Foreword:

For centuries musicians have experimented with strings for their instruments. Through experimentation and intuition they have used the acoustic method I describe, but without the tools we have today to quantify these concepts.

Now with the help of inexpensive and readily available tuners and apps we can put some numbers and certainty to our intuition, even with the lack of detailed information from the string manufacturers. We can understand in simple terms the strings we use, without the engineering formulas and complex calculations that are defined by string length ratios and string mechanical properties.

This method will let musicians alter or experiment confidently with stringing on their instruments. Instrument makers can use this method to improve and develop new instruments with different stringing options.

The Acoustic Modulus of a string can easily be derived with simple measurement and arithmetic. It is the breaking frequency of a one meter long string of any particular material regardless of its diameter.

As an example, you can adjust the stringing and pitch of your instrument by tuning the string to be 6 semitones lower than the breaking frequency, which results in a 50% stress tension from its maximum load capacity (tensile strength). Other tension ratios can be attained by choosing another semitone decrement. A three-semitones step down from the breaking frequency put the string at about 70% of its maximum tensile strength. The feel of the string (stiff or soft), and the capacity of the instrument to hold the tension, will guide you about the thickness of the string you need, and how to best make your instrument if you are an instrument maker. It is as simple as that. You can easily quantify these 2 factors with the simple arithmetic of the String Resonance Strength Method.

Part One describes the method you can use to establish the String acoustic modulus from an unknown string in addition to string density and Modulus of Rupture (tensile strength).

Part Two explores important acoustic considerations of string characteristics.

Part Three defines string materials for non-metallic instrument strings.

Appendix: home made testing apparatus

How best to string various instruments, especially folk and traditional instruments, can be a challenging question. So many string types and compositions are available from string manufacturers with sometimes limited information on the string package or on their websites. New string materials like Nylgut, Nyltech, Savarez strings, carbon and fluorocarbon strings, Sugar strings, gut or polymer strings infused with various fibers, or powders inside, all have different acoustic properties that are crucial for proper stringing and affect the sound of an instrument. Or, perhaps you have strings that are breaking, or too tight, or too loose, or dont sound the way you want them to? What can you do? What are your options? Is there a method to assist with proper string selection?

There is a practical, hands-on approach outside of the well-known Taylor formula we use today. Instead, we can use an older empirical approach that was practised by musicians and made popular by Marin Mersenne (1588-1648). Watch this great Youtube video about Marin Mersenne (1588-1648) string inquiry (https:/youtu.be/Psfkpfstj8E or search for Marin_Mersenne_part1.avi). That video will introduce you to Marin Mersenne and his study of string acoustics.

By breaking and weighing strings as Marin Mersenne did, we can derive the most important properties that characterize strings. Those characteristics are tension, tensile strength ratio, size, length, density, and how they relate to the pitch of a string. Knowing that, we can refine or alter the stringing to meet our needs, without damaging our instruments. That method and how it can be used is what I wish to share with you.

First we need to define two parameters for this method. Secondly, we will use simple calculations that relate to the pitch of a string just before it breaks (maximum tension that the string can hold). Thirdly, in part 2, we will relate how changing tuning by semitones correlates with tensile strength ratio, string diameter and string density.

Two related properties

This method is centred on the concepts of String Resonance Strength (SRS) and String Acoustic Modulus Resonance (SAM). Those terms are neglected and/or absent in physics books and engineering reference data. They are inter-related.

String Resonance Strength is the maximum frequency possible at a string length before it breaks. This will be true regardless of its diameter. Breaking frequency is related to the length, density and the strength of the string (maximum loading force capacity).

String Acoustic Modulus Resonance is a reference frequency established for a one meter long string reaching its acoustic strength. Unit is frequency in Hertz at one meter or Hz/m.

If you know the pitch at which a given string (size, length, material) breaks you know how high you can tune the strings. From the String Acoustic Modulus we can simply derive the breaking frequency of a string of any length by dividing that String Acoustic Modulus by the length of the string. A one meter string that has a breaking frequency of 250Hz will have a breaking frequency of 500Hz at 0.5 meter. (250Hz 0.5m = 500Hz). That same string at 0.75m will break at 333Hz, at 2m it will break at 125Hz. These are the String Acoustic Strengths for that string at different lengths.

There are many kinds of string materials (many types of nylons, steel, brass etc...) for musical instruments, each having their own String Acoustic Modulus. The table below shows the variability that exists among the six non-metallic string types I broadly identified.

