shown on Figure 2, there is a first and second section of string 34 not contributing to the sounding length. The first and second sections of string are between the tuner 16 and the nut 18 and the saddle 24 and the tremolo block 28 respectively.

Figure 2 shows a first, initial configuration for the tremolo mechanism and a second configuration for the tremolo mechanism. The resulting displacement of the string 34 is shown in the first and second configurations. The first configuration is shown in light grey. The second configuration is shown in black. A user can move the tremolo mechanism from the first configuration to the second configuration by applying an upwards (away from the guitar body 12) force to the tip of the tremolo arm 26. The force direction is indicated by F in Figure 1. The application of force moves the tremolo arm 26 away from body 12. As the tremolo arm is connected to the distal end of the bridge plate 20 and because the bridge plate is pivotably mounted to the body 12, the movement of the tremolo arm away from the body 12 causes the bridge plate 20, saddle 24 and tremolo block 28 to be rotated about the pivot point 22, through a first angle, denoted in Figure 2 as Θ. The applied force and movement can also be in the opposite direction corresponding to negative values of F and Θ.

Stringing the guitar involves first securing the string 34 between the tuner 16 and the tremolo block 28 and then turning the tuner 18 to tune the string 34 to pitch. When first secured, the portion of string which will later come to rest between the nut 18 and the saddle 24 has an initial length l_a. On tuning to the appropriate pitch, the string is lengthened by a distance, denoted ΔΪ, to have a final length between the nut 18 and the saddle 24 of 2 = l_a + AI. The portion of string between the nut 8 and the saddle 24 is free to vibrate, and therefore the length of this string is a vibrating length. The string vibrates upon being struck thereby producing a sound. A section of string 34 between the nut 18 and the tuner 16 does not contribute to the vibrating length and has length i_nat. A section of string 34 between the saddle 24 and the tremolo block 28 does not contribute to the vibrating length. These sections are denoted by dot-dash lines in Figure 2. The string slides across the nut and saddles during tuning up. The bridge plate also moves. This means that the initial length l_Q does not exactly match the initial distance from the nut to the saddle unless a locking saddles and a locking nut are used. The final length of the string I = _-0 +Δί is the exact distance from the nut to the saddles after tuning. A precise value of ._fl can be worked out retrospectively, for example by using the Young's modulus.

In the initial, rest configuration the vibrating length of the string 34 between the nut 18 and the saddle 24 is I = 1₀ +Δ1. In the second configuration, due to movement of the tremolo arm and tremolo mechanism, the vibrating length of the string is extended by a small amount δ and its length can be written as I = l₀ +ΔΖ + δ. An increase in length of the vibrating length of the string increases the tension of the string, and therefore increases the sounding pitch. A decrease in length of the vibrating length of the string decreases the tension of the string, and therefore the sounding pitch. In a non-locking tremolo design, the string will slip across the nut to approximately equalize tension on either side of the nut. The portion of string responsible for the original sounding length actually slides to occupy a different length l_t = 1₀ + Δ1 + δ„. As described elsewhere, this may have an effect on the change in pitch of the string. It is also possible to exert a force in an opposite direction. A user can apply a downwards (towards the guitar body 12) force to the tip of the tremolo arm 26. This application of force moves the tremolo arm 26 towards the body 12. As the tremolo arm is connected to the distal end of the bridge plate 20 and because the bridge plate is pivotably mounted to the body 12, the movement of the tremolo arm towards the body 12 causes the bridge plate 20, saddle 24 and tremolo block 28 to be rotated about the pivot point 22, through an angular distance, in an opposite direction to that described above e.g. a negative value of Θ. A movement of tremolo arm towards the body 12 results in a decrease in the vibrating length of the string 34 by a small amount, therefore decreasing the tension of the string 34 and the sounding pitch.

In use, a user attaches the set of strings to a guitar and tunes the strings to their respective pitches. To tune a string, the strings must be stretched from their zero tension initial length i_a to their final rest length under tension, the rest length being I = IQ+ AI. Once tuned, the matched strings (strings 3, 4, 5 and 6) are under substantially equal engineering strain. The matched strings are also under substantially equal tension. A sound having a first pitch is produced by striking one of the set of matched strings. Tremolo arm movement causes the sound of each string to change pitch to a second pitch. The change in pitch of the sounds between the second pitch and the first pitch is equal across all matched strings. In this way, a user may play a first chord involving the matched strings and the chord will stay in tune on operation of the tremolo mechanism (the relative intervals between the notes being kept constant on tremolo arm movement).

For the most musically satisfying result, the variation between the change in pitch of the first string and the change in pitch of the second string should ideally not vary by more than 10 cents for a mean pitch change of a minor third (300 cents) for the two strings. The design described demonstrates relative changes in pitch of up to 21 cents for a mean pitch change of a minor third (300 cents) on four strings, and this is musically acceptable.

The difference between the ideal variation in pitch changes and the pitch changes achieved by the custom string set may be due to the gauges of wire available for use in the string set. The custom strings described are manufactured using gauges of wire that are available in increments of 0.001 inch and therefore have windings of an integer number of thousandth of an inch. The difference may be reduced if more gauges are available. The difference may also be reduced by adjusting saddle heights to fine tune intervals. The experimental data presented does not include fine tuning by adjusting saddle positions.

Musical intervals vary in different systems of tuning. For example, a minor third in just intonation corresponds to a change in pitch of 315.64 cents but in equal temperament a minor third corresponds to a change in pitch of 300 cents. As a further example, equal temperament major thirds are 13.7 cents way from just intonation major thirds. Both tuning systems are considered appropriate in certain contexts. If some notes are flat by this interval and some sharp by this interval then the total range would be just over 27 cents. This total range is larger than the range of 21 cents in the experimental table.

If the saddle positions and distance of string behind the nut are the same then the maximum experimental error, or variation, in engineering strains across strings should be around 2% in some embodiments. It should be noted, however, that ideal engineering strains are slightly different from one another when compensating for the different lengths of string behind the nut and the different saddle heights on Fender Stratocasters meaning that the difference in ideal engineering strains for different strings may vary by around 2% of their mean values.

The ideal specification would be different again for Floyd Rose (locking nut) designs with the ideal engineering strains for wound strings differing by around 12% percent depending on the radius of the fretboard.

To achieve the above ideal engineering strains with equal tensions, the core diameters may vary by a maximum of up to 20%. In contrast, standard string sets such as the D'Addario EXL120 have core diameters that vary by around 69%.

The force required for whole tone bends for standard EXL120 D'Addario strings vary by over 50% between D and low E. On the design listed here this is closer to 16% although even better agreement would be possible if more gauges of wire were available (as these have been based on gauges in increments of 0.001 inch).

Figure 3 is a plot of theoretically calculated pitch deviations for a set of strings described with reference to Figure 1 and Figure 2, as a function of Θ. The strings are manufactured by Newtone Strings and are secured to a Fender Stratocaster. For comparative purposes, Figure 4 is provided which shows a comparative plot for a standard set of strings.

