A Primer on the History and Technology of Strings

Ephraim Segerman

String Materials

Materials used in ancient times

Depictions of stringed instruments (lyres, pillar-less harps and long lutes) start to appear on Near East artifacts from about the 30th century BC.¹ The earliest known mention of a string material in Europe or the Near East was in Homer’s Odyssey (9th century BC), where strings for the lyre were specified as ‘twisted sheep-gut.’² An old Greek legend related that hemp was used originally, and was replaced by gut.³ Ancient writers mentioned the use of the vegetable materials bast, hemp, flax and lianas.⁴ The strings on a surviving Egyptian long lute from the 16th or 15th century BC appear to be of gut.⁵ In the Far East, evidence of stringed instruments starts with a long zither that appeared around the 11th century BC, and the primary string material there appears always to have been silk.⁶

Subsequently used non-metallic materials

Instrument string materials mentioned in European sources from the 5th to the 14th century were surveyed by Page.⁷ Most indicate that gut was used. In most cases, this was unambiguous since intestines were specified, but other terms used were less specific (animal insides or viscera) or apparently specified other things (sinews, nerves or skin). Strings can be made of tendons (sinews), and a 16th century collection of recipes includes one for making them from tendons taken from the backs of horses, pounding them in a cloth with a mallet, and twisting them with animal glue to bind the separated fibres together.⁸ Chemically, tendons are made of collagen, the same as gut, so their properties should be similar. The knowledge of anatomy in medieval times did not distinguish between nerves and sinews the way we do, and for our purposes, they were the same. The use of skin (or leather) as a string material is unlikely to have ever been common because of its comparatively poor strength and of the difficulty in making such strings uniform enough to sound well. It is likely that most of the reports of skin and sinews as string materials were based on the strings appearing to be made of these materials (more like them than like an animal's intestines), but it was actually gut.

Recently (in the 21st century), competition for sheep’s guts from sausage making has caused shortages and soaring prices. Another material made of collagen with properties very similar to gut, beef serosa is often now used as a cheaper substitute for sheep’s gut. Serosa is a membranous material that keeps the gut in place in the animal by suspending it from the peritoneum.

Medieval instruments strung with gut included lyres, harps, rotes, fiddles (including giges and rebecs), hurdy gurdies, citoles, lutes and gitterns. The animal that the gut came from was almost always sheep, and sometimes more specifically rams or wethers (castrated rams). Occasionally wolf gut was mentioned, usually with the oft-copied warning that having wolf gut and sheep gut on the same instrument invites trouble. In the 20th century, ropes made of pig's gut was often used to support the counterweight in sash windows.

Gut strings were usually made by the player until the craft of string making became professionalised, (the earliest clear evidence of which comes from the 15th century in Munich, Germany).⁹ A 13th century instruction stated: ‘take the intestines of sheep and wash them cleanly, then place them in water or lye for half a day or more until the flesh comes away easily from the material of the string which is like a sinew. Then take the flesh from the material cleanly with a quill or with a clean finger. Next put it in strong lye or red wine for two days. Then take it out and dry it with a linen cloth and join 3 or 4 together according to the quantity that you wish to have, then twist them until it is enough. Next extend it over a wall and allow it to dry. When it is dry, take it below and put it in a place that is neither too humid nor too dry, because excessive dryness easily destroys them - as does dampness. Then keep them for use.’¹⁰

The Arabs learned the art of making silk from the Chinese early in the Middle Ages. By the 9th century, silk as well as gut was used on Arabic lutes. By the mid-6th century silk production had spread to Byzantium, by the 13th century to Greece and Italy, and by the 15th century to the rest of Europe.¹¹ In Page’s survey, two sources from early in the 14th century mention silk stringing. One specified it for an expensively decorated fiddle, and the other mentioned it (with gut and metal) generally as string materials. Since then silk (the strands bound together by gum Arabic according to 19th century sources) has been an occasionally available alternate to gut for bowed strings. From late in the 18th century, silk floss became standard for the cores of wound strings used on guitars.

In the middle of the 20th century, nylon floss replaced silk for the cores of guitar wound strings, and nylon monofilament replaced gut in unwound guitar strings. Since then, nylon and other plastic materials as well as metal ropes have replaced gut as the cores of most bowed strings. Polyester strings have a density close to that of gut, and polyvinylidene fluoride strings have a density much greater than gut, and monofilament strings of these types are used on lutes and guitars to give better tone than nylon monofilament strings.

One of Page’s medieval sources indicated that the strings on fiddles were either of guts or the hairs of horses. The source was written in 15th century England, giving a commentary on Psalm 150:4 by a 14th century author.¹² A 14th century Welsh source indicated that horse hair strings were used on the some instruments.¹³ Horse hair is still being used for strings on some folk instruments.