String Acoustic Modulus Resonance for various string materials I tested. (breaking frequency at 1 meter)

Note: Modulus averages depending on material density and tensile strength variations.

Hard Nylon: Low density, high strength, Nylon 6 family: 6.6, 6.10, 6.11, 6.12 can be used on fretted and string instruments like guitar, ukulele and harps. They strings come in various fashions (rectified, colored, polished), with many brand names.

Synthetic Gut: (heavy, high density nylon/polyesters): nylons or polyesters loaded with additives (various carbon/glass/other fibers; or mineral/synthetic particles) to increase/lower density. Various names are Savarez, Nylgut, NylTech, any gut-like strings.

Fluorocarbon: a kind of polymer with a carbon fluoride bond used in many applications for hardness and heaviness. Like nylon there are many varieties with different names.

Gut: Natural gut strings have been a reference for traditional musical strings but there is no standard for it. There is a great variability of gut strings and each manufacturer has their own (patented/secret) methods. In general there are three grades of gut strings never mentioned in product descriptions. Grade 1 is high tensile strength used in high treble strings. Grade 2 is medium tensile strength used between high treble and bass strings. Grade 3 is low tensile strength used for bass strings. In each of these grades there are significant variations as per the study done by Bell and Firth in 1986*.

Synthetic Fiber: (nylon fiber, floss) micro filaments not twisted made of the afore mentioned nylon.

Natural Fiber: (various plant/animal fibers) silks, hair, wool, cellulose fiber from hemp, bamboo, flax. ** Mostly from Franz Jahnel study 1981. Large variability. Those fibers were used in the past and have been forgotten. They can still be used if instruments are made to accommodate those natural fibers. Usually they are low tension, with a quieter sound but may be with a more charming sound?

By knowing the Acoustic Modulus or acoustic strength of a string material and its density you can find out the tensile strength ratio with simple calculations and easily-done measurements.

Sample calculations using the String Acoustic Modulus method

I have a Nylon Tynex sample string, 80cm long and 1.00mm thick that weighs 0.670gram.

What is its density in gcm^3 (weight volume)

0.670g * (0.05cm^2 *pi* 80cm) = 1.067g/cm^3 or 1067Kg/m^3 (* means multiplied by)

What is its linear mass at 1 meter (weight to unit length - 1 meter)

0.670g * 0.8m = 0.838 gram per meter

If you don't know the diameter or mass of a string you need a small sensitive digital scale that reads to 1/100 or 1/1000 gram which can be found online for $25 to $50. A dial caliper for measuring string diameter accurately is necessary, which can be bought under $50.

When you know the density of a material, the linear mass for 1 meter (100cm) of that string can be calculated directly like this: 1.067 * (0.05^2 * pi *100) = 0.838g/m, so you dont need to weigh that kind of string again. (Linear mass = Density times volume of a one meter long string)

What is the breaking tension of that string?

We can work this out it from the String Acoustic Modulus. By testing the Tynex nylon, I determined the String Acoustic Modulus to be 268 Hz/m. Use these simple arithmetic steps and formulas to find out the breaking tension of the string.

String wave velocity = Frequency times wave length. Wave length is twice the string length ~ or ~

String velocity = frequency * 2 * string length, = 268Hz * 2 * 1m = 536m/s

Tension = string velocity^2 * Linear mass  (^2 means value squared: a^2 = a  * a)

Tension in milli-Newton = 536^ *2 * 0.838g/m = 240,754 mNewton or 240.8 Newton

Converting to Kg = 240.8N / 9.81 = 24.54 Kg

What will be the breaking frequency (SRS) at 60 cm playing length for the string we used above?

String diameter: 1mm, Nylon Tynex, Density: 1.067, Breaking tension: 24.54Kg, SAM: 268Hz/m

String Acoustic Strength for that 60cm string = String Acoustic Modulus strings length

268 Hz / 0.6m = 446.7 Hz = between A4 and A#4 sharp.

Breaking tension stays the same regardless of length as the frequency is inversely proportional to the length of the string. That 1mm diameter string, 60cm long, will break as the tuning reads to about A4 ~ A#4 sharp. That is the String Resonance Strength for that given length.