The x-axis of the plot of Figure 3 displays angular displacement, Θ, due to tremolo arm movement as described with reference to Figure 2. These values are positive and negative corresponding to tremolo arm movement away from the guitar body and tremolo arm movement toward the guitar body, respectively. The y-axis gives a measure of pitch deviation of the string. The measure is calculated, for each string, with reference to the fundamental frequency of the 3^rd string (a reference frequency), using the following formula:

(2)

String index, n, is chosen such that strings 1 , 2, 3, 4, 5 and 6 correspond to strings (high to low) E, B, G, D, A and E respectively. These strings have sounding pitches ¾, B₃, G_3| D_3| A₂ and E₂ following securing, tuning and when the tremolo arm is in its rest configuration. The rest configuration corresponds to zero angular displacement Θ. Relative tunings of the 1^st, 2^nd, 3^rd, 4^th- 5^th and 6^th strings with respect to the 3^rt string correspond to standard tuning intervals of +900, + 400, 0, -500, -1000 and -1500 cents respectively.

On moving the tremolo arm through an angle Θ, the frequency of sound produced by a string changes from its first fundamental value to a second value For a range

of angular displacements Θ, the second value is calculated. Each string has a solid

curve representing the variation of the interval over angular displacement with reference to the fundamental frequency of the third string, as calculated using the above formula using the value of shown in Table 3 and predicted values of

Figure 3 also shows five dashed curves. The dash curves of Figure 3 are drawn using the interval predicted for the 3^rd string (G₃) transposed by a constant value corresponding to the standard tuning intervals as listed above, of the other strings to the third string. For example, the highest dash curve Figure 3, corresponds to the first string. The first string has a relative tuning interval from the 3^rd string of +900 cents. Therefore, the first dash curve is drawn by shifting the 3^rd solid curve upwards by a distance of 900 cents. Likewise, the second dash curve, corresponding to the second string, is equal to the third solid curve shifted upwards a distance of 400 cents. The third dash curve is, by definition the third solid curve shifted by 0 cents and therefore is not shown. The fourth dash curve corresponds to the fourth string, which has a relative tuning interval from the 3^rd string of -500 cents. Therefore the fourth dash curve is equal to the third solid curve shifted downward by a distance of 500 cents. The fifth and sixth dash curves, corresponding to the fifth and sixth strings are equal to the third solid curve shifted down by 1000 and 1500 cents respectively.

Marked on the right hand side of Figure 3 are theoretical pitch deviations from the standard tuning intervals (indicated by horizontal lines) and the predicted intervals (solid lines) at angular displacement of 0.085 radians corresponding to 4.27 degrees. These values are also presented in Table 2. Also shown in Table 2 are experimental values measured on the guitar by measuring pitch intervals on a Logic Pro X tuner. The tuner gives results that are consistent to plus or minus 6 cents. As is observed, the theoretical and experimental values are in agreement within 20 cents (or less than 7%) which is within the experimental error that stems from precision of measurement of the point of contact between string and saddle with respect to the pivot.

Theoretical values are calculated using experimental setup parameters provided in Table 3 for American Fender Stratocaster, with the exception of τ. Parameter r is derived from a destructive test on the custom strings involving cutting a length of string, unwrapping the winding from the core and measuring the ratio of the sum of mass of core,

and mass of winding, to mass of winding, as displayed in Equation 18.

Table 2: Comparison between theoretically derived and experimentally measured pitch intervals for custom string set, for a tremolo arm movement corresponding to 0.085 radians angular displacement. The pitch intervals are also plotted in Figure 5.

Table 3: String Parameters for American Fender Stratocaster.

Several measured parameters are shown in Table 3. Sounding fundamental frequencies v_f are given to 4 significant figures and correspond to the frequency of the string sounded without any tremolo arm movement (in the rest configuration). Vibrating lengths l_f have an uncertainty of ±1 mm but with differences between vibrating lengths accurate to 0.1 mm. The length of string between nut and tuning peg has an uncertainty ± 0.5 mm. Radial distances from pivot to centre of a string at point of contact with the saddle R_a have uncertainty ± 0.1 mm. Angular displacement of said radial distance R_a from body is Φ and has uncertainty of ± 0.01 radians. Ratio of the total mass of the string to the mass of the core is represented by τ and has uncertainty of ± 0.01. Values of density and Young's modulus for steel core are taken from a reference textbook.

Figure 4 shows comparative plots for a standard set of strings, D'Addario EXL120 "Nickel Wound" Super Light 9-42 (round wound with nickel plated steel) strings secured to a Fender Stratocaster. Physical dimensions of these strings are also provided in Table 3 with the mass ratio parameter, τ, (obtained by destructive testing) given in Table 4. The three lower (in pitch) strings are round wound with nickel plate steel. The three high pitch strings (E, B, G) have a circular core and the three low pitch string (D, A, E) have a hexagonal cross-section core. A results table for these strings is provided in Table 4.

Table 4: Comparison between theoretically derived and experimentally measured pitch intervals for a standard set of string, D'Addario EXL120 "Nickel Wound" Super Light 9- 42, for a tremolo arm movement corresponding to 0.085 radians angular displacement. The pitch intervals are also plotted in Figure 5.

The x-axis of the plot of Figure 4 displays angular displacement, Θ, due to tremolo arm movement as described with reference to Figure 1. These values are positive and negative corresponding to tremolo arm movement away from the guitar body and tremolo arm movement toward the guitar body, respectively. The y-axis gives a measure of pitch deviation of the string. The measure is calculated, for each string, with reference to the fundamental frequency of the 3^rd string, using Equation 2. Values of are derived as described with reference to Figure 3.

On comparing Figure 3 and Figure 4, it is observed that predicted pitch deviations for 3^rd, 4^th, 5^th and 6^th strings are more similar for the custom string set. Therefore, four note chords played over these strings are predicted to remain close to being in tune over a much wider range of tremolo arm motion.

There is large number of possible tunings for any string set. A custom string set designed for a standard tuning (lowest strings E_2l A₂₎ D₃, G₃), as described above, will also provide equal relative pitch intervals when tuned to different tunings with the same pitch intervals as the standard tuning (musical interval of a perfect fourth). For example, the standard tuning string set may be tuned to E flat tuning wherein all strings are lowered by a semitone. Another choice for relative tuning is drop tuning where only the sixth string E₂ is re-tuned, in this case to D2. In this case, matched change in pitch may be achieved using a different string specification for the sixth string. For example, the sixth string may have a higher winding diameter.

Sounding Frequency and Strain on Unwound Strings Tension in a plain or unwound string is given by the equation:

(3)

where A_Q is the cross-sectional area of the un-stretched string, l₀ is the rest length of the portion of the string that has a length of i_f = l_Q + Al after being extended under tension, T, and E is the Young's modulus. Note that this assumes that the Young's modulus measured for nominal stress (constant cross-section) is valid. In reality the linear regime may be exceeded to some extent and this may occur due to the string cross-section contracting in diameter as the string is lengthened, due to the Poisson effect and due to non-linearity in the stress-strain graph for the particular material used.

The mass per unit length does not remain precisely constant when the string is stretched. Defining the mass per unit length of the unstretched string as μ₀, the mass per unit length after stretching will be:

(4)

where p₀ is the density and A_a is the original cross-sectional area of the string when un-stretched. The sounding frequency, v_f, is given by:

(5)

Theory for Wound Strings

The D, A and low E (lowest pitch) strings on the electric guitar are usually wound. The most common case, known as round wound strings, consist of a core (which may have a circular or hexagonal cross-section) wrapped with cylindrical wire. This is done in order that the degree of stretching involved in getting them up to pitch is large enough to give the desired approximately linear behaviour (where the pitch is close to being largely independent of the amplitude of vibration) and to prevent excessive resistance to flexing (which would lead to inharmonicity). The G string was also usually wound in the 1950s (when the Fender Stratocaster tremolo was first invented) but most players have used unwound G strings since lower tension string sets become popular from the 1960s onwards. The winding acts to increase the mass per unit length (hence allowing for lower resonance frequencies).