Metal strings

The earliest evidence for metal stringing is from late in the 12th century. The author described the rapid and intricate style of playing in Ireland and its imitation in Wales and especially in Scotland. He mentioned the names of instruments used in each place (what these names specifically referred to is not clear today, but the harp was most likely amongst them) and that they used copper alloy strings (the word used meant bronze, but it could have been any other copper alloy). It is not clear in that source whether the metal replaced or was an alternative to gut.¹⁴ Two French sources from early in the 13th century and one a century later mentioned harps with silver or gold strings. A tradition of using metal strings on the harp didn’t take root in France, but it lasted till the late baroque in Ireland. The only other medieval instrument with a tradition of metal stringing was the psaltery, with many sources starting from the middle of the 13th century saying the strings were of copper alloy or silver (before then, and occasionally afterwards, gut stringing was indicated). Outside of Europe, the use of copper alloy strings by the 13th century was reported in China.¹⁵

During ancient classical and early medieval times, iron wire was made and valued for its strength as wire and its stiffness when made into nails. It was wrought (hammered into shape) because it was too stiff to pull through a draw plate, as copper alloys and precious metals were thinned to the desired diameters. Wrought iron wire is too uneven to use for strings. In the 14th century, German wire makers harnessed water power to be able to draw iron wire. Iron wire is much stronger than the metal string materials previously used (copper alloys and precious metals), so they can be tuned to higher pitches. Thus using both iron for higher pitches and copper alloy strings for lower ones allows a wider range of pitches to be played in an instrument with a limited range of string lengths. This contributed directly to the development of the harpsichord and clavichord around 1400. Iron strings are more stable and durable than other string materials, so they also have advantages for instruments needing only a limited pitch range. This contributed to the development of the cetra in Italy in the 15th century, which was a modified revival of the old citole and progenitor of the cittern.

Around 1580, a type of iron wire appeared that was considerably stronger than previously made, so it could be tuned half an octave higher, even as high as gut. It was made using a secret process by Jobst Meuler in Nuremberg. New instruments such as the small English cittern and the orpharion were developed to exploit the range this wire offered, and some existing instruments, such as the violin and archlute, occasionally used it. Evidence from Praetorius indicates that after 1600, Meuler refined this process for even greater strength. He apparently was much better as a metallurgist than as a politician. In 1608, a rival was granted a monopoly on all wire making from the Imperial Court in Vienna, but in 1610, the Nuremberg Town Council supported Meuler’s claim that he invented a new kind of wire (in great demand) that others couldn’t make, and so was outside the monopoly specifications. In 1621, Vienna granted the rival a more powerful monopoly, after which a resolution of the Town Council had to be passed each time to allow Meuler to fill an order. He apparently then gave up his business, keeping his secret.¹⁶ The next time that a wire became available that could tune as high as Meuler's wire (as high or higher than gut) was the 19th century, when steel piano wire was developed.

In the second half of the 20^th century, a spin-off of the space programme was a steel alloy wire much stronger than piano wire, and it has been available from an American string maker for the New Violin Family, which needs it.

String Constructions

Twist in gut strings

A typical string twisting machine of the Renaissance (and later) had a trough over which the strings were twisted. It collected the drips from the wet gut and let them run off, and it was deep enough so that the gut could droop low in it before twisting and not get too tight from contraction when fully twisted. On one side of the trough were two twisting hooks on shafts that had gears of the same size on them that meshed with a larger gear on a third shaft that had the turning handle. This gearing made the two hooks rotate the same amount in the same direction when the handle was turned in the opposite direction. On the other side of the trough was a stationary peg around which the gut was threaded on the way from one hook to the other.

The end of the strand of gut was tied into a loop that went into one hook, loosely threaded around the stationary peg to the other hook and then back again around the peg to the first one. This threading continued until the number of threads between the hooks was enough (from previous experience) for the desired final string diameter, and then the strand was tied into a loop that went onto one of the hooks, and the remainder was cut off. After the twisting, the end loops were removed from the hooks and stretched over a pair of pegs in a drying rack.

The degree of twist on a string can be measured by the amount the string contracts as a result of twisting. It can be observed directly by noticing the angle between the visible gut fibres on the string surface and the string direction. A typical low-twist string has an angle of less than 30 degrees, and a typical high-twist string has an angle over 45 degrees. The number of turns of twist needed to give a particular amount of twist is inversely proportional to the initial string diameter, so a thinner string requires more turns than a thicker one. A particular number of turns puts more twist into a thicker than a thinner string. If maximum twist is desired, the twist needs to be topped up regularly on the drying rack as the string dries and shrinks in diameter. One of the pegs may have to be moved to allow for the further contraction. While twisting, one can tell that the twist is maximum when a portion of it (usually near one end) starts taking the shape of a corkscrew. One could continue twisting such a corkscrew until the whole string is a tight helix, but such a string usually breaks before reaching a useful tension.

When a string is twisted, a strand near its surface has a longer path per turn than a strand near the centre. Under tension on the rack, the surface strand is under greater tension, so it migrates towards the centre (pushing lower-tension strands outwards) along some of its length, tending to equalise the strand lengths and tensions of fibres. This is made more efficient if the strands regularly moved between the regions in the centre and the outside of the string, as would happen if the string is made of several strings twisted together, each with some twist to start with (i.e. a roped string).

