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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?

Shape of a harp

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1  
I guess Railsback curve also applies to harp. –  user13838 Nov 10 '11 at 8:21
9  
An observation: classical pianos seem to have this shape too, turned on its side, whatever it is. –  alex.jordan Nov 10 '11 at 8:26
1  
string calculator –  pedja Nov 10 '11 at 14:19
    
@alex.jordan I've seen a joke somewhere: "The harp is a vertical naked piano". –  Vÿska Sep 16 '13 at 14:53

4 Answers 4

up vote 8 down vote accepted

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.

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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.

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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.

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There is another variable to consider: the mass of the string per unit length. Wrap your string in heavy wire, and it sounds a lower pitch: look inside any piano. –  Lubin Sep 12 '12 at 19:55
    
@Lubin That is called the density! (The effect is more than density. Obvious the physical equation is simplified but density is what you are talking about and I did include that. –  AbstractDissonance Sep 13 '12 at 4:17

Another factor which many of you are missing is the fact that harp strings are made of different materials in different areas of the harp. I am a harpist myself by the way. The three types of strings are (from the large bass end to the small high end) wire, catgut, and nylon. These different materials require different lengths (and diameter and tension as well).

The long wire strings are different lengths because the body of the harp is angled down, while the neck (top part) is relatively straight here. These strings are changed primarily by tension rather than length or diameter.

The center portion (gut strings) has the most dramatic length change. The diameter is also significantly varied as you compare strings from different octaves. Finally, the top portion (nylon strings) has the least variety in length, with the neck and body almost parallel. I do not now exactly why this is (the diameter is not greatly altered). My guess is that the tension makes the greatest impact on these strings. I can tell you that turning the tuning key results in a larger change on the nylons than the catgut, but not as much as on the wire strings.

I hope this is helpful! The shape of a harp is fascinating and surprisingly logical and mathematical.

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