The tension in wound strings is, to the first approximation, produced by the stretching of the core, as the (helical or spring-shaped) winding is angled and doesn't stretch along its length as significantly. It is worth proving the limited extent to which the winding contributes to the overall axial tension of the string.

Assuming the winding behaves like a helical spring consisting of N_a active coils of wire diameter d_w, with a mean diameter D_w, modulus of rigidity G, the spring constant will be:

(6)

Assuming the spring starts at rest such that all the adjacent wires are wrapped to be touching with no loading, the rest length of the spring/windings on the sounding length of the string will be l₀= N_ad„ and the spring force on stretching the this portion of the spring/windings to increase the length by Δl will be

(7)

where is the cross-sectional area of the wire used in the winding. The

most common choice of material in electric guitar strings is steel, and the winding is often nickel plated steel. In order to approximate typical values we will assume steel music wire construction such that the Young's modulus is E = 207 Gigapascal (GPa) and the modulus of rigidity is G = 79.3 GPa. Remembering that D_w is the mean diameter of the winding around the axis, the core diameter is defined as

In practice the core diameter is larger than the diameter of the wire

in the winding, so that < d_core and thus d_wfD_w < 1/2, implying that

(8)

The corresponding tension for the core would be given by Equation (3) as (9)

This proves that the tension contributed by the winding is less than one percent of the tension in the core and may be assumed to be negligible in most practical applications. It is possible to construct a spring that is pre-tensioned (for instance through rotating the wire during cooling) and theoretically this could give the winding a stronger contribution to the overall tension force but doing so would reduce the benefits of using wound strings.

Mass per unit length of wound strings

Defining the total mass per unit length of wound strings as μ^, this will equal the sum of the mass per unit lengths of the core, μ„„ and the mass per unit length of the windings giving:

where the factor τ is the mass per unit length of the string divided by the mass per unit length of the core. Expressing the mass per unit length of the core after stretching as

(11)

where is the density of the core when un-stretched and the cross-sectional area of the core when un-stretched is

(12)

where d_core is the maximum core width and is a factor which depends on the

cross-sectional shape of the core. For a cylindrical core, is the core diameter and

for hexagonal cross-section cores, is defined as the distance between opposite

points on the hexagon. The factor is given by:

(13)

Remembering that D_w is the diameter of the centre of the winding with respect to the axis of the core, the mass per unit length of the spring/windings (after stretching) is given by:

w

in the winding, allowing the overall mass per unit length of the wound string to be given as:

This ratio of the mass per unit length of the string in relation to that of the core will be in the range T > 1 for wound strings. The above equations are also valid for unwound strings of circular cross-section by substituting

Hexagonal cores wound with a circular cross-section winding have

Issues such as the deviation of the core and winding from idealised shapes mean that, in practice, it is more accurate to establish an accurate value of τ for a string destructively, based on cutting a length of string, unwrapping the winding from the core and measuring the ratio: (10)

where is the mass of the core and is the total mass of the same

length of string.

Sounding frequency and strain on wound strings

Assuming that the tension supported by the winding is negligible, we may return to Equation 3 considering just the core such that:

Now using this to calculate the frequency using Equation 1 and Equation 16 gives:

(20)

(21)

Squaring both sides and solving the resulting quadratic equation gives the engineering strain as:

In practice, the dependence on τ in Equation 22 and 23 means that the ratio of the core width and winding diameter can be used as a very effective control parameter to achieve particular relative tuning intervals between wound strings under tremolo arm usage. The greater the diameter of the winding in relation to the overall string width, the larger the value of

required to bring the string up to pitch, and therefore the smaller the pitch change for a given tremolo arm usage. The density and/or ratios of densities can also be used to control the sensitivity of the string to tremolo use. Strings of higher density have greater engineering strains in tuning to a given pitch and therefore feature smaller pitch changes for a given tremolo arm movement. However, in spite of some string sets being advertised with the name "pure nickel", the low Ultimate Tensile Strength of nickel (maximum values being less than 1.2 GPa so roughly half that of typical values for the steel used in strings) means that steel is used for unwound electric guitar strings and the cores of all wound strings from such sets (with the pure nickel being used exclusively in the winding). This still leaves the ratio of winding density to core density as a relevant control parameter.

Theoretical tremolo arm pitch deviations

Consider again a string initially tuned up to sounding length lf = l_Q+ Aland fundamental frequency v_f when no pressure is applied to the tremolo arm. Suppose that the length the string is then changed by a distance 8 using the tremolo arm to give a new sounding length of l_t = 1₀ + ΔΙ -\- δ. it is important to note that, in a standard

(non-locking) nut design, the string will slip across the nut to approximately equalise the tension either side of the nut. This is clear from the fact that pressing the string behind the nut to increase the tension there also produces a corresponding pitch increase in the main portion of the string (and the main portion of the string returns to the original pitch when the pressure behind the but is released assuming the nut is in good condition). This means that, when the sounding length changes by <3, a fraction of that length change is taken up by the portion of string behind the nut. The portion of the string responsible for the original sounding length (which is responsible for calculating changes in tension and density) actually slides to occupy a different length

(24)

with ._nHt being the length of string behind the nut. The resulting fundamental frequency, labelled v_t, will change, using Equation 4 to give: (25)

hence the sounding frequency changes by the ratio:

(26)

Since dominates the resulting equation giving a first

approximation of The change in cents (where a cent is defined

as one percent of a musical interval of a semitone) is given by:

(27)

With reference to Figure 2, when the tremolo arm rotates by an angle of Θ around the pivot axis then the total change in sounding length of the string will be given by:

(28)

where are the initial and final positions of the point of contact

between the string and the saddle from the pivot in polar coordinates. Due to slippage at the nut, the length change responsible for changes in tension and mass per unit length is then:

(29)

Custom Wound Strings with Close to Equal Pitch Deviations

Tensions for different strings in the same set tend to be within about 50% of one another in order to prevent warping of the neck and for a consistent playing experience. Different wound strings in the same set will therefore tend to have similar values for from Equation 19. To a rough first approximation they will have a

very similar sensitivity to a given angle of tremolo movement if M/l_Q is the same for the different strings and this implies that they should share approximately the same cross- sectional area, A_eore, for instance by having as many cylindrical cores as possible sharing the same diameter. This also implies that the strings would then have close to the same value for the stress meaning that the stress will be approximately

the same fraction of the Ultimate Tensile Strength. A benefit of this is that the low E string may be prevented from having a stress below a recommended range for optimum tone quality. From Equation 22 it is clear that for different strings constructed of the same core material, the value of should be the approximately the same.

Labelling the number of cents between the fundamental frequencies of the strings as "interval", the ratio between the fundamental frequencies of the strings will be given by

(30)

Consider the case of an unwound string (string number m) having a core

diameter, and a wound string (string number n) of core diameter also equal to but

with windings constructed using wire of diameter This may be combined with

Equation 17 which gives the mass ratio r^in terms of the core diameter and the diameter of the winding to produce a quadratic in the ratio which may be

solved to give:

string.

In reality the winding diameter used will have to be slightly higher due to the winding becoming oval. The effect of the different distances behind the nut, l_nut, and the different positions of the saddles with respect to the pivot for different strings will also impact the results and can be compensated for (e.g. decreasing dw for a particular string to increase the pitch interval produced for a given control change).