Roped gut strings

An early 12th century source exploring moral associations with the psaltery stated: ‘A string is dried, twisted (torquetur), and then stretched; so must Man dry himself from fleshly desires, be plaited with virtues (virtuitibus torqueri) and stretch himself with love ...’.¹⁷ The plaiting refers to the virtues, not the string, but this shows that the idea of roped gut would not have been far-fetched before the Renaissance if the players felt that it was useful and practical. It is very likely that at all times, musicians would have twisted together several available thin strings to make a desired thicker string that was otherwise not available.¹⁸ Rope construction became useful when lutes added open-string range using 6th courses, and it became practical when string making became professionalised, using twisting machines like that described above. One can make a roped-gut string on such a machine by first making two or more identical strings, and then putting the end loops of all of them on each of the two hooks and twisting again. If the first twisting has few turns, the resulting string has a smooth surface, not showing the regular bumps expected from rope construction, and there is not much range expansion. With maximum twist in the first twisting, the string ends up with a bumpy surface, which can be polished smooth, but one gets much more range expansion.

There is evidence that such a type of gut string, which was particularly uniform and elastic, was available from Munich early in the 16th century.¹⁹ It was not widely used, probably because of its cost. After then it was (expensively) available from Spain (affluent from American gold), where a thriving string-making industry developed in Barcelona.²⁰ An indication of how costly that could be is in the accounts of the French court in 1543, which include 180 Livre tournois for 4 viols for Henry II and 9 Livre tournois for a set of strings for one viol, i.e. a set of 5 gut strings cost a fifth that of an average viol made for a king. A sudden decline of vihuela activity appears to coincide with the massive bankruptcy of the Spanish court in 1557, which bankrupted many of the South German merchants who had been running the economy.²¹ The vihuela decline could well have resulted from these strings becoming unavailable, suggesting that the original Munich string makers had been running the Barcelona industry, and they were amongst the casualties. During the time of the Spanish availability, these strings acquired the name ‘catlin’ in England, probably from ‘Catalan’ (Catalonia being the region Barcelona is in). From the 1570s onwards, string-making centres in France, Germany and Italy produced such strings. They then were reliably available at affordable prices so new instruments could safely be developed to exploit them. Roped gut strings then showed the bumps, especially on very thick strings. These were called ‘cordone’ in Italy.

Twisted metal strings

In the twisting of gut, the wet strands are very flexible and slippery, so they present little resistance to sliding past one another and changing path during the twisting process. This allows all of the twisting to be accomplished by turning one end (the twisting of both ends in the twisting machine is just a means of halving the number of turns required and keeping the width of the trough down to half the length of the final twisted string). This flexibility and slipperiness is not available with twisting metal wire or the vegetable materials in normal thick ropes. In these cases, the twisting occurs between the rotating hook and a moving block around which the strands are guided. The twist angle is kept constant by making the block retreat from the hook as the section of rope lengthens. How close the block is kept to that advancing roped section determines the angle that the strands go into the rope, and so the block movement controls the tightness of the twist. The other end of each strand is constrained by a second moving block that provides the tension that keeps the rope taut. The twist in the rope creates twist in the strands between the two blocks, and there must be provision in the second block to relieve that twist which can cause metal strands to break or other kinds of strand to corkscrew. The second block needs to move towards the hook during twisting to keep the tension in the strands from changing.

The earliest evidence for twisted (or roped) metal strings is from the middle of the 16th century, when it was used on French citterns. Later in that century, maximum-twist brass strings became available in England, and led to bandoras and orpharions with the slanting frets and bridge fan-shaped. Twisted brass strings were used extensively in the 17th century. In the 18th century, they were used on citterns and mandolins, but by then, maximum-twist strings, which are the most difficult to make without breaking the strands, appear to have been no more available. Twisted steel strings are used extensively nowadays as cores of wound strings.

Overspun or wound strings

When a string has a core that supports all of the tension, and that core is covered by some substance that is wound onto it to increase its weight, it is called a ‘covered’, ‘overspun’ or ‘wound’ string. This kind of string was invented around the middle in the 17th century. In the first half-century after that, it was not widely used except on instruments that could then tune lower than had previously been possible for their sizes. During the 18th century, for strings other than those involved in the expansion to a lower pitch range, players started using them on other strings (such as the violin g) where the quicker response they offered was needed.

In making a wound string, the core is rotated under tension either between two coupled rotating hooks or between one rotating hook and a bearing on the other end that lets the core rotate freely with the twist created by the rotating hook. The covering material is tied initially to the core loop in the hook, and it is traditionally hand fed with constant resistance (to ensure even tightness) as the rotating core pulls it on. The operator’s other hand is often placed over where the covering material meets the core to dampen vibration and guide the winding. When enough of the core is covered, the covering material is fixed to the core, with the rest cut off. When winding on gut, the final fixing is often done by tucking the cut wire under the last turn and breaking the wire there with a quick tug, while when winding on metal, the fixing is often done by previously flattening the core in the region of the end so the winding there is resistant to unwinding. In modern machines, the feeding of the covering, moving in step with the advancing covered portion of the string, is mechanised, and it is often speeded up and stabilised by feeding from both sides at the same time, putting on two turns of covering for each turn of the core.