In general strings should not be tuned to more than 70% of their breaking point. (Details of this later in part 2 of this series). Tuning a string down by a semitone reduces its tension by about 10% of the semitone above. You can estimate that about 3 semitones down it will bring its tensile ratio close to 70%. That will be about F#4 (370Hz).

You can find the exact tension by using our formulas above.

String velocity = 370 * 2 * 0.6m = 444m/s

Tension in milli-Newton = 444^ *2 * 0.838g =165200 mNewtons or 165.2 Newtons

Converting to Kg = 165.2N / 9.81 = 16.84 kg

Exact Tensile Strength ratio is 16.84 / 24.54 = 0.69 or 69%.

That is still a bit high, so tuning to F4 or E4 would be safer. If you are designing a new instrument you can reduce the length of the string to lower the tensile strength ratio to less than 69%. Aim at 50% of the maximum tensile strength, that is about 6 semitones below the breaking frequency.

Establishing the String Acoustic Modulus

So how do you find the String Acoustic Modulus at the first place? You need to break a string (any reasonable length) like Marin Mersenne did. Record the frequency and playing length of the string as we did by finding the string resonance strength at a given length. When you know that frequency you multiply it by the length of that string and voila - it is as simple as that. Check that with our last above example. We said it will break at 446.7 Hz at 60cm. I did the experiment and it worked. So 446.7Hz 0.6 m = 268 Hz at 1 meter. That is the String Acoustic Modulus for Nylon Tynex for any diameter.

(SRS) String Resonance Strength is the maximum frequency possible at a given string length before it breaks. This will be true regardless of its diameter. Breaking frequency is related to the length, density and the strength of the string (maximum loading force capacity, modulus of rupture).

(SAM) String Acoustic Modulus is a reference frequency established for a one meter long string reaching its acoustic strength. Unit is frequency in Hertz at one meter or Hz/m.

Frequency: number of periods (waves) in one second

String wave velocity: Frequency * 2 * string length

Density: weight / string volume

String mass: density *  string volume OR Linear Mass * string length

String linear mass: string mass / string length (for 1 meter length)

String tension: wave velocity squared * string linear mass

SAM: SRS * string length

SRS: SAM / string length

Tensile Strength: tension at break / string section area #

Modulus of rupture: Maximum Tensile Strength #

Length * 1.059463 lowers the note by a semitone ( flat, string is longer)

Length * 0.943874 increases the note by a semitone ( sharp, string is shorter)

# Tensile Strength and Modulus of Rupture are different in physical engineering. Tensile strength is the resistance to a force tending to tear a material apart, measured as the maximum tension the material can withstand without tearing. For acoustic string evaluation related to pitches the difference is negligible.

During a frequency break test the highest frequency possible is the one before the string starts to break. For conceptual reasons, we can assume that maximum tensile strength is interchangeable with modulus of rupture when dealing with strings for musical instruments.

Unit system used has to be consistent for length and weight.

Use the system that is most convenient to you.

Reference Data and Real Measurements

Real measurements of your string will most likely give lower results than reference data (when they are available). Tensile strength done in laboratory are averages and most always exceeds their actual use because of compression points, bending, friction and attachment methods. Also, there may be a 10% variance in the diameter and density and over 20% variance for the tensile strength for the same nylon, synthetic materials or various gut string types. For metal strings you need to have the exact reference specifications for the various type of spring steel, bronze, brass and aluminum strings. There is a wide variety for each of these materials with large differences in strength/density characteristics.

These variances are significant for musical string applications with high tension. A sure way to know for musicians or instrument makers is to actually break a string for that specific sample and record its acoustic strength as we did here. Reducing your own reference data by 10% to 20% (about 1 or 2 semitones) and add 10% for each friction point on the string (at least 2 bridge support points for a total of 30% to 40% is adequate for most strings). This strength reduction is known as the coefficient of safety in engineering. Different materials and application methods will alter the coefficient of safety needed. It is mostly done by experimentation. It is sometimes called the maximum allowable stress.

Heat waves

Most materials lose strength at higher temperatures. During a heat wave in temperatures around 30 C (86F) and above your strings will stretch and lose about 20% to 30% of their strength. If you have high tension strings they may break. At higher temperature the wood on your instrument will be weaken and bend beyond its intended functionality. The sound board will lose its springiness and it will not recover. It will stay that way. While hot, the wood cells will slide and change their shapes because of the constant stress from the strings. Sound board and strings are very thin and it does not take very long for this to happen. Avoid high temperature from heat waves, exposure from sun, fire-places, car confinement, behind windows etc... If you cannot, loosen the strings by about 3 semitones (about 30% of tension). With less tension the strings and instrument will not be over stretched.