Assuming that the steel core has

and that the nickel plated steel windings have and the cylindrical core has strings can be

designed that have an unwound 3rd string (m = 3) of diameter inches

and have three wound strings (n = 4, 5, 6) each with a core of 6 inches but

with the diameter of the windings, designed using Equation 31. Taking standard

tuning so interval = 500 cents, 1000 cents and 1500 cents for the 4th, 5th and 6th strings respectively, this results in: and

each to two significant figures and hence inches,

inches to two significant figures.

A set of strings was therefore designed that show close to equal pitch change intervals for the G_3| D₃, A₂ and E₂ strings. These were constructed by Newtone strings to the specification given in Table 1. The values of d_w were increased where necessary (by up to 22%) in order to utilise available gauges of wire and to compensate for various approximations made during the derivation of Equation 31 (such as ignoring the fact that windings become oval during manufacturing).

The result is a set of strings with a significantly lower fraction of the mass in the windings (r approximately 28% smaller) than usual on the D₃ string, and a significantly greater fraction of mass in the windings (τ approximately 30% greater) than usual on the E₂ string.

The theoretical pitch deviations for this set of strings is shown in Figure 3. It is clear that the predicted pitch deviations for the 3rd, 4th, 5th and 6th strings are much more similar for this custom string set, meaning that four note chords are predicted to remain close to being in tune over a much wider range of tremolo arm motion. Carefully calibrated sets could be designed to achieve even better agreement between these strings although the results will be sensitive to variables such as the saddle heights, distances of string behind the nut and availability of precise gauges of winding wire. The relative tuning of the 1st and 2nd strings is unchanged due to this being a fundamental property of unwound steel strings tuned up to the standard pitches. The strings were tested for their pitch deviations and then removed, cut at points of contact with the tuners and saddle, the windings unwrapped and cores and wraps measured on mg accurate scales as before allowing for the actual value of the ratio rto be assessed using Equation 18. Experimental measurement of the custom string set loaded onto the Fender Stratocaster is compared with theory in Table 2. Agreement between the theoretical and experimental pitch deviations is within 20 cents over pitch a pull up of order 300 cents and demonstrates a musically useful level of matching between pitch changes obtained for tremolo arm use for the G₃, D₃, A3 and E₂ strings. It should be noted that the value of τ obtained from Equation 18 using the specifications are 1.99, 3.38 and 6.21 for the D₃, A₂ and E₂ strings respectively. These are larger than the values obtained using destructive testing of the masses (utilising Equation 19 as given in Table 2) by 0.93%, 4.2% and 7.9% respectively. This is expected as it is known that windings flatten (become oval in cross-section) by somewhere between 7% and 10% during the winding process.

It is interesting to note that the low E string shows the biggest discrepancy between the theory and experiment in two of the three measured string sets. In calculating the theoretical results it was assumed that the distance of string stretch was based on the point of contact between the string and saddle. In doing so we ignore any effects due to the finite thickness of the cores and windings of strings, the slightly rounded shape of the saddles and the slightly rounded profile of the string nearby and any slipping of the strings at the saddles. It should be noted that basing the string stretch on the mid-point of the string would result in reduced agreement between experiment and theory for the unwound strings (increasing the predicted pitch deviations for the G string for instance). On the other hand adding the thickness of windings into the calculation of the string stretch (giving larger string stretches for the thicker wound strings) would improve the agreement between experiment and theory (giving 296, 306 and 306 cents for the D₃, A₂ and E₂ strings respectively if the winding thickness given in the specifications is simply added in the distance to the pivot,

Figure 5 shows a comparison of theoretical and experimental pitch deviations in cents (with respect to the open string pitches) for a full tremolo arm pull up on Fender Stratocaster with D'Addario EXL120 Super Light 9-42 strings and with the custom strings set. The results are taken from Table 2 and Table 4. Temperature Dependence

The vibration frequencies of metal strings on musical instruments are known to decrease with increasing temperature, necessitating re-tuning. In the case of steel (or steel cored) guitar strings this is principally due to the thermal coefficient of expansion of the steel in the strings which is in the range 11 x 10^-6 m/mK < a < 13 x 10^-6 m/mK. Here the units refer to meter of expansion per meter of initial length, per change in temperature in Kelvin (or change in Celsius). After a change in temperature of dt Celsius, the section of string formerly of length l₀ when not under any tension would therefore have a rest length of l₀+ S_a when not under any tension, where

(32) with a being the thermal coefficient of expansion of steel. The very small decrease in Young's modulus with temperature is neglected, changing the extension from Δl to ΔΪ + δ_{c cnan}g_es the sounding frequency by the ratio

when taking the limit

In the case of a temperature change, it is assumed that the sounding length of the string, If, doesn't change, but that the length of that portion of string if all tension was removed would be increased from l₀ to l₀ + δ _n due to the thermal expansion. The extension in the string (defined as the difference between its length under tension and its length if all tension was removed) therefore changes from Δl to — δ_α -p_nj_s results in a ratio of sounding frequencies of

(33)

where v_aand v_f are the frequencies after and before the change in temperature respectively. Using Equation 32, the interval in cents due to the temperature change by dt degrees Celsius thus becomes:

(34)

This suggests that the effect of temperature depends on the engineering strain required to get the string up to pitch, with larger engineering strains indicating a smaller deviation of sounding pitch with temperature. This agrees with the fact that high E (E ) strings are known to drift to the smallest extent with temperature while low E (E₂) strings drift to the largest extent with temperature in standard electric guitar string sets. It is also clear that the E₂ in a set designed with close to equal engineering strain for multiple strings will stay in tune to a greater extent than for the more temperamental E₂ in a standard string set. Four strings in the custom set would therefore be expected to stay in tune with each other and all to drift out of tune by the same number of cents due to temperature changes.

Figure 6 shows theoretical and experimental pitch deviations in cents after an ambient temperature increase of 8.3°C for D'Addario EXL120 Super Light 9-42 strings and after an ambient temperature increase of 6.3°C for custom strings made by Newtone Strings to the author's specification. The theoretical prediction is based on Equation 34 with values taken from Table 2 and Table 4.

The results of Figure 6 are obtained by leaving the guitar in a room at a cool temperature (averaging 15°C for these experiments) for over an hour and tuning the instrument, and then bringing the instrument into a room with a warm temperature (averaging 23°C) and re-measuring the pitch 5-12 minutes later. The experimental data for the change in sounding pitch in cents was noted using the tuner in Logic Pro X. The temperature at the location of the guitar for each measurement was read from a Tenma 72-7712 thermocouple thermometer. When the D'Addario strings were used the temperature change between the room was 8.3 °C while for the test on the custom strings, the change in temperature was 6.3 °C, hence the larger theoretical pitch change shown for the D'Addario strings.

For these measurements, the tremolo screws were tightened so that the bridge plate was flush with the body in order to minimise the effect of movement at the bridge. Nonetheless, the results show that the experimentally assured pitch changes are only around a half to a third of those predicted by the simple theory as the theory doesn't account for the flexibility of the instrument body and its response to changes in string tension, temperature and humidity. It is clear that strings with lower engineering strains are generally correctly predicted to have a higher sensitivity to temperature changes and thus the low E string (E₂) on the standard D'Addario descends in pitch to the greatest extent with a temperature increase. The low E (E₂) in the custom set is indeed improved in this regard with the strings with close to equal engineering strain staying close to being in good relative tuning to one another with temperature changes. By way of a specific example, keeping the engineering strains equal to or larger than 0.245% keeps the experimental pitch deviation to 12 cents or less when changing ambient temperature by 6.3 degrees Celsius.

