Mathematics of the Harp

After my second harp post, I thought I’d keep going with some mathematics. In particular I want to answer the question: why does a harp have that shape?

A modern electric lever harp (photo: Athy)

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 (in kg per metre).

For a string of diameter d and density ρ, we can calculate μ = A ρ, where A = π (d / 2)2 is the cross-sectional area. Nylon has a density ρ of 1150 kg/m3. For a nylon string of 1 mm diameter, we get μ = 0.0009 kg/m. Putting 0.448 m (17.6 inches) of that string under a tension of 140 newtons (31.5 pounds force), we get a frequency of (1 / 0.896) √ 140 / 0.0009  = 440 Hz. That is, the string plays the note A.

The important thing here is that the frequency is inversely proportional to the length. Over an octave the frequency doubles, which means that the string length halves. A 36-string harp covers 5 octaves, therefore if all the strings were made of the same material under the same tension, the longest string would be 32 times the length of the shortest. Doing some calculations in R for strings of diameter 0.8 mm under a tension of 140 newtons (31.5 pounds), we would get the following harp, which has strings ranging from 7 to 224 cm in length (note that the strings run from A to A, and the C strings are red):

Image produced in R. Click to zoom.

You can see quite clearly that, starting at the treble end, there is an exponential growth in string length. That makes for a terribly unwieldy instrument, and creates all sorts of problems in playing. In practice, we make bass strings thicker (and usually of heavier material) and we vary the tension as well – although, to make life easier for the harpist, we want the properties of the strings to vary reasonably smoothly. If we re-do our calculations with string diameters varying linearly from 3 mm to 0.6 mm, and tension varying linearly from 210 newtons at the bass end to 60 newtons at the treble end, we get a much more realistic-looking harp:

Image produced in R. Click to zoom.

We can flatten out the curve at the top a little by changing the way we vary the strings (after all, a guitar manages to span 2 octaves with all the strings being the same length). However, we cannot eliminate that curve completely – it is the inevitable result of spanning so many octaves, combined with the mathematics of exponential growth.

Left: an exponential curve (red) and a similar polynomial (dashed). Right: the quotient of the two functions (green) compared to a straight line (grey). Click images to zoom.

Mathematically speaking, the modifications to the strings have the effect of dividing an exponential function by some kind of polynomial (as shown above). Over a short range of x values, we can find a polynomial that fits the exponential well, and gives us strings of the same length. Over a wide range of x values, however, the exponential wins out. Furthermore, exponential growth is initially slow (sub-linear), so that (starting at the right of the harp), growth in string length is slower than the linear shift provided by the sloping base, which means that the top of the harp curves down. After a few octaves, growth in string length speeds up, and so the top of the harp curves up again.

A similar situation arises with the strings of a piano, although these are usually hidden from view:

Inside of a piano (photo: Alexandre Eggert)

And to finish, here is one of my favourite classical harpists in action:

Marching for Science #5

Further to my previous comments on the Science March, the graph below shows the (somewhat dubious) attendance estimates from Wikipedia for various cities (excluding vague counts like “thousands”), compared to the power-law predictor 0.47 D1.49 P0.78, where D is the fraction of the relevant state voting for Clinton last year (from Wikipedia), and P is the city population (also from Wikipedia).

The population P predicts 56% of the variance in turnout (not surprisingly), and D an additional 7%. Both factors were significant (p = 0.000000055 and p = 0.014 respectively). Prediction could probably be improved by using metro area population numbers for the cities, by using metro area election results (rather that state results), and by adding factors indicating the number of other marches in the relevant state (Colorado Springs, for example, was rather overshadowed by Denver) and the presence of universities (Ann Arbor, for example, is a university town). But the basic messages seem to be: Democrat voters do not like Donald Trump and Large cities attract large crowds. It would be interesting to compare the numbers here against other recent political marches which focused on different issues.

The Harp Twins

Someone recently pointed me at Camille and Kennerly Kitt, the so-called “Harp Twins” (above). I admire anybody who “thinks outside the box,” and these young women have clearly left the “box” of traditional harp-playing several light-years behind.

Their rather eclectic oeuvre includes film, game, and TV tie-ins (from e.g. Lord of the Rings or The Legend of Zelda); rock, folk, and pop classics (like “Hotel California” or “House of the Rising Sun”); metal (from bands like Iron Maiden or Metallica); and other music (such as “Amazing Grace” and “Scarborough Fair”). They have just started releasing their own compositions. The chart below summarises their releases by genre (data taken from Wikipedia, so probably incomplete).

Stories of the Past and Future

Inspired by a classic XKCD cartoon, the infographic above shows the year of publication and of setting for several novels, plays, and films.

They fall into four groups. The top (white) section is literature set in our future. The upper grey section contains obsolete predictions – literature (like the book 1984) set in the future when it was written, but now set in our past. The centre grey section contains what XKCD calls “former period pieces” – literature (like Shakespeare’s Richard III) set in the past, but written closer to the setting than to our day. He points out that modern audiences may not realise “which parts were supposed to sound old.” The lower grey section contains literature (like Ivanhoe) set in the more distant past.