Recommendations

I suggest a string should be at about 50% (plus-minus) 10% of its maximum tensile strength tension and that is about 6 semitones from the breaking frequency (see part 2). Some instruments, especially harps, require higher tension for the treble range. Wound bass strings could have high tension on the core section too as this helps to bring brightness to the sound if that is desired. For those strings I would recommend not exceeding 70% of the tensile strength and I prefer not to exceed 65%.

Also, it is not recommended to momentarily stress a string more than 80% of its maximum tensile strength unless you want to break it. As the string reaches its maximum tension, its elastic limit is compromised and the string starts to deteriorate with micro fissures which will lead to failure over time. An over-stretched string will not keep its tuning for very long and will affect the sound with off harmonics until it breaks. There is a common saying that the string sounds best just before it breaks. This is really not true. A well designed instrument has a proper string length for each pitch without over-stretching the string.

Strings should break at friction points because that is where there is the most stress (tension and compression) but sometime they break in the middle of the string. This means there was a very small diameter variance there or that the internal integrity of the material was compromised during the manufacturing process or was damaged by excess bending or over compressed at one time. If you have a broken string and it breaks often at a pitch, it will be easy to run the calculations and you can better understand why.

Test your broken strings

Take the time to explore these concepts. I hope this method will motivate you to explore new possibilities in stringing and also in instrument/string making. After all, your instruments tonality is partly about strings and partly about design features of your instrument. Dont throw away your broken strings. Test them. Learn from them. What can we do to our strings and instruments so they can respond better to each other the way we would like? Be curious like Marin Mersenne (1588-1648). And what about string stretching? It is an easy measurement to take and that can be done without breaking strings or during a string breaking test. Do a science fair project with your kids, grandkids or like you used to do in Middle/High school some time ago. If you do, let me know I will be very curious. But above all have fun with strings and play music.

Correlation between semitones and tensile strength ratio

Below is a table of tensile ratio reduction from breaking point by lowering the pitch by semitones. I used a nylon string in the table but the reduction to the tensile ratio is the same for any strings.

If you look at the table from bottom up then you will see that the tension quadruples as you tune a string an octave higher. As you decrease the pitch by semitones you lower the tension by about 10% of the previous tension. Depending on the instrument design and sound board construction there is a range where the string will sound better. Usually it is between 35 to 60% but bass strings can go as low as 25%. For highest notes it should not be more than 70% for most instruments.

Effect of changing string diameter sizes

A nylon string, 21 cm long, tuned to C6 (1046 Hz) with different diameters produces the following results:

What this shows is that regardless of the string diameter the tensile strength ratio for that frequency stays the same, although the tension really changes from 16.5Kg to 66Kg. As the string gets thicker it has more capacity to hold the tension but its ratio to breaking point strength stays the same. This means that if you have a string that breaks often, getting a thicker one will not solve the problem.

By doubling the diameter of the strings the tension becomes about four times greater. String deflection from plucking will be less and will have a harder feel. It is characterized by the Tension per Length unit ratio T/L%. That ratio is often referred to as the Feel number. Different units will give a different value it is a comparative value. It tells us the potential force (stiffness) stored in the string per unit length. Quadrupling the tension of a string on an instrument can be dramatic!

By reducing the diameter by 10% as done from 2mm to 1.8mm in the table above we reduced the tension by about 20% and the tensile ratio stays the same. Increasing the string diameter by 10% will increase the tension by about 20% affecting the feel (playability) of the string. It will drive the soundboard with a greater intensity for a better or worse tonality and that is up to the musician to decide what sounds and feels right. As you change the feel of the string you also change its tonality. Musicians will have to make a compromise between feel and tonality if they do not coincide. A well-balanced instrument will have even tonality characteristics on all the strings, while having a comfortable string feel. The relationship between sound board and string tension is unique to each instrument and is not easily modelled mathematically.

Effect of changing string material (strings tensile strength and density)

Changing string material is changing the string tensile strength, density and stiffness. These determining factors characterize the string performance.

Let us look at the following strings with different materials with the intent to keep the same tension by adjusting the diameter of the strings. This may be a requirement for your instrument, or a technique to use if you want to keep the same tension feel for the string. All strings 18 cm long tuned at C6 (1046Hz).