Approximate Formulae for Force and Distance required for bends as applied to feel

Figure 7 shows a schematic diagram of a conventional string bend as viewed from a direction perpendicular to the plane of the body. The string has an open sounding length l ₀ +Δi when initially tuned up to pitch. Fretting in order to half the sounding length, and then bending by displacing the string through an angle of a by applying a force F_B gives the approximate sounding length:

(ί₀ + Δ1 + δ_E)/2 and the stretching of the string raised the tension to T_B. Considering the string bend illustrated in Figure 7 and again taking the assumption that the windings provide zero tension, the force required by the player to bend a string at a fret halfway along its length to make an angle of a with respect to its rest position is:

F_B = 2 T_B sin a (35)

where T_B is the tension of the core of the string after the bend. It should be noted that the twelfth fret is not exactly halfway along the length of the string in practice as "unbent" fretted notes have a slightly raised tension on being pressed down onto a fret. Denoting the change in length of the string due to the bend as:

(36)

(37)

In the limits of low engineering strain 1) and where the length added to the string by the bend, δ_E is much smaller than the length of the string {δ_B/l_f « 1) it may be shown that, for a note fretted at the 12th fret and bent to change the sounding frequency from ν_Ό to v_B, the lateral force required by the player is approximately:

7

and in practice this approximation for F_B is accurate to better than 1% in comparison to a full derivation (i.e. by not taking the low strain limits).

The lateral distance required to raise the pitch when performing a conventional string bend on the twelfth fret is given by y_B:

(40)

and in the limit δ_B/l_f << 1 Equation 40 becomes:

This approximation is accurate to better than 1% in practice. It should be noted that, assuming the string is moved through 2 mm to contact the fret for an unbent note, the initial engineering strain percentage value would be increased by only around 0.002% from Equation 37. The position of the 12th fret is halfway along the string to an accuracy of 0.3%.

It is clear from Equations 38 and 41 that the force required to bend a string through a given pitch interval depends on the initial tension and thus the string gauge (in addition to the starting engineering strain

_wnj_cn \_{s a} function of initial frequency and mass ratio τ ). The distance required to bend the pitch doesn't depend on the initial tension or string gauge if all other factors are equal. If different strings are to have equal force required for equal pitch interval changes for a conventional string bend then it is not enough to simply match the tensions, as the value of

also has a strong impact.

This is the same requirement as for matching sensitivity to tremolo arm use (give or take the small differences in the position of the strings on the saddle with respect to the tremolo unit's pivot point). In the limit

it should be noted from Equation 23 that

and thus the force required for a pitch bend through a given interval will be approximately proportional to the initial sounding frequency so the force required (and lateral distance of movement required) doubles when going up an octave for unwound strings of the same lengths, tensions, densities and Young's moduli. As with matching sensitivity for tremolo arm movements, matching the force and distance of movement required for a conventional string bend through a given pitch interval on different strings (of the same material) can only be achieved if at least one of the strings being matched is wound. This requires matching using the mass ratio

_as j_mp|j_ec| by Equation 22, give or take small corrections due to the ratio being different for different strings on typical (non-locking) necks and

due to the point of contact with the saddles being at different positions with respect to the pivot at the bridge. It is not necessarily desirable that the forces should be perfectly matched, however, as player preferences may suggest that larger forces should be required for bending the lowest strings (with higher tension low strings being popular in so called "heavy bottom" string sets). Matching the distance of movement required for conventional strings bends, y_B, on the other hand, is desirable to provide consistent feel across as many strings as possible.

While performing conventional bends on guitars with the tremolo set up for floating operation, it is common for the raised string tension, T_B, to displace the bridge by stretching the tremolo springs. This would lead to larger than predicted values of y_B unless the spring constant and position of the tremolo springs were taken into account.

In order to avoid this complication the guitar used in the test was fitted with an ESP arming adjuster. This device features an additional spring operating in compression mode and was set up such that whole tone conventional pitch bends were not able to move the bridge significantly. In this way the results presented are directly comparable to fixed bridge or "hardtail" instruments (including Fender Stratocasters with the tremolo screws tightened so that the bridge plate lies flat to the body). Distances of motion were measured by performing whole tone bends by hand (checking the tuning on a digital tuner) and the distances with digital callipers (SPI 300 mm).

The theoretical predictions for the lateral force, F_B, and lateral string displacement, y_B, required for a conventional pitch bend of a whole tone at the 12th fret are shown in Table 5 and Table 6 for the standard strings and for custom strings, respectively. Also shown is the experimentally measured distance of string motion in each case. Dimensions and variables are taken from Table 1 and Table 3. The values of τ were taken from destructive testing of the strings and by utilising Equation 18. The cross- sections of the cores in m³ were calculated according to was the measured mass of the core

obtained by destructively removing the winding of the string cut to the length

Table 5: Theoretical calculation of force F_B and distance of movement ye required across the 12^th fret for whole tone bend {ν_Β/ν_π = 2^ίζ^) for Fender Stratocaster with

D'Addario EXL120 "Nickel Wound" Super Light 9-42 strings (round wound with nickel plated steel). Experimental measurement of the distance of movement is also shown.

Table 6: Theoretical calculation of force F_B and distance of movement y_B required across the 12^th fret for whole tone bend for Fender Stratocaster with

custom string set. Experimental measurement of the distance of movement is also shown.

The theoretical and experimental values for y_B found in Table 5 and 6 are also shown in the plot of Figure 8. Figure 8 shows y_B, required across the 12th fret for whole tone for a Fender Stratocaster. Results are given for both a standard set

of strings (D'Addario EXL120 "Nickel Wound" Super Light 9-42 strings) and the custom strings. It is demonstrated that the theoretical predictions of y_B give similar behaviour to the experimental results verifying the theory, but the distances of motion required are under-predicted by a little more than the error bars in some cases. This may be due to the player applying a component of force directly into the fretboard (perpendicular to the frets) in addition to the force parallel to the frets. This force is required to maintain a secure grip on the string and will help to sharpen the string (through bending the string into a curved shape between the 11th and 12th frets in the case of a 12th fret bend and through twisting of the string around its axis). Such a force can be used to increase the sounding pitch typically by around 10 cents thus explaining the errors of the order of 5%. A slight displacement of the bridge (in spite of the ESP arming adjuster) during the measurements could be another factor. Some non-linearity of the stress-strain relationship in the steel also cannot be discounted in the current measurements.

Figure 8 in particular verifies that the design of custom strings set out in Table 6 has been successful in giving close to equal lateral distances of motion for conventional string bends for the G₃, D₃, A₂ and E₂ strings. The standard string design shows significantly larger distance of bend required for the D₃ string in particular. Having a D₃ string that is significantly easier to bend is important not only for bends sounded on the D₃ string, but also makes bends on the adjacent strings easier. This occurs because the strings are only separated by around 9 mm and thus the D₃ string must be forced out of the way when sounding whole tone bends on the G₃ and A₂ bends as whole tone bends on the A₂ string in the direction of the E₂ risk running off the edge of frets while whole tone bends on the G₃ string are usually performed towards the D₃ string by choice. Pulling the G₃ towards the B₃ string is also possible but the B₃ remains more difficult to move out of the way.