Before the mid-20th century, the vast majority of covered strings were made of round metal wire close-wound on gut. An occasional practice in the 19^th and 20^th centuries was to grind the surface of such a wound string smooth. In a close-wound string the covering is wound so that each new turn is in contact with the previous turn. Before the 19th century, there often was a string made with space between windings (called demi-filée in France), usually located as the next string lower than the lowest all-gut string. It provided the smoothest tonal transition between an all-gut string and a close-wound string. It is not clear whether the winding was on the surface of a smooth core or in the grooves of a bumpy roped core.

Evidence of metal wound on iron or silk first appeared in the second half of the 18th century for use on plucked instruments. Silk wound on silk is currently traditional for the stringing of the Japanese koto (and has recently been imitated as gut on gut by an American lute maker). Praetorius reported that the heavier strings on the geigenwerck were ‘made from thick brass or steel wound with fine parchment‘. The parchment here was apparently to facilitate bowing by the parchment-covered rosined wheels. Since the parchment did not significantly add weight, it is possible that it was only in the vicinity of the wheel.

Since the middle of the 20th century, strings with several layers of thin flat metal winding (fed by machines) have become standard on bowed instruments. Between the core and the first layer, and usually between each metal layer, is a layer of plastic floss. The floss prevents buzzing of loose metal layers when the core stretches and thins. Since the windings don’t stretch when the core stretches, no spaces between turns then develop (as happens with round-wound strings, which bind to the core). This inhibits tuning instability with a gut core due to moisture diffusion between the windings. There is a small loss of power due to energy absorption of the floss layers, but there is usually reserve power in bowing. Such flat winding has not been adopted for classical guitar stringing because all of the power possible is needed.

String Technology

Mersenne-Taylor Law

This law allows the calculation of any one of four variables if one knows the values of all of the other three. These variables (with the symbols representing them) are the vibrating string length (L), the fundamental frequency of the sounding pitch (f), the string tension (T) and the string mass per unit length (m_L). The relationship is fL = (1/2)sqrt(T/m_L), where two adjacent symbols means they are multiplied, a slash / means that what is before is divided by what is after, a bracket ( ) or [ ] means that what is inside must be treated as a unit when operations outside of it are performed, and sqrt represents the square root of what is in the following bracket ( ). If the string is a uniform cylinder, then m_L = ρA, where ρ is the density of the material and A is the cross-sectional area. Now the stress on the string material (S) is defined as T/A. The maximum stress that a material can take before it breaks is defined as its tensile strength. Also, the area can be expressed as A = πD²/4, where D is the string diameter.

In each of these equations, all of the variables involved need to be expressed in consistent units. In practical circumstances, it is often convenient to use inconsistent units. For instance, I find it convenient to express vibrating string length (L) in cm, string diameter (D) in mm, string tension (T) in Kg of force and density (ρ) in gm per cubic cm. Then with this combination of units, the Mersenne-Taylor Law can be expressed as fLD = 5588sqrt(T/ρ). If the string is not a uniform cylinder (as with a wound or bumpy roped string), the D represents an ‘equivalent diameter’ that is the diameter it would have if it were a uniform cylinder of the core material that had the same mass per unit length.

Wound strings

Before the 19th century, there is no evidence that string makers commercially offered wound strings. The winding of metal wires on gut or metal cores was done by players, instrument makers or specialist purveyors of musical goods. Some evidence suggests that the variety of wire diameters used by a particular string winder was very restricted.

Various people have offered equations for calculating the equivalent diameter D of a close-wound string from the core diameter D_c, the core density ρ_cthe initial wire diameter D_w and the wire density ρ_w. The equation usually takes the form D² = D_c²[1+π(ρ_w/ρ_c)(D_w/D_c)M], where M depends on the approximating assumptions made. None are particularly accurate, but high accuracy is rarely needed. The form we at NRI use is M = sqrt[A+B(D_w/D_c)²], where in our theoretical model, A = (D_w/q)² (where q is the distance along the string axis for each turn) and B = 1/(π)².²² It works well with open-wound strings, and with close-wound strings when D_w is much less than D_c. For winding copper on steel when D_w and D_c are nearly the same, we experimentally found that A = 2 and B = 0.9 works well.

Relative tensions of different strings

We usually find that strings at higher pitches are thinner than strings tuned to lower pitches, but how the thickness varies with pitch depends on complicated factors such as the instrument’s response at different frequencies, the feel of the string to the player and how we listen to the sound. A tension profile of the strings is often a useful way to describe the variation. On instruments where all the strings have the same length, and melodies are expected to move amongst most of the strings, the tension has tended to be the same for those strings involved. This rule held for 6-course single-nut lutes, but the tensions tended to fall off on the low basses added onto the later ones. It has always held for viols, except for the lowest strings on the smaller sizes of Renaissance ones, which had somewhat lower tensions to get the high-twist gut to sound decently. It held for fiddles until the middle of the 18th century, when the Italians adopted a wound 4th on the violin and lowered its tension, probably to maximise quickness of response. This practice of dropping tension with each lower string spread with the popularity of Italian opera, and it survives on the violin today.²³ In the 19th century, this was also often followed by cellos except that sometimes the 4th string had a higher tension than the 3rd, almost as high as the 1st.

