Why isn't a harp in a logarithmic shape?

Question

I was watching a harp, yesterday, and thought about the mathematics involved. I know that music is closely related to logarithms, because having a string or pipe twice as long produces the same note.

As an octave contains 7 notes, I would expect the strings 8 to be twice as long as the string 1, the string 9 to be twice as long as the string 2, etc.

Instead of that, the strings are strained between an oblic plane and a "S" shaped holder that reminds me a little of a $\frac{\sin(x)}{x}$ function.

I see two possibilities:

• This shape is precisely calculated to produce the right notes.
• This shape is only a matter of esthetic and the right pitch of the notes are mainly produced by the string diameter and tension.

Is there a mathematical reason to make the harps with such a non-logarithm shape?

Answer 1

You would expect it to have an exponential shape. Which it does, more or less, until the bottom end of the scale $-$ where practical considerations rule out 5-metre-tall harps.

Answer 2

the length would grow exponentially, if all the strings would be the same diameter and weight. however, they are not, to avoid 7 meter tall harps. the tension varies as well, because it's much harder to pluck a thicker string then a thinner one under the same tension.

Answer 3

The shape has little to do with the pitch. Pitch only deals with the string length to tension ratio. Obviously then the shape can effect the length BUT a harp could be "square". What it might require then is some very loose or tight strings... which then will, on practical matters, require very thick or thin strings.

Because a harp is "fretless" it allows one to vary the lengths of the individual strings (which would otherwise cause issues with the temperament). Hence the lengths of the strings can be adjusted to counter the above mentioned issues. I.e., increasing length allows one to increase the tension but have the same pitch. Of course there may be a special "curve" that maximizes all benefits of intonation for a harp but since a harp is fretless it wouldn't be all that useful, say, if one could do it on a fretted instrument (which you can't to any significant degree).

$$\begin{align*} &f=\frac{\sqrt{T/u}}{2L}\\\\ \text{CP 12TET}\implies &f=440\cdot2^{p/12} \end{align*}$$

which can be solved to relate the pitch in terms of tension, density, and length. You can figure out some "optimal" shapes (in the sense of the equation, not necessarily in practice) by, say, setting one of those variables as constant. If length is constant then it will produce the box pattern and you can see the required tensions and densities.

You could also "plug in the length curve for an actual harp and see how the tension and density are affected.