Stringing Analysis Spreadsheet

INTRODUCTION: General Information, the ‘whys and wherefores’

I first started making harps before the days of personal computers, when even electronic calculators were only beginning to become widely available. The ability to instantly perform a series of calculations for numerous variables which might pertain to the dynamics of a harp was something that could only be dreamed of, hence most of my early research and learning was done empirically. Yet even after a small amount of practical experience, some basic truths or rules of thumb soon became apparent.

Generally I found that the key factor was to get the right string length for the pitch of a given note: if this was suitable, the instrument would work successfully, whether the gauge of stringing was lighter or heavier. However, changing the thickness of the strings obviously has a significant effect on the forces acting on the structure — if the stringing is too heavy, the increased tension can cause the harp to break. The gauge also affects the quality of tone and playing dynamic: too thick a string sounds dull, and too thin a string will become overloaded when plucked and start to “wow” (change pitch through the duration of the note).

It was these aspects that I wanted to analyse and compare across the various harps I had made. Therefore, some years later when technology allowing one to quickly perform a mass of tabulated calculations became available, I started to design this spreadsheet. Some aspects — such as the stress and tension of a string at a certain length and pitch — could be calculated to give absolute values, whereas others (e.g. how a string feels to a player when they pull it) were more abstract.

It seemed worthwhile to construct a table of information which could be used for comparative purposes, if only to give some idea of how the above factors might be affected by variations in stringing practise. The reason for devising this spreadsheet was to provide a mathematical overview of how different string lengths and parameters might behave across the range of the instrument (and from one design of harp to another). NB: It is not intended as a design tool per se, but rather as a means of analysing the effect that varying such parameters might have on the harp.

Stress, Tensile Strength, Tension and Load

To get a good sound from a string it must be stressed, and the more it is stressed the brighter it will tend to sound. This is a basic rule of thumb no matter what the gauge of the string is. It is also the stress of a string, rather than its tension, that governs whether it will break or not. People often confuse stress with tension and although the two are related they are not actually the same thing.

Two factors govern the tension of a string: one is the amount of stress within it, and the other — relating to its thickness — is its cross–sectional area. Putting a thicker string on a harp increases its physical tension but, being thicker, the string will also be stronger. Conversely, using a thinner string will lessen the tension but the string will also be weaker. Changing the thickness of a string changes its tension, but this will equally change its strength; thus the two cancel each other out.

When the pitch of a string is raised by tightening it with the tuning key, it is the stress within the string that is increased. Since this stress (and not the string's diameter) is the significant factor to consider when assessing the pitch at which a given string might break, the first and most important thing to work out on the spreadsheet is the stress produced in the string.

Stress is a directly quantifiable property, generally expressed in units of megapascals (MPa), and can easily be calculated from a string's length, density and pitch. But just knowing the stress acting in a string won't necessarily tell you whether it is likely to break or not. Different string materials have different tensile properties, thus you need to compare the stress produced with the strength of the string material. This is referred to as the Tensile Strength and it will vary from one type of wire to another (e.g. steel is stronger than copper). To get an idea of the degree to which the string is being stressed in relative terms, we can divide the calculated Stress by the Tensile Strength and multiply by 100 to give a percentage figure, here noted as %Stress.

The %Stress gives an indication of how likely a string is to break, but this should not be regarded as a definitive answer. The figures given for a material’s Tensile Strength should only be used as a guide, and leeway allowed: in practice, the actual qualities of a wire may vary from the theoretical for a number of reasons. These include the variances that occur in the manufacture (slightly different alloys and purities) and also the fact that the actual process of drawing a wire down to the desired gauge alters its strength and hardness. In reality it may be found that a wire which has been drawn down further may have a higher Tensile Strength figure than it previously did.

The other thing to realise is that the likelihood of breakage is increased at points where extra stress occurs due to the string being bent around an object, e.g. where it comes out of the soundboard or goes around a bridge or tuning pin (this is especially true if the string is bent abruptly over a hard edge). So although it may be thought desirable to have the upper strings highly stressed, in practice it would probably be unwise to have the %stress figures go too high; anything much above 70% is likely to be close to the limit. Fortunately, the lower pitched strings do not need to be stressed as highly to produce a reasonable quality of sound. If they did, the lowest bass strings on some of the larger harps would have to be so long that the top of the instrument would exceed the height of the room’s ceiling.

An indication of the stressing that makers have used in the past can be gained from looking at some of the extant historical Irish harps. The treble strings of several of these calculate to give around 50% stress (assuming a hard brass wire was used), though a few examples may have the odd string up around 80%. (Note however that some of these high strings may have been made of iron or steel for the sake of expedience.) The bass strings of these historic instruments are, as expected, of a much lower stress, and it’s not unusual to find that the stress figures for the lowest bass strings of a large harp are less than a fifth of those found at the top.

Where knowing the magnitude of the string’s tension, rather than its stress, becomes important is in assessing the effect that the forces exerted by the strings may have on the structure of the instrument (both the soundboard and frame), and how this might change with any alterations made to the stringing. From an engineering standpoint, the force acting on a structure is often referred to as Load, which in this case also equates with the string tension.

Once the stress of the strings has been worked out, the string’s cross–sectional area can be incorporated into the calculations to provide a value for the physical tension and thus for the load exerted. Overloading a structure will obviously result in breakage, and sudden or marked changes in the loading distribution are best avoided or minimised too. I believe that the soundboard is likely to vibrate more freely if it is stressed more evenly, and a concentration of potentially destructive forces into a small area is not a good thing for the instrument's structural integrity either.