On instruments where the different strings have different lengths, the majority of the strings tend to follow the rule that the tension is proportional to the length raised to a particular power. In this power law, for any two strings 1 and 2, the ratio of their tensions T₂/T₁ = (L₂/L₁)^p, where L₂/L₁ is the ratio of lengths and p is the power. For the modern harp, p is about 0.6 on the unwound strings, and for the harpsichord, a p that is about 0.3 fits the top 4 or more octaves of most original stringings in the Rose & Law handbook.²⁴

Special cases are when p = 0 and p = 1. When p = 0, the tension is constant, independent of length. This happens most commonly on one string when one fingers it up and down a fingerboard. It also happened between strings on some early keyboard instruments. When p = 1 the tension is proportional to the length. This is the tension-length principle that seems usually to apply to corresponding strings of different members of a family of plucked or bowed instruments with parallel nuts and bridges.

Displacement

When we pluck a string, just before releasing the string, the string at the plucking point is displaced by a distance d from its resting position. The plucking point is at a distance X from the bridge. The vibrating length of the string is L, and the displacing force is F. If d is much smaller than X (as is the case in normal playing of musical instruments), then dLT = FX(L-X). If the amount that the string is stretched is ΔL, then ΔL/L = d²/[2X(L-X)] = [X(L-X)/(2L²)](F/T)².²⁵

Elasticity and Hooke’s law

Hooke’s law states that in an elastic string, a change in strain is proportional to the change in the stress that causes it. The change in strain is the fraction of its length that it stretches (ΔL/L), the change in the stress (ΔS) is the change in tension divided by the cross-sectional area (ΔT/A). The elastic or Young’s modulus (E) is defined as the stress divided by the strain, or E = (ΔT/A)/(ΔL/L). It should be called a stiffness modulus because the more elastic or stretchy a material is, the smaller the modulus is. The elastic modulus of a bumpy roped-gut string is about half that of a high-twist one, which is about half of a low-twist one.

The elastic constant really is a constant (no matter what the stress or strain is) if the atoms and molecules of the material keep their same set of neighbours during the stretching. If as a result of the stress, the arrangements of neighbours changes by creep, we get more stretching from a change of stress. This is called inelastic stretching if it is not reversible when the stress is stopped (as it is with elastic stretching). This occurs with most materials when the stress is near the breaking point, and with many (like brass, gut and most plastics) at other stresses when there is creep while the neighbours change, but after a while, they eventually get locked in.

With gut there is also an in-between situation which can be called semi-elastic stretching, which is most noticeable when some tension returns a while after one detunes a string to apparently no tension. This reverse creep results from the stable paths of some strands being different at high stress than at low stress.

Pitch instability

The pitch at which a string sounds can be dependent on how hard it is played, i.e. on the amplitude of the vibration. The pitch distortion at each instant is (from the above) ∆f/f = (1/2)(E/S)(∆L/L). The average ∆f/f over the vibration cycle is half the maximum, or ∆f/f = (1/4)(E/S)(∆L_max/L), where the term ∆L_max/L is the maximum elongation in the cycle. That maximum elongation in the plucked string is that of the displacement before the string is released, so ∆L_max/L = d²/[2X(L-X)]. The energy put into that elongation is all the string has, and as the the vibration of the body of the instrument sucks it away, the vibration amplitude on the string decreases and the pitch drops. In the worst case, if the ear’s initial judgment of pitch is not confirmed immediately afterwards, the perception is only that of a twang (this is usually experienced with a string at very low tension). For the bowed string, it is the maximum displacement during its motion, which is at the centre of the string (where it would be measured), so ∆L_max/L = 2d_max²/L. On an unfretted bowed instrument, pitch instability can be corrected (as is often done on the cello C) by fingering flat to stay in tune in very strong playing.²⁶

Highest pitch for a string

Some early instructions for tuning an instrument start with tuning the highest string as high as it could go without breaking. This was an expensive choice since a string near breaking has more creep than at lower stresses, so the judgment of how high to tune involves a judgment of how long the string can be expected to last. Modern opinions I’ve heard state that, to make the string last long enough, the highest a gut string is tuned is roughly a tone lower than the pitch at which it would break if one kept tuning it up. For the highest iron string on a harpsichord, it is about one semitone lower, while on a clavichord (where the string gets battered more), it tends to be more like two semitones.

The stress condition for string breaking is the tensile strength, S_max = T_max/A. The tensile strength of a twisted string depends on the amount of twist since the strands have to stretch more then the total string. For the highest string, the lowest twist that makes it an acceptable string is usually chosen to maximise strength. The tensile strength of a plain metal string usually depends on how hard it is drawn. The harder it is drawn, the more brittle it becomes. That brittleness makes thicker strings more prone to breaking when tied around pins at their ends than thinner ones, so thinner strings can safely be drawn harder than thicker ones. The concept of a particular tensile strength for a particular metal is valid in the context of the highest stress on an instrument because the diameters of pins and the thinnest string diameters happen to be similar.