The custom set of strings may offer the advantage that a desired pitch change for at least two strings can be achieved by moving the at least two of the strings through the same lateral distance.

The lateral distance of movement required to achieve a conventional pitch bend for the strings is such that for each of the first and second strings the same ratio of fundamental frequency to the fundamental frequency requires a lateral distance of movement to be substantially the same.

Non-linearities

Non-linearities result in the pitch of strings being noticeably higher (more than 15 cents sharp) when the string is vibrating at larger amplitudes as the tension is increased during the high amplitude points in the vibration cycle. This effect can produce undesirable pitch glides during loud playing. On conventional string sets, the low E (E₂) string experiences the most severe nonlinearities due to its low engineering strain (meaning the stretching of the string during high amplitudes increases the engineering strain by a larger fraction). The low E on the custom strings described here (with approximately equal engineering strain) will be improved in this regard because the initial engineering strain required for tuning up the low E string is higher than the low E on standard string sets. Advantageously, if the engineering strains are equal then the relative tuning of strings may improve after changes in temperature. If the engineering strains are equal then the distance required for changing the pitch on tuners of the same design may be substantially more equal (so strings will require equal rotations of the tuning peg for equal pitch adjustments).

As described above, a set of strings may be configured, by selecting at least one property or parameter of the cores and windings, to produce a matched change in pitch between the first and second string when moving the first string through a first lateral distance and when moving the second string through a second lateral distance, wherein the first lateral distance is substantially equal to the second lateral distance. In addition, by selecting at least one property of the cores and windings, a set of strings can be configured to have substantially equal tensions.

In further embodiments, a set of strings may be configured, by selecting at least one property or parameter of the cores and windings, to produce a matched change in pitch between a first string and a second string by applying a first lateral force to the first string and applying a second lateral force to the second string, wherein the first lateral force is substantially equal to the second lateral force. In further detail, the lateral force (F_B) required to be applied to a string to produce a matched pitch bend may be substantially equal between two or more strings of a string set. One method of achieving such an equal or matched lateral force is to have substantially the first and second string at substantially equal tensions and to move the first and second strings through substantially equal lateral distances. As lateral force is related to tension and lateral distance (see for example equation 35), a matched lateral force between strings may be achieved when the tension is unequal. In particular, a matched change in pitch between a first string and a second string may be achieved by having the first and second string at different tensions and moving the first and second strings through different lateral distances, such that lateral force applied to the first and second string is substantially equal. In this case, a difference in tension between the first and second string compensates for a difference in the lateral distance moved by the first and second string so that the lateral force remains substantially equal.

A lateral force applied to a string required to achieve a conventional pitch bend for the strings is such that for each of the first and second strings the same ratio of pitch change to respective fundamental frequency to be obtained requires a lateral force to be substantially the same.

As discussed above, at least one selected property may be the diameter of the cores of the first and second string. A related property, is cross-sectional area of the string. In further embodiments, the at least one property of the cores and windings selected to provide a matched change in pitch is the cross-sectional area of the string.

In further embodiments, a first unwound string made of circular wire is provided and a second wound string with a hexagonal core is provided. In such embodiments to provide a matched change in pitch between strings, the circular core and hexagonal core require slightly different diameters to match the cross-sectional area of the string, and hence, the linear density of the string.

In further embodiments, an unwound string is provided that has a cross-sectional area that is matched to a cross-sectional area of a core of a second string. In the above embodiments, different parameters for a custom string set are described. In further embodiments, a user interface is provided that allows a user to determine design parameters for a set of strings based on a set of input parameters. The design parameters include at least one property of the cores and windings of one or both of the first and second strings that is selected such that a change in pitch of the first string matches a change in pitch of the second string. The user interface allows the user to firstly select an instrument from a pre-determined list of standard instruments. These can be different instruments or different models of instruments, for example, different models of guitar. The user is then prompted to enter values for a set of input parameters. The user could enter values for core and winding gauges rather than simply on a single gauge number for a wound string. In some embodiments, if the entered parameters do not match a standard set of parameters then a set of design parameters are produced. A large manufacturer could stock large numbers of core and winding combinations or custom makers could build them to order.

The interface allows the user to enter the details of their instrument. A set of strings is designed for them that has as close as possible to equal pitch deviations for specified strings for a given tremolo arm movement, or equal sensitivity for conventional pitch bends. The set of strings designed could then be ordered and their string specification could then be saved.

The input parameters may include one or more sizes or dimension of the instrument the strings are designed for use with. The input parameters may also include one or more materials of the strings.

The input parameters may include:

- number of strings;

- string pitches (a user may choose from a number of standard options such as standard tuning and Drop D tuning and may then customise the values from there); scale length (inches or mm);

- stretchable length behind conventional nut (0 for locking nut) (inches or mm);

- stretchable length between the bridge saddle and tailpiece (0 unless the bridge is separate from the tailpiece);

- fingerboard/Bridge radius (inches or mm);

- height of middle strings from the pivot plane (inches or mm).

The user may also choose from a number of standard options such as Fender Standard Stratocaster with standard nut, Fender Standard Stratocaster with Original Floyd Rose, PRS with standard nut etc.. By selecting a standard option values for the above parameters will be selected or populated for further customisation.

Further input parameters may include:

string core shape (circular or hexagonal for instance)

wound string (yes corresponding to a wound string or no corresponding to a unwound string)

core material (steel or nylon etc.)

winding material (nickel plated steel, stainless steel, pure nickel, aluminium, phosphor bronze etc.)

matching strings (an option to specify which strings are being matched) match condition (an option for specifying whether tremolo pitch intervals, lateral force for conventional bends or lateral distance for conventional bends is being prioritised in the matching).

The interface also includes a calculate button that is selectable by a user. This button prompts an algorithm to run that produces optimised design parameters for a custom string set based on the values given for the above input parameters. The algorithm is based on the above input parameters, the string design parameters or output parameters that are determined include:

the core gauge (thousandths of an inch);

the winding gauge (thousandths of an inch).

A further option is provided to bring up the string design parameters for standard string sets such as an old style normal set or an old style balanced tension set or a set designed for Fender Standard Stratocaster optimised for in tune tremolo etc. or to edit the string design parameters provided by the algorithm.

The user is also presented with outputs that include the values for various other output parameters. These values are calculated using theoretical calculations (based on the input values for the input parameters and/or the determined string design parameters) for:

the tension (N or lbs);

the relative tuning for a tremolo arm move (cents); the distance of lateral movement for a whole tone conventional pitch bend (inches or mm);

the lateral force for a whole tone conventional pitch bend (inches or mm). The output would include graphical display of the tensions, relative tuning for a tremolo arm move (cents), the distance of lateral movement for a whole tone conventional pitch bend (inches or mm) and the lateral force for a whole tone conventional pitch bend (inches or mm) along for comparison with the data for a standard set of strings. The above input parameters, design parameters, and outputs are provided as representative examples. Due to relationships between values of input parameters and determined design parameters, it will be understood that different sets of input parameters and design parameters can be used.

In some embodiments, a system is provided for implementing the above method of designing a set of strings for a musical instrument. The system has a computer processor and memory configured to run user interface software. The memory may store the software and other parameters for use by the software. The system has user interface hardware, including, a display for displaying the user interface and an input device, for example, a keyboard, a mouse or touchscreen, to allow a user to provide the input parameters to the user interface. In some embodiments, the system includes a server with a network connection, for example, an internet connection. In some embodiments, the processor configured to perform determination of design parameters is provided on a remote server and accessed by a user via a web interface.