Obviously, the tension of a string affects how tight it feels to the player's fingers, but this is also governed by other factors. These will be expanded upon shortly.

String Stiffness and Feel

The next relevant aspect is what gauge of wire to use, and how this affects the tone: if it’s too thick it will sound dull. I had come to the conclusion that this was probably because the diameter of the string had an effect on its stiffness, and that as it increased in gauge it began to behave more like a stiff rod than a flexible wire. The forces that produce the modes of vibration in a rod and in a wire are quite different and do not act on the oscillation in the same way. Excessive thickness is not only likely to dampen out the small high frequency vibrations of the upper harmonics (which give the string its brightness) but also begin to introduce inharmonic partials. However, a certain degree of thickness is necessary to provide the string with sufficient mass to give the instrument some dynamic power.

It therefore made sense to try and come up with something that could provide an indication of the degree of stiffness of the string. These would obviously be affected by both the gauge and the length, but how these factors would quantify in terms of the stiffening effect on any given string was less clear. In the end, the ratio I decided to use for the spreadsheet was to divide the string's diameter by the square root of its length. Although a direct ratio might seem more appropriate, to correlate with stiffness, I found that incorporating an exponential function provided a more useful figure for comparative purposes. The idea was not to give a definitive value but rather to provide an indication of the relative stiffness of a string (the larger the number the greater the stiffness) as something that can be used for comparison — string to string — across the harp, or from one instrument to another.

NB: There are other factors that affect the stiffness of an individual string, mainly to do with the type and hardness of the specific wire used; but attempting to calculate these is beyond the scope of this particular spreadsheet. However, it is as well to be aware that different types of wire will have varying qualities of stiffness and that (for instance) a hard steel wire will be stiffer than a similarly sized one made from a softer brass.

“Feel” is also difficult to quantify, but is perhaps best described as the lateral force the fingers feel when the string is pulled. Obviously the greater the tension in the string, the tighter it will feel and the harder it will be to pull; but length also plays an important part. A long string will yield more to the pull than a short one at the same tension. Therefore for small amplitudes of vibration, where the string is only pulled slightly, dividing the string's tension by its length may give a reasonable indication of the lateral resistance felt by the fingers. For this reason I thought it worth including calculations for the string’s load divided by length (called Ld/L) in the spreadsheet, the purpose being to give a comparative indicator of how the feel might vary across the harp, or from one instrument to another.

However, Ld/L figures are only an indication and do not give a definitive answer for the true feel of a string. The degree of elasticity within the string itself also plays an important part, and a string made of a more “stretchy” material may feel less taut, even though its tension and length are the same. As the amplitude of the string's vibration becomes larger, the amount of elasticity within the string begins to play an increasingly important part, not only in the way the string feels, but also in how it sounds. Although a detailed assessment of the effect that a string's Ld/L ratio and degree of elasticity has on its behaviour is outside the scope of this particular spreadsheet, it is still worth some further comment here.

When a string is plucked and made to oscillate, the string’s actual length must increase a small amount during the cycle to accommodate the lateral movement. This tiny increase in the length will have the effect of raising the stress in the string slightly; therefore the level of stress in the string will also fluctuate by a minute amount with the cycle of the oscillation. But stress has a direct relationship to the pitch of the string, and also to the way its upper partials and harmonics are formed, so the sound of the string will also be affected by this fluctuation. As long as the degree of fluctuation in the stress remains small this should not present too much of a problem, but the more the amplitude of the vibration increases, the more the stress will be affected. If the string is pulled too hard it will begin to sound false (inharmonic) and may even “wow”.

Not over-pulling is obviously important, but it’s not fair to the player if the strings are so under-powered that they are obliged to merely skim the strings for fear of sound distortion. Also this gives you a harp with little dynamic potential.

Another aspect affecting tone is the point at which the string is plucked. If this is done near to its end, the wire physically cannot be displaced as much as when plucked near the centre; therefore the string is not pulled into a state of increased stress, and instead will tend to spread its energy away from the fundamental oscillation and into producing more upper partials. This gives the string a sharper (harder) sound, even though it lacks the power and sustain which would otherwise occur.

There is a technique in mainstream harp playing which specifically utilises this effect, known pres de la table (plucking as low as possible on the string, close to the soundboard). Playing pres de la table on a wirestrung harp will certainly reduce any wow from the strings, but it would be a poor instrument that required this simply to reduce string distortion. A player may choose to utilise this technique in a piece for effect, but it should be by choice and not something forced upon them through necessity.

Therefore, aspects of string stiffness and feel must be considered alongside those of playability. It is a question of attempting to find a suitable balance.

A Note of Clarification

I would like to emphasise that this spreadsheet should not be viewed as a design tool but rather as an analytical aid: if there were some perfect and absolute mathematical answer, all harps would be fundamentally the same. However — fortunately — harps can vary to suit people’s differing requirements and tastes. Certainly amongst the many historical wire–strung harps I have examined there is a huge diversity in string lengths and scales, and this makes them all the more interesting. Therefore this table is not intended as any sort of magic–answer device, for numbers mean little in themselves without the experience to interpret them.

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Submitted by Michael Billinge, 24 June, 2013.

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