The highest string is usually a uniform cylinder, so the Mersenne-Taylor law at breaking can be expressed as (fL)_max = (1/2)sqrt(S_max/ρ). An excellent measure of the highest safe stress that musicians find acceptable in a particular culture is the product of the highest frequency (which requires knowledge of both the nominal pitch and the pitch standard) and the vibrating string length. With the string length expressed in metres, we call this maximum the ‘fL product’. Different string materials would have different fL products because they have different tensile strengths and different densities.

Highest pitches for gut strings

Praetorius’s Syntagma Musicum (1619,20) provided information on his pitch standard (about a’ = 430 Hz), as well as nominal pitches and string lengths (from scaled drawings) for a wide variety of instruments. The ones with the biggest open-string pitch range were most likely to have the highest strings as high as they could comfortably go. These were the lute, the Paduan theorbo, the large 5-string fiddle and the viola bastarda. The fL products were 211, 209, 207 and 209 metres/sec respectively, making 210 a good general estimate.²⁷ Using this fL product, the left half of Table I lists the longest string lengths for most of the highest pitches one might encounter in the three most commonly used pitch standards in Praetorius’s time (Italian corista and English consort pitch were the same as German chorthon).

In the middle of the 19th century, when orchestral woodwinds acquired the power to raise pitch standards to sound more brilliantly, many violinists had to tolerate an fL product of over 220 on their gut 1st strings. Violin 1sts broke too often, and later in the century the compromise arrived at was a’ = 440 Hz, when the violin 1st had an fL product of 216. It is highly unlikely that modern gut is any weaker than early gut (as some musicians have claimed) since the fL product for some treble harp strings can go over 220. Modern baroque violinists, used to long-lasting steel 1sts on modern violins in their training, accept the breaking rate provided by an e” fL product of 203 resulting from the modern baroque pitch standard of a’ = 415 Hz.

Highest pitches for metal strings

If we assume that the string lengths of the bandora depicted by Praetorius were unchanged from the time when the fan-shaped fretting was invented, and that such fretting resulted from a maximum open-string pitch range, then the highest 1st course iron fL product was 157, about 5 semitones lower than gut.²⁸ Then Meuler’s wire appeared, with an fL product of 222 on Praetorius’s orpharion with a g’ 1st course (6 semitones higher), and when Meuler refined his methods he achieved an fL product of 276 on Praetorius’s theorboed lute (almost 4 semitones higher yet), a figure not far from modern piano wire. It is not impossible that his secret process produced a similar product.

We can expect that the other wire makers in Nuremberg tried to duplicate Meuler’s wire. They refined their production of iron to reach an fL product of about 180, as measured on old harpsichord wire. Surviving bits of old harpsichord wire reveal that the strengthening additive was not carbon (as in modern steels) but phosphorus, which only works on carbon-free iron. They seem also to have improved the strength of brass to provide an fL product of about 150.

Pitch distortion

Pitch distortion is the sharpening of a string’s pitch by stretching it during stopping. This is a common way of adding vibrato to the playing of guitars and hurdy gurdies. It also limits how low metal strings of a particular type can be used on fretted instruments. The string stretching leads to an increase in tension. From Hooke’s law, ΔT/T = (E/S)(ΔL/L). From the derivative of the Mersenne-Taylor law, the amount of sharpening is Δf/f = ΔL/L+(1/2)(ΔT/T) = (ΔL/L)[1+(1/2)E/S]. The term E/S is usually much larger than 1, so the 1 can be ignored (for a steel violin 1st, E/S is around 140). So Δf/f = (1/2)(E/S)(ΔL/L). To the author’s ears, the maximum tolerable Δf/f (pitch sharpening) is about 0.02, a third of a semitone. When this limit is approached, the next lowest string has to have a lower elastic modulus. The usual sequence of metal string types early in the baroque was Meuler iron (1580 to 1620), iron, brass, and twisted brass (low to high).

Lowest pitch due to pitch distortion

A particular type of string with a particular length can become unsatisfactory when tuned lower than

a limiting pitch. If the problem is pitch distortion, Δf/f = (1/2)(E/S)(ΔL/L), as stated above,. To express the stress S in more useful variables, we substitute for it by the Mersenne-Taylor formula: S = 4ρf²L², and get the pitch distortion Δf/f = (1/8)[(E/ρ)(f²L²)](ΔL/L). (ΔL/L) can be minimised by fine adjustment of the interaction between the strings and the fingerboard, and in any case is constant for an instrument. The maximum tolerable pitch distortion Δf/f is also constant, so we can deduce that the fL at the bottom of the acceptable range is proportional to sqrt(E/ρ).

This allows us to calculate the bottom of the range of one material from that of another from their relative elastic moduli and densities. For early iron, which we have an estimate from Praetorius’s bandora of an fL product of 157 for the top of the range, an estimate of the bottom of the range, of about 5 semitones lower, can be made from the range of the iron strings on the early French cittern. Another estimate can be made from the 18th century guitarra battente that used all inon strings with an open-string range of a sixth (9 semitones). From published values of E and ρ, the values of sqrt(E/ρ) are about 5100, 3100, 2800, 2140 and 2100 m/sec for iron or steel, copper or brass, silver, gold and low-twist gut respectively.²⁹ Thus compared to the lowest pitch for iron or steel on a given string length, the lowest pitch of copper or brass is 8.6 semitones lower, of silver 10.4 semitones lower, of gold 15.0 lower, and of low-twist gut 15.4 semitones lower. The low-twist gut string becomes unacceptable at a much higher pitch than any metal string because of inharmonicity.