Here only the diameter of the string and its tensile strength ratio is affected. The 1 mm nylon string can be replaced with a 0.9mm gut like string or a 0.75mm Fluorocarbon string. Of course by changing one string material with another you will affect the tonality of the instrument.

There are also some unexpected results.

a) - Many believe that gut like strings are stronger than Nylon strings as gut like strings have a greater static tensile strength than Nylon. However, gut like strings have a greater specific density than Nylon. The increase in density is not proportional to acoustic strength. So while it is true that most gut-like strings have a stronger static tensile strength, their String Resonance Strength (SRS) is less than nylon as seen in the Tensile Strength ratio column of the table above. The gut-like strings will break before the nylon strings. That is one reason for using nylon strings at the very upper range of concert harps because gut-like strings are not durable at that range.

b) Density is not proportional to static strength and a greater static strength may not have a greater Tensile Acoustic Strength. What is surprising here is the Fluorocarbon string will almost double the tension if you kept the same size string as the nylon one. Something to think about!

Tensile Strength and String Acoustic Strength are indicators of noticeable difference in strings tonality. This applies to all metallic alloy, natural or synthetic strings. Soft metallic alloys like brass, bronze, aluminum have low tensile strength and require judicious attention to the instrument design.

Composite/wrapped/wound strings density variation and String Acoustic Modulus (SAM)

For composite strings, things get more obscure because of the variability of the components in material, size and layers. Adding mass with wraps or added particles to the core is like increasing the density of the core. It doe not matter how elaborate those layers or how special the wraps or particles are. Their main purpose is to increase the mass of the string incrementally and it can be achieved in so many ways with synthetic and alloy materials. The wraps add mass but do not affect the tensile strength of the core.

We can follow the same procedures as we did for a single filament and work out the tensile strength and the String Acoustic Modulus. Measure or derive the diameter of the core by taking the wrapped string apart at one end. Secondly, weigh the string including the wrap and calculate its density as if the string was only the core size. Thirdly, break it on your monochord or sonometer to find out the breaking frequency (String Acoustic Strength) or the tension. Then do your calculation as we did previously.

Complexity and dynamic limitation of composite strings

To illustrate the complexity of composite strings let us look at theoretical wound strings I calculated for the articles I wrote for the Folk Harp Journal in 1994-1997.

Wound string data using Imperial units in inches for length and pounds for tension

All strings at F2 (87Hz), length 48 inches, diameter in inches as listed, all tensions remaining the same.

Every time you change an element of the composite string is like creating a new material with a new tensile strength and/or density. Each composite string will have its unique String Acoustic Modulus that you can calculate now. However, you can use different materials in different sizes combination to obtain the same result as we have in the table above. This is counter intuitive but all these strings have the same acoustic modulus at about 160 Hz/m. They will all work but they will sound different.

All these strings have very different tonalities and behave differently even though their numeric characteristics are similar. Tonality is very unique to the core material and how it is being affected by adding wrap mass in many different ways. It changes the bending, stretching, flexibility of the core and consequently it changes the harmonic patterns, intensity and reverberation of the sound, giving it a unique identity. Tonality of an instrument is very complex and subjective by culture and tradition and is beyond the realm of simple analysis. It is mostly an empirical search by musicians, instrument and string makers. The method outlined here may be helpful to resolve issues, refine solutions or even explore new instrument specifications.

Musical string instruments have evolved with ideological mythologies. Makers made instruments and tried to find strings to best match them. Historical instruments are not designed for modern strings although we can or should try to make strings that match the specifications of historical instruments. Our instruments are evolving partly because of new string materials that are more reliable, durable, cheaper with new tonalities and new complexions that are more sensitive to our technological age. The search for what strings can best match our instruments for the tonality one wants to hear and ease of playing is still going on and defies cultural traditions.

The table below shows the relationship between reducing the pitch by semitones, and the breaking frequency (pitch). This is true for any scale starting from any pitch.

* Tensile strength and Modulus of rupture are different in physical engineering. Tensile strength is the resistance to a force tending to tear a material apart, measured as the maximum tension the material can withstand without tearing. For acoustic string evaluation related to pitches the difference is negligible. During a frequency break test the highest frequency possible is the one before the string starts to break. For conceptual reasons, we can assume that maximum tensile strength is interchangeable with modulus of rupture when dealing with strings for musical instruments.