Above design parameters were determined for the custom string set. In some embodiments, an iterative approach is required to arrive at suitable values for design parameters for the custom string set. The following provides some non-limiting examples of equations that can be used to directly obtain optimum values for the one or more properties of the strings, in accordance with embodiments. The following provides detail on approximations that can be taken in determining said values.

Equations for designing multiple strings (such as guitar strings) to match the pitch sensitivity under tremolo arm use and/or conventional pitch bends are presented here. The evaluation of relative sensitivity of designs of guitar strings was described above. The design process for achieving a desired sensitivity is iterative. In the following, direct equations for calculating optimum values of d_w and are presented for achieving a desired sensitivity.

The equations for designs allowing matched tremolo arm sensitivity are presented, as are the conditions for simultaneously achieving matched tensions at pitch. Also presented are the equations for matched sensitivity to lateral displacement for achieving conventional bends along with the conditions for simultaneously achieving matched tensions at pitch and therefore matched lateral forces required by the player.

It is also shown how strings may be designed where the force required by a player for achieving a given conventional pitch bend may be matched for multiple strings in general, and how this may be achieved while simultaneously matching the tremolo arm sensitivity (at the expense of being able to freely specify matched tensions at pitch). These equations allow for matching the described parameters for any tuning, even for strings of differing length, core shape, core Young's modulus, core density, winding construction etc.

I. MATCHING TREMOLO ARM PITCH SENSITIVITY ON MULTIPLE STRINGS

Restating Equation 20, the distance of stretch when a tremolo arm is moved through an angle of 8 is:

and this can be shown to be:

According to this theory, strings with different nest angles, Φ, (for instance due to curvature of the bridge matching the curvature of the neck) wifl haw distances of string stretch that do not stay in a constant ratio as 0 changes (due the bracket containing In

principle this mokes perfect matching of the tremolo based pitch variation impossible; but in practice the values of φ for different strings arc simitar enough, and the value of tan(0/2) small enough in comparison to tan φ to allow for very good matching.

The ratio of sounding frequency for a string before tremolo arm move

{given by the limiting behaviour of Equation 2G for small fractional changes of length) is:

(45)

This implies that to match the frequency ratio through which two different strings go Ihrough during on identical change in

we have to match the ratio of for the two strings. Noting

that froni Equation 23:

τ is the ratio formed from the total mass of the string divided by the mass of the core, we may combine this with Equation 44 to work out the ratio of values of r for two strings that share the same sensitivity to tremolo arm motion as

where the superscript (n) for string number n and the superscript (m) for string number m. Strings designed with this ratio for the τ should thus share the same relative pitch before and after a move of 0 radians (and the bracket containing the tangent functions is very dose to 1 meaning that the relative pitches arc very close to being the same at all angles of practical interest)), giving

Normally where Λ" is the number of semitones between the mth (higher

pitch} and nth {lower pitch) string.

A. Diameters of windings for round-wound strings

Designing a string to achieve a particular value of τ can be achieved for cores or windings of any cross-sectional shape. In particular, the theoretical relationship between this value and the diameters of perfectly circular windings is known and

while this is an approximation {as the windings become slightly oval during manufactur- ing for instance), wo can combine the resulting equations along with polynomial fitting to experimental data to obtain a robust method of string design. The theoretical value of r calculated from the diameterij is Equation 17 (here given a subscript d to denote that this is based on the diameter calculation and will be larger than the value obtained by destructive testing of a real string with these design parameters):

such that the cross-sectional area of the string is is. the maximum

core width in the case of a hexagonal core. In practice hexagonal cores become somewhat rounded during manufacturing and the specification is usually given in terms of the narrowest part of the wire, , rather than the widest part, For real-world hexagonal core

strings, the approximation may be used to work out the widest width,

for available gauges, and the approximation gives improved results Tor the

cross-sectional area of hexagonal cores.

Equation 40 can be solved to obtain the theoretical value of d_a for obtaining a desired value of giving:

In practice the experimental value of τ obtained by destructive testing of the masses in a manufactured string, τ, differs from the theoretical values of calculated directly from the

design specifications using Equation 40, An example of thie data is shown in Table L

TABLE I. lite mass ratios obtained by destructive testing of the masses En manufactured string;, T, and the theoretical values, TVJ, calculated directly from the design specifications using Equation is the diameter of the windings and is the diameter of the circutar core.

A least squares polynomial fit (calculated with the ployfit function in MATLAB) was used to obtain a functional relationship between the experimental and theoretical values from Table I, giving

This means that a desired value of τ may be obtained for a string of given core diameter by specifying the winding diameter through substituting Equation 52 into Equation 51. B. Matching the tension

The tension is given by Equation

Matching the tensions of string number m and string number n implies:

and therefore

As noted after Equation -1ΰ, the ratio of must bo matched for two (strings to ba\O

matched sensitivities to tremolo arm movements and so

and using Equation 44 we obtain:

giving

where the following approximation is accurate enough for use in practice:

C. Designing strings for matching both tremolo sensitivity and tension

The first step in designing a set of strings is to choose a string for matching which will bo labelled as string number m. As an example, this may be a 0.016 inch diameter unwound G string on a guitar. This will have a known value of

the case of an unwound string). Wound strings designed to operate at a sounding pitch with matching sensitivity may then be built using Equation 59 to deduce the core

diameter and then calculating the mass ratio r from Equation 48 and building strings to achieve this mass ratio. This can be achieved by substituting the desired value of τ into Equation 32 to calculate Td and substituting this into Equation

to deduce the winding diameter in the case of round-wound strings. Strings can then be constructed by choosing from available string gauges. Optimising the sensitivity means choosing the best match for available gauges of wire within acceptable limits of the resulting tension at

pitch. The tension at pitch ie roughly proportional to the core diamotcr squared so. as an example, limiting the search to cores delivering within 10% of the tension being matched means searching for cores diameters within around 5% of the optimum value.

High precision measurement of the string height above the pivot, ft, is difficult (not least, due to the finite width of the string), so small adjustment of this parameter from string to string may be required to achieve optimal matching between experiment and theory. Π. MATCHING CONVENTIONAL PITCH BEND SENSITIVITY ON MULTIPLE STRINGS

A. Lateral distance for conventional bends

Restating Equation 41, the lateral distance required to achieve a conventional pitch bend is:

(60)

where vu is the sounding frequency of the unbent string and v_B is the sounding frequency of the bent string. Different strings thus require the matching of to achieve equal

sensitivity for this condition. The equation for specifying the ideal value of r for matching the sensitivity for the lateral distance of conventional pitch bends for string n to string m becomes:

(61)

and the ideal core for matching tension becomes:

(62)

By matching both the tension of the string and the sensitivity of conventional pitch bends to lateral displacement, the lateral force required from the player is also matched. B. Lateral force tor conventional bends

It is also possible to match the force required for a bend without matching the tensions. Restating Equation 38, the force required for a conventional pitch bend is:

It can then be shown that balancing Fa for a given pitch interval requires:

There arc many ways of achieving this condition without needing to match the tensions or distances of lateral movement. One choice is to set the cores diameters as being equal (for example giving constant for different strings if all other parameters

arc equal). Alternatively, it is possible to simultaneously obtain matched sensitivity for tremolo arm moves and the player force required for conventional pitch bends through using Equation 48 to obtain τ and then substituting this into Equation G4 to get the core diameter as:

A skilled person will appreciate that variations of the enclosed arrangement are possible without departing from the invention. Accordingly, the above description of the specific embodiment is made by way of example only and not for the purposes of limitations. It will be clear to the skilled person that minor modifications may be made without significant changes to the operation described.

