Harp strings and design

Having previously blogged about the mathematics of the harp, I thought I might say some more about harp design issues. It’s an interesting question that involves both physics and human factors. For simplicity, I’m going to talk about just one string, one playing the note A at 440 Hz. Of course, a real harp will have between 21 and 46 other strings.

The physics of vibrating strings gives us Mersenne’s laws, which tell us that the frequency of a string of length L is (1 / 2L) √ T / μ , where T is the tension force on the string (in newtons), and μ is the density per unit length of the string (in kg per metre).

The diagram below shows the required tension force (in newtons) for a nylon string of various lengths and diameters to play the note A at 440 Hz (click to zoom). A newton corresponds to roughly the gravitational force on 100 grams.

The first, and most obvious, design factor is that too much tension causes the string to break. Setting a design limit of 90% of the expected breaking strength means that the string must be less than 560 mm in length. Interestingly, this limit is independent of the diameter of the string.

The string must also be playable. A string that is too floppy or too tight cannot be effectively played. A rough guide is that the tension should be at least 35% of the expected breaking strength, which means that the string must be at least 350 mm long. Additional limits, which I’m ignoring here, relate to how much room the string needs to vibrate.

Thirdly, the frame can only take so much. If the frame of a 40-string harp is built to withstand 10,000 newtons (roughly the gravitational force on 1000 kg), then the average string has a limit of 250 newtons. This restricts us to the design space on the diagram outlined in red.

Finally, a harp has levers or pedals which shift the strings to be sharp or flat (the basic harp strings correspond only to the white keys on a piano). Those devices set further limits on the design space for strings.

A modern electric lever harp (photo: Athy)

It is interesting to relate this to my other interest, that of solar cars. They are vehicles, which means that they must hold a driver (and new guidelines on World Solar Challenge driver space have just been announced). They are solar, which means that their upper surface must hold a solar panel of specified size. And they race, which means that their aerodynamic drag must be as low as possible. This necessitates a variety of compromises, just as with the design of a harp. The problem is much more complex however; the space of possible solar car designs has many more dimensions than two.