Permanent set elongation - molecules alignment within elastic limit plastic deformation.

Elastic limit workable/safe force applied within Hookes law/ Poisson ratio stress factors.

Yield point above elastic limit, material deformation with tearing then failure if stress is constant.

Fracture point Same or lower than Yield point at break time for most materials.

Ductile materials refers to material that have plastic deformation such as soft metals (gold, copper) and natural or synthetic polymers (cellulose, proteins, plastics) before they break. They can absorb mechanical energy.

Brittle materials refers to material that have little plasticity. They are limited in absorbing mechanical energy. They break without or little deformation like ceramic, cast iron, glass.

In part one I referred to 6 broad types of plucked string material which can be grouped in two different categories, synthetic and natural fibers (both are polymers). Natural fibers are easily identified by the manufacturers in their product descriptions. It is not the same for synthetic fibers made from industrial polymers, mostly nylons or polyester. Brand names drive the market for synthetic fibres. Product descriptions for string instruments are often poetical and flowery, with little real data.

[*A brand name* strings are based on an exclusive new core material, along with advances in manufacturing techniques, resulting in the true warmth and complex overtones of gut core strings with the stability and quick response of synthetic core strings. * A brand name* has also demonstrated quick break-in time, exceptional projection, and unusual durability.]

Nylon was used in the early days of synthetic silk/gut strings for instruments. As the table below suggests we now have many nylons with different density and with different properties. Gut-like synthetic strings in various concoctions of nylon, polyester and fluorocarbon are now produced with the major brand names. I tested a few of them and they gave similar results since they shared similar properties.

Density and tensile strength are the main identifiers and the most important features for string instruments. The main difference between the hard, low-density nylon and synthetic gut-like strings can be identified to a certain extent by their density difference with a float test. A flame test will also provide further certainty to the identification of your sample.

Flame test

Hard, low-density nylon will barely burn with a small (bud-like), clear flame, no black smoke

Loaded nylon will burn with a bit bigger flame and some residual stuff

Fluorocarbon string does not burn, will melt and turn black

It is not a nylon if you see a significant burning, yellow flame and/or some black smoke. It could be a polystyrene, polyester, acrylic or rayon.

Float test

Have a series of small jars with water and white table sugar (sucrose) solution of different concentrations so the densities of the solutions are at specific values. For example: add 15g of sugar to water for a total volume of 100ml solution. When fully dissolved the density of the solution will be 1.15g/cc or 115g per 100cc. If the sample floats its density is less than the water solution.

Additionally you can also do a simple Melting test. A variable temperature electric wood burning tool can be used with a long tip. The very tip will be cooler than the main body. You can compare different samples and see which one will melt first as you move the samples away from the tip toward the main body which is hotter. If you have reference samples or industrial data online you can work out the approximate melting point of your sample.

Hard, low-density nylon stretches much less than synthetic gut and fluorocarbon strings. When stretching exceeds 10%, the effective diameter of the string and its linear density are reduced. Take that into account if you need accurate tension calculation.

The table below indicates the many options available to make a special nylon.

From: https://omnexus.specialchem.com/polymer-properties/properties/density

                                                        PolyAmdide : Nylon industrial classes                                                                             

What about gut strings:

The early manufacturing of gut strings was full of industrial intrigues as reported by Franz Jahnel in his book Manual of guitar technology English edition 1981. Dominance in that industry was supported with many patents that made strings superior by introducing additives in the production process with various bath concoctions of potash, pumice powder, glycerine, oils, talc, gelatin and/or adding threads fibers of hemp, flax or silk. Now the industry is much more mechanized with better quality control but variability is still a major issue in gut strings. Franz said there are basically 3 grades of gut strings for tensile strength made now but they are rarely specified by manufacturers. Here they are:

Grade 1 at 450 N/mm^2;  Grade 2 at 380N/mm^2;  Grade 3 at 315N/mm^2;

Elongation at break is about 15% and density varies between 1.25 to 1.35 g/cm^3

The table below shows that gut strings still have significant variability in strength and density.

What about Natural Fiber:

Various plant/animal fibers have been used throughout the ages to string instruments. Ordinary folks used what they had. Their strings and instruments did not survive the passing of time. Silk, hair, wool, fiber from hemp, flax (linen) have been heard to be used as strings. They also altered the strings with special concoctions similar to those used on gut strings to make them more resonant, playable. Large variability in these string characteristics were obvious and challenging for instrument makers and musicians.