Inharmonicity

What limits how low gut strings of a particular type can be used is the loss of harmonics in the sound, which makes it sound very dull and lose focus. The loss of harmonics is due to phase cancellation because the harmonics are out of tune with each other and the fundamental. The equation that expresses how much inharmonicity there is is f_n/(nf₁) = 1+B(n-1)², where f_n is the real frequency of the harmonic called the nth mode, f₁ is the frequency of the fundamental or 1st mode (and nf₁ is the frequency the nth mode would have if it were in tune with the fundamental). B is the inharmonicity constant, which is B = (π²/32)(E/S)(D/L)². When B is small but not negligible, the fact that the higher harmonics are slightly sharp gives the sound a distinctive colour, as is the case with the steel strings of the piano and dulcimer. It presents problems in the accurate tuning of octaves. Also, the suppression of the highest harmonics can either avoid shrillness or reduce brilliance, depending on the circumstances.

Lowest pitch due to inharmonicity

When the lowest-string problem is inharmonicity, we can substitute for S in the equation for the inharmonicity constant and get B = (π²/128)(E/ρ)[D²/(f²L⁴)]. If we confine ourselves to families of instruments, we can assume the tension-length principle (tension is proportional to open-string vibrating length). Then for a constant lowest acceptable inharmonicity constant, f⁴L⁵ is a constant for that type of gut, at least for that family of instruments. With this relationship, larger instruments at lower pitches have a larger overall acceptable pitch range than smaller instruments, which is consistent with historical experience.

An octave string in an octave pair should replace some of the harmonics lost in inharmonicity, so more inharmonicity should be acceptable than if it were a single lowest string. Praetorius’s lute had a very large open-string range of 2 octaves and a fifth, and the lowest string can be expected to be at the limit for an octave pair. His viola bastarda type of viol had the open-string range of 2 octaves and a fourth, and can be considered to be at the limit for a single lowest string, at least for viols. For each, from the f and L, the constant f⁴L⁵ can be calculated, from which either the lowest pitch can be deduced for any string length or the shortest string length deduced for any lowest pitch. Praetorius’s data on the other viols, as well as other data on English viols (with the same sizes, but different pitches), are consistent with the limits derived from the viola bastarda. Praetorius’s data on the fiddles are consistent with it as well. It is likely that the values of the f⁴L⁵ constant calculated above (for a single lowest string and one in an octave pair) provide good enough estimates of the lower limits of roped-gut strings generally.³⁰

The right half of Table I lists the shortest string lengths for most of the possible lowest pitches one can encounter in the three most commonly used pitch standards in Praetorius’s time (Italian corista and English consort pitch were the same as German chorthon). In addition for the pitches for roped gut or catlin strings, there is a column of pitches a fourth higher that represents the lowest pitches for a high-twist lowest string.

Effects of moisture content on gut

Gut is a hygroscopic material, so water evaporates from wet gut and and is absorbed from the air by dry gut. At the point when there is no absorption or evaporation, the moisture content (the % of its weight that is water) is in equilibrium with the relative humidity (the % of the maximum water content that the air can hold at that temperature). When under tension, the twist presses the fibres tightly together, which resists the absorption of water (which tends to swell them), so the equilibrium moisture content of the gut decreases at all relative humidities.

Traditionally wound strings get buzzy when high humidity swells the gut, which stretches the metal winding, and then when the humidity gets low again, the winding becomes loose and rattles. If such a wound string is in a sealed packet, it is advisable to not break the seal until the string is put on the instrument and tuned up, so that there is less chance of the core swelling enough to cause buzzing.³¹ There is a c.1900 specification for a double bass EE string that involves first winding with iron before winding with silver-plated copper, the iron presumably there to resist swelling of the gut core.

When an instrument is moved from an environment of lower to higher humidity, the pitch of a gut strings tends to go sharp. That is because the string swells, and since the fibres are twisted, this lengthens the path between their end fixings, stretching them and thus increasing the tension. If the humidity is particularly high and time is given for the string to reach equilibrium, the weight of added water lowers the pitch because of the added mass per unit length.

High moisture content considerably decreases the tensile strength of gut. This is because the moisture lubricates the creep that eventually leads to breaking. If the temperature of a instrument’s environment lowers at night, an unchanged amount of moisture in the surrounding air leads to a higher relative humidity, and thus higher moister content in the gut. Keeping the instrument in a well insulated fairly airtight case reduces the likelihood of overnight string breakages.

