Possible Applications

Example 1 — Identify a suitable metal for a particular length and pitch

We know the length of a string and we also know what note it is to be to tuned to. We need to find out what string material would be suitable.

l = 0.068 m
ƒ = 1760 Hz

When we enter these values in the upper row and click on the calculate button we find that no additional values appear in the lower rows.

Values for other parameters must be entered in order to proceed further. We enter values of 8.75 g/mm3 for density (density of 85/15 brass C23000) and 0.40 mm for diameter (typical diameter for the upper strings on a metal-strung harp) and calculate again. The values for the mass per length µ and the mass m of the string now appear in the orange rows.

m = 0.075 g
µ = 1.1 g/m

When we enter these values in the top row and calculate again we find that the amount of tension required in order to produce the desired note is approximately 6.44 kgf.

Enter a value of 550 MPa for YS, the yield strength of 85/15 brass at H08 temper and calculate again, we find that the tension required falls between the minimum and maximum practicable tensions for 85/15 brass.

Repeating these steps using the values provided for sterling silver in Table 1 leads to a very different outcome. The maximum practicable tension on the string would be 6.28 kgf (assuming that the strength of wire is equal to the ultimate tensile strength of the material, 490 MPa) and the amount required would be 7.65 kgf. If we enter a value of 6.28 kgf for T we find that the highest practicable value for ƒ would be 1596 Hz (G slightly sharp) almost a full tone lower than A at 1760 Hz so the string would inevitably break before reaching the required pitch, a silver string would not be suitable here.

Similarly if we repeat the steps using the values provided for iron we find that the string would have to be tuned much lower than the practicable minimum, the result would be poor tone quality.

Example 2 — Replace one type of metal with another

We want to replace the lower brass strings on a harp with silver strings to improve the tone in the bass register but we want to keep the same overall tension on the instrument.

Starting with the lowest string, which needs to be tuned to G 98 Hz. Its current parameters are:

l = 0.62 m
d = 1.1 mm
? = 8.8 g/cm3 (red brass)
m = 5.185 g
µ = 8.363 g/m
T = 12.59 kgf

Entering a YS value of 267 MPa (soft red brass) we can see that the tension on this string is well below the minimum practicable, resulting in poor tone.

Entering a value of 115 GPa for Q we can also see the coefficient of inharmonicity for this particular string is relatively high at 0.001718.

Entering a value of 2 in the ‘Partial no.’ field we can see that the second partial of this string tone would deviate from the fundamental by 13.65 cents, making tuning difficult.

Delete the value for diameter, update density to 10.4 g/mm3 and click on the ‘calculate’ button, the required diameter for the new material now appears in the lower rows.

Entering a YS value 165 MPa (for soft silver) we can see that the tension on the string is well within the practicable range, in theory resulting in improved tone.

Entering a value of 75 GPa for Q we can also see that the coefficient of inharmonicity is reduced and the deviation of the second partial has been more than halved. At 0.000799 this level of inharmonicity is still significant but falls within acceptable margins (see Inharmonicity). The level could be reduced further by using a thinner string, reducing the diameter of the string to 0.85 mm would bring the coefficient down to 0.000566, but bear in mind that the string’s lateral resistance would be considerably reduced.

Example 3 — Switch between monofilament and wound strings

We want to replace our bass strings with wound strings and keep the same tension on the instrument.

Repeating the steps in example 2 we know that the linear density or mass per unit length required to keep the same tension is 8.363 g/m.

Looking at Table 3 we can see that PB047 or PB049 would be the closest match.

To reverse the process to change from wound strings to monofilament strings.

Enter values for the linear density of the string (values given in Table 3 for d’Addario phosphor bronze wound strings should be representative of most brands), length, and frequency.

In this case µ would be equal to 8.09 g/m for a string equivalent to PB047, l 0.62 m and ƒ 92 Hz. Click ‘calculate’ and copy the value returned for T. Enter the density of the new material (10.4 g/cm3 for silver) and calculate again, the required diameter will be returned in the lower rows.

Example 4 — Identify the yield strength of a material

We want to find out the yield strength of a particular material or a batch of wire.

Simply tune a string slowly up to the point where turning the tuning pin appears to have little or no effect on the pitch and wait for an hour or so. Pluck the string gently and measure the frequency with an electronic tuner.

Enter all the known values into the relevant fields and then enter different values for YS and re–calculate. Repeat until the values for ‘Maximum Tension’ and tension ‘ T ’ match.

To find the optimum tension for a particular string material in order to predict where the ‘sweet spot’ is going to be for any length repeat the above steps having tuned the string to the point where it sounds its best. Note that the frequency of the ‘sweet spot’ for a given material will vary slightly depending on the length/diameter ratio.

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Submitted by Paul Dooley, 21 March, 2013. © 2004-2013 Paul Dooley

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