You should test your supplies for accuracy. Those fibers used in the past have been forgotten/neglected but with better acoustic understanding and modern tooling these fibers could still be used if instruments were made to accommodate those natural fibers. Using the String Acoustic Method for string evaluation would facilitate their usage. Folk instruments are usually lighter instruments with lower tension and a gentler sound.

What about metallic wires, piano, guitar and alloyed metal?

Availability is specific to industry needs, and stainless steel piano/guitar/spring wires each have their own standards. Regardless of those standards and availability, they all have intrinsic limitations due to manufacturing processes, density, hardness/ductility and strength. You can only buy what is manufactured for other purposes.

Typically all metallic wire types have a unique composition and manufacturing drawing/rolling process which affect the wire acoustic characteristic. Below is a data-sheet from what was sold to me as piano wire. Each size of that specific wire has its tolerance and strength specification. The same applies to most quality wires and alloys. That information is not always passed on to customers so you need to characterize it using the Acoustic Modulus method or a traditional engineering test.

You need to differentiate your needs between core material and wrap material. Core material needs to be strong to hold the tension of the string. It should be either steel or bronze (brass is weaker than bronze so it is best to use bronze whenever it is available). For wrap material strength is not an issue so it can be a soft alloy like brass, copper, aluminum or a steel alloy. The wrap material should be clear coated (insulated as in copper wires for transformers or motors, known as magnetic wire) so strings do not tarnish over time.

Acoustic instruments need to be properly coupled with their string characteristics. string elongation, string size unevenness, string material variability in density and strength, and the many varieties available as showed in industrial data. Academic studies tell us that it is difficult to use our idealized formulas with certainty. Switching from an engineering method to an acoustic resonance strength method as proposed here will assist musicians and instrument makers to make practical and effective adjustments based on acoustic outputs with limited string information, and without the usual engineering complexities.

I programmed  a special calculator to work out string selection using the Acoustic Strength Method and the Acoustic Modulus that you can download on the Calculator page.

Please send me an email for any suggestions, application or feedback regarding this method. Thank you.

Testing can easily be made using minimum tooling: micrometer/calliper, digital milligram scale, digital luggage scale or weights, digital tuner or smartphone app and a discarded string instrument. It is all that you need to have to do a string test with the Acoustic Modulus Method. The apparatus and set up I made below allow me to be more efficient and consistent with my tests. It is not that sophisticated except for the signal analyzer software (oscilloscope/spectrum analyzer using your soundcard) which these days are not expensive to obtain for your computer or your smartphone.

Below is the apparatus I made for this study which allowed me to measure the 4 variables independently, length, mass, tension and frequency. For the stretching test I made a mark on the string at the fret location and measured the distance before and after some stretching. After each tension test (load ) I plucked the string to measure the frequency as recorded in the data sheet below. The unit can be dismantled and put in the box that is on the bench for storage. I used a round bar to fix the modular parts of the apparatus.

Below is the screen shot of the oscilloscope analyzer software from Virtins Technology Multi-Instrument. Signal input is from a 10mm piezo glued on the left adjustable string fret just below the string and it is directly connected to the microphone input of the laptop. (Make sure that the signal is not more than 2 volts), start with the microphone setting on low). The longitudinal hardwood fret grain is vertical for maximum transmission of the string transverse wave to the piezo sensor. The bottom graph shows the first 3 harmonics of the signal, very clean and sharp. The software allows me to program mini calculators to display wave characteristics from the raw signal such as the Q factor, damping ratio, log ratio and harmonics.

Below is my data sheet. The RED dotted cells are my input data, the rest (greyed cells) are calculated/derived values from my input. The graph at the bottom right is the plotted data from the Stretch Reading above. That sample test was from a Super Nylgut AQ103 compared to Nylteck from a previous test. I did not do multiple tests of the same string so the data is not statistically accurate but the idea for those limited tests were to get some approximation of the significance of this method. This method can really be an asset for small scale application and small artisanal shop. All tests were done with a vibrating length of 50cm. After each tension reading test, the string was relaxed and stretched again. The Hooke K constant, Young MOE and maximum elongation calculated are theoretical from the limited data input and should not be taken as an established value. What really matter for this study was to find the breaking frequency accurately for that string specimen and explore how this method can derive some meaningful engineering values.