Effect of creep on sound

When a material creeps under stress, it appears to move very slowly in a steady way, as the word ‘creeping’ implies. Things are quite different on the microscopic scale. The micro-regions vary in the stresses they feel, and the mechanism by which creep occurs is by micro-regions at the highest stress suddenly slipping a bit with respect to neighbouring regions. This only happens if weak shearing surfaces that allow such slips are available. This slip movement reduces the stress on that micro-region, but it also redistributes the stresses amongst its neighbours, creating new regions of highest stress, leading to other such slips. Strings made of materials like gut, plastics and brass stretch inelastically (by creep) when first tuned up, and then stabilise. The stabilisation is due to disruption of the possible shearing surfaces by structural disorders resulting from previous movements. The breaking of a string usually starts from the breaking of a few load-bearing micro-region or fibres, creating greater stress on the remaining ones, eventually leading to an unstoppable cascade of similar breaks and such slips.

Each sudden slip creates a tiny sound vibration. Similarly, sound vibration can trigger a slip earlier than it would otherwise happen. When that happens, the slip absorbs some of the energy of the vibration, so strings that are inelastically stretching absorb some of the sound. This is why new brass strings mounted on a harpsichord need to settle in before they will provide the full sound expected, and why players on gut strings notice a dulling of the sound a short time before it breaks.

String pathology

Many early instruction books indicated that the way to test the quality of a gut string was to stretch it between the the fingers of two hands and to pluck it with a spare finger on one of them. If one clearly sees two string images, then it is a good string, but if one sees much fuzz or waves between the two string images, it is a false string. The two parabolic string images are where the string spends more time in its cycle than elsewhere, near the limits of its excursion. This test measures the uniformity in mass per unit length of the string. If there is a small region of more mass per unit length (or less) than the average, it reflects the disturbing wave created by the pluck, creating extraneous waves that can be seen. When a false string is on an instrument, these extraneous waves interact with the normal waves, causing wave phase cancellation (like in excessive inharmonicity), reducing the strength of harmonics and dulling the sound.

Medieval wrought iron wire was unacceptable as a musical string material because of such lack of uniformity. Drawn metal wire is uniform in cross-section (matching the hole in the draw plate), and so can make good musical strings. The modern method of using a centreless grinder for polishing gut and plastic strings confers high uniformity to these strings. The previous method of mechanically polishing gut strings, by rotating a number of them beneath a flat abrasive pad, sometimes led to another string pathology, that of a oval rather than circular cross-sectional shape. A misshaped hole in the draw plate can do the same to a metal string. Each string vibration is in a plane, and the relevant stiffness to bending during the vibration is provided by the thickness of the string in that plane. Thus the frequency of vibration of the string depends on the direction of the vibration. In practice, the direction of vibration is not stable, and we get a range of frequencies that interact with each other, dulling the sound.

The stiffness to bending of a gut string that is relevant is the stiffness under the tension at which the string is used. If the fibres are loosely bonded to one another, the string can be very flexible to bending, but when tuned up, the tension in the string acting on the string twist presses the fibres together, making the stiffness to bending little different from a string in which the fibres are well bonded. Many musicians feel that bending flexibility should be associated with warmth in sound, and so are attracted to gut strings with poor bonding. All of the early evidence on this issue indicates that high stiffness, implying good bonding, was preferred. This includes Dowland (1610): ‘if they finde stiffe, they hould them then as good’, Burwell (c.1670): ‘hard’, Mace (1676): ‘Stiffness to the Finger’ and Galeazzi (1791): ‘elastic and strong, not limp and yielding’. Poor bonding is a sign of age.

Visual appearance of gut

The early evidence on the visual appearance of gut strings use the words ‘clear’ or ‘transparent’. This means at least translucence, where one can see some light from a candle transmitted through the string when held between the fingers. If no light is transmitted, the opacity could be due to light-absorbing pigments or to scattering of all the light in other directions before it could get through. What scatters light is the number and magnitude of sharp changes in the index of refraction of the material along the light’s path.

Any air spaces between the fibres would scatter strongly, as can be seen in poorly bonded strings, which can be very opaque. If one fills those spaces with another material like oil, the closer its index of refraction is to that of gut, the more transparent the string becomes. Some early sources indicate that oils were used in gut string manufacture. Soaking normal gut strings in oil increases transparency. As some have proposed, placing a heavy metal powder or heavy metal salt between the fibres (to increase string weight like a wound string) cannot be what the early sources described.

Aged strings

Oils situated between fibres tend to evaporate or change their properties with time, making oiled gut strings more opaque with time. With age, the gut itself turns brown (possibly by oxidation) and loses some strength. There are some brown opaque old strings that survive in museums and elsewhere. They are most probably from no earlier than the 18th century, so we can expect that the gut was originally bleached white by sulphur dioxide, which was usual by then. Such treatment is a traditional preservative for dried fruit. Before such bleaching, the major problem with the lasting of gut strings was bacterial decay. Currently, the most desired early instruments are ‘restored’ original instruments that show their age, and in this spirit, unbleached opaque gut strings are often preferred by early musicians.

Aged iron strings do not lose their innate strength, but the development of rust on the surface can initiate micro-cracks that can propagate inwards and cause the string to break. Brass is an unstable material on the long term, with the zinc that is in solid solution in the copper microcrystalline grains tending to migrate out of the copper to aggregate at the grain boundaries. This is called ‘dezincification’, and it limits the life of brass strings (as well as brass instruments).

Table: Gut String Limits of Pitch and String Length from Praetorius