Claims

CLAIMS:

1. A set of strings for a musical instrument comprising:

at least a first and second string, wherein the first string comprises a core and a winding and the second string comprises at least a core;

wherein each string is configured to be held at a selected tension in a musical instrument to produce a tone of a respective different desired pitch; and

wherein at least one property of the cores and windings of one or both of the first and second strings is selected such that a change in pitch of the first string matches a change in pitch of the second string for a variation of the string tensions away from the selected tensions.

2. A set of strings as claimed in claim 1 , wherein the musical instrument comprises a guitar.

3. A set of strings according to claims 1 or 2, wherein the variation of the string tensions is caused by operation of a tremolo arm mechanism.

4. A set of strings according to any preceding claim, wherein the at least one property comprises the diameters of the cores of the first and second string.

5. A set of strings according to claim 4, wherein the diameters of the cores of the first and second strings are substantially equal.

6. A set of strings according to any preceding claim, wherein the first and second strings are configured such that the engineering strain of at least the first and second strings is substantially the same when the strings are held at said selected tensions to produce the desired pitches.

7. A set of strings according to any preceding claim, wherein the at least one property is selected such that the selected tensions to produce the desired pitches are substantially equal for the first and second strings.

8. A set of strings according to any preceding claim, wherein the at least one property comprises a ratio of string mass per unit length or string to winding mass per unit length of string.

9. A set of strings according to any of the preceding claims, wherein at least one of the cores of the first and second strings has one of a circular, hexagonal cross- section or braided core.

10. A set of strings according to any of claims 3 to 9, wherein:

the pitch of each string is dependent on a ratio of tension to mass per unit length for the string;

each string is configured such that the variation in pitch caused by operation of the tremolo arm mechanism is primarily determined by properties of the core and the engineering strain applied in tuning the instrument;

and the mass of the windings per unit length of the string for the first and second strings is selected to provide the desired different starting pitches for the different strings when under the selected tension.

11. A set of strings according to any preceding claim, wherein the matching of the changes in pitch for the strings is such that at least one of:

for each of the first and second strings the ratio of the change in fundamental frequency to the fundamental frequency is substantially the same.

12. A set of strings according to any preceding claim, wherein the material of the core of the first string is the same as the material of the core of the second string, and/or wherein the material of the winding of the first string is the same as the material of a winding of the second string.

13. A set of strings according to any preceding claim, wherein at least one of the cores comprises steel.

14. A set of strings according to any preceding claim, wherein at least one of the windings comprises nickel or nickel-plated steel.

15. A set of strings according to any preceding claim, wherein the first and second strings have a core diameter in a range 0.1 mm to 4 mm.

16. A set of strings according to any preceding claim, further comprising at least one unwound string having a string diameter that is matched to a core diameter of the first string and/or second string.

17. A set of strings according to any preceding claim, further comprising at least a third string comprising a core and a winding and configured to be held at a selected tension in the musical instrument to produce a tone of a different desired pitch to that of the first and second strings, wherein the core and winding of the third string are selected to match the change in pitch of the third string to the first and second string, wherein at least one property of the core and winding of the third string is selected such that the change in pitch of the third string matches the changes in pitch of the first and second strings for the variation of the string tensions away from the selected tensions.

18. A set of strings as claimed in claim 17, wherein the pitches of the first, second and third strings are E₂, A₂, D₃.

19. A set of strings as claimed in claim 17 or 18 comprising at least three unwound strings, wherein the three unwound strings produce associated tones at their selected string tensions in the musical instrument comprising pitches: G₃, B_3| E₄.

20. A set of strings according to any preceding claim, wherein the variation of the string tensions comprises a variation of at least 40 % away from the selected tensions.

21. A set of strings according to any preceding claim, wherein said variation of the string tensions are substantially the same for at least the first and second string.

22. A set of strings according to any preceding claim, wherein a lateral distance of movement required to achieve a conventional pitch bend for the strings is such that for each of the first and second strings for substantially the same ratio of pitch change to respective fundamental frequency to be obtained requires a lateral distance of movement to be substantially the same.

23. A set of strings according to any preceding claims configured to produce a matched change in pitch between the first and second string by applying a first lateral force to the first string and a second lateral force to the second string, wherein the first lateral force is substantially equal to the second lateral force.

24. A set of strings according to any preceding claim, wherein a lateral force applied to a string required to achieve a conventional pitch bend for the strings is such that for each of the first and second strings the same ratio of pitch change to respective fundamental frequency to be obtained requires a lateral force to be substantially the same.

25. A set of strings according to any preceding claim, wherein the at least one property comprises the cross-sectional area of the cores of the first and second string.

26. A set of stings according to claim 25, wherein the cross-sectional area of the first and second strings are substantially equal.

27. A set of strings according to any of the preceding claims, further comprising at least one unwound string having a cross-sectional area that is matched to a cross- sectional area of the core of the first string and/or second string.

28. A musical instrument including a set of strings according to any preceding claim.

29. A method of matching variations of pitch for a set of strings in a musical instrument comprising:

installing a set of strings according to any of claims 1 to 28 in the musical instrument, wherein each string is installed to be at its selected tension to produce a tone of a respective different desired pitch, and the strings are such that for at least some of the strings changes in pitch for a variation of the string tensions away from the selected tensions are matched.

30. A method of designing a set of strings for a musical instrument, the set of strings comprising at least a first and second string, wherein the first string comprises a core and a winding and the second string comprises at least a core, wherein each string is configured to be held at a selected tension in a musical instrument to produce a tone of a respective different desired pitch, the method comprising:

inputting values for one or more input parameters;

determining values for at least one property of the cores and windings of one or both of the first and second strings using the inputted values for the one or more input parameters, such that a change in pitch of the first string matches a change in pitch of the second string for a variation of the string tensions away from the selected tensions.

31. A method as claimed in claim 30, further comprising:

producing the set of strings in accordance with the determined values for the at least one property of the cores and windings, optionally and at least one of the input parameters.

32. A system for designing a set of strings for a musical instrument, the set of strings comprising at least a first and second string, wherein the first string comprises a core and a winding and the second string comprises at least a core, wherein each string is configured to be held at a selected tension in a musical instrument to produce a tone of a respective different desired pitch, the system comprising:

a user interface, for example a web interface, for receiving input parameters from a user;

a processing resource configured to use the input parameters to determine values for at least one property of the cores and windings of one or both of the first and second strings, such that a change in pitch of the first string matches a change in pitch of the second string for a variation of the string tensions away from the selected tensions.

33. A computer program product comprising computer-readable instructions that are executable to perform a method of designing a set of strings for a musical instrument, the set of strings comprising at least a first and second string, wherein the first string comprises a core and a winding and the second string comprises at least a core, wherein each string is configured to be held at a selected tension in a musical instrument to produce a tone of a respective different desired pitch, the method comprising:

receiving input value data for one or more input parameters; determining values for at least one property of the cores and windings of one or both of the first and second strings using the received input value data for the one or more input parameters, such that a change in pitch of the first string matches a change in pitch of the second string for a variation of the string tensions away from the selected tensions.