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:


The Man Who Knew Infinity: a book review


The Man Who Knew Infinity by Robert Kanigel (1991)

I recently, and somewhat belatedly, read Robert Kanigel’s The Man Who Knew Infinity, a biography of the brilliant Indian mathematician Srinivasa Ramanujan. A partly fictionalised film based on the book was released in 2015 (see Scott Aaronson’s review of the film here).


Whewell’s Court, Trinity College, Cambridge, where Ramanujan lived when he first arrived in England in 1914 (photo: Cmglee)

Ramanujan had one of the greatest mathematical intuitions of all time (he himself credited his insights to the goddess Namagiri). However, his brilliant guesses were as likely to be wrong as right. Furthermore, Ramanujan often neglected formal mathematical proofs, so that the work of separating the many diamonds from the occasional paste was frequently left to collaborators (like G. H. Hardy, who invited Ramanujan to England, and who wrote several joint papers with him). There are still results in Ramanujan’s journals which have neither been proved nor disproved (see this perspective on Ramanujan by Stephen Wolfram).


One of Ramanujan’s formulae for π

Interest in Ramanujan seems to have peaked at around the year 2000, according to Google Ngrams (although this does not include the influence of the recent film):


Google Ngrams search for Ramanujan’s name in books

I found Kanigel’s book a very enjoyable read. There is extensive biographical detail, albeit with a few misquotes, and with apparent confusion at times about the language of a century ago (e.g. the word “cult,” used in a technical sense to mean “a particular system of religious belief,” referring to the Brahmin version of Hinduism which Ramanujan followed). Kanigel does not quite succeed in fitting Ramanujan into a larger context – I would have liked a bit more discussion of Ramanujan by other mathematicians. And I cannot help but wonder what would have happened had illness (probably chronic hepatic amoebiasis, although Kanigel suggests tuberculosis) not killed Ramanujan at the tragically young age of 32. I guess nobody can imagine what further mathematics we might have seen.

See here and here for other reviews of the book.


The Man Who Knew Infinity by Robert Kanigel: 3.5 stars


L.E.J. Brouwer, fifty years later

Luitzen Egbertus Jan Brouwer (27 February 1881 – 2 December 1966) was a Dutch mathematician who founded intuitionism and made important contributions to topology, such as his fixed-point theorem, which states that every continuous function f mapping a compact convex set into itself has a fixed point a [i.e. f(a) = a]. A consequence of the theorem is when a crumpled sheet of paper is placed on top of (and within the boundaries of) a copy of itself, at least one point on the top sheet lies over the corresponding point on the bottom sheet.

Brouwer had a huge impact on mathematics and logic in the Netherlands, influencing people such as Arend Heyting (student), Dirk van Dalen (grandstudent), and Henk Barendregt (great-grandstudent).

The Dutch Royal Mathematical Society (Koninlijk Wiskundig Genootschap) is organising a special event marking 50 years since Brouwer’s death. The event is on 9 December in the Amsterdam Science Park. It looks to be an interesting event. Details here.


On fairy tales

“About once every hundred years some wiseacre gets up and tries to banish the fairy tale,” C.S. Lewis wrote in 1952. The wiseacre of our time seems to be Richard Dawkins who, two years ago, told the world that fairy tales could be harmful because they “inculcate a view of the world which includes supernaturalism” (he had said similar things in 2008). In a later clarification, he added that fairy tales could “be wonderful” and that they “are part of childhood, they are stretching the imagination of children” – provided some helpful adult emphasises that “Do frogs turn into princes? No they don’t.”

But many scientists grew up with, and were inspired by, fantasy literature. For example, Jane Goodall tells of growing up with the novel The Story of Doctor Dolittle (as I did!). In fact, many science students and professional scientists avidly read fantasy literature even as adults (as they should). The booksthatmakeyoudumb website lists, among the top 10 novels read at CalTech and MIT, Harry Potter, Dune, and The Lord of the Rings. And Alice in Wonderland was written by a mathematician.

This is a science blog, so I have a strong emphasis on scientific truth, which tells us many important ecological and physiological facts about, for example, frogs. Without science, we’d all still be struggling subsistence farmers. But there is actually more than scientific truth out there.

There is also mathematical truth. Are the links in this frog network all equivalent? Yes, they are – but that is decided by mathematical proof, not by scientific experiment. It is in fact a purely abstract mathematical question – the background picture of the frog is actually irrelevant.

And there is ethical truth. Is it OK to eat frog’s legs? Science does not give us the answer to this (although logic can help us decide if our answer is consistent with our other beliefs), but fantasy literature often helps us to explore such ethical questions. Tolkien’s The Lord of the Rings is one superb example. Would you “snare an orc with a falsehood”? Would you attempt to take the One Ring and “go forth to victory”?

There is metaphorical truth. A frog may, in spite of what Dawkins says, be a handsome prince – there’s more to the universe than can be seen at first glance. Or, as Antoine de Saint-Exupéry put it, “What is essential is invisible to the eye.” Children often learn this important fact from fairy tales.

And there is even religious and philosophical truth. Does the frog-goddess Heqet exist, for example? Does the universe exist? Is there a spoon? The methods of philosophy are different from the methods of science, and some amateur philosophers simply state their beliefs without actually justifying them, but philosophy is actually very important. Science itself is based on certain philosophical beliefs about reality.


The Blue Mountains Water Skink


The endangered Blue Mountains Water Skink, Eulamprus leuraensis (photo: “Sarshag7”)

I have previously mentioned my interest in ecological niche modelling and amphibians. The cute little skink above, native to the Blue Mountains near Sydney, is sadly endangered. The black circles in the map below show online occurrence records for the skink. These range in altitude from approximately 530 to 1,170 m.

The blue area shows a predicted potential range for the species, based on MaxEnt modelling using those occurrence records and BioClim climate data. The model does not take into account the skink’s need for sedge and shrub swamps with permanently wet boggy soils – there are readily available online land cover datasets, but these have insufficient spatial resolution to identify the 30 or so swamps in which the skink is found. The predicted potential range for the skink is consequently very much exaggerated, and covers 1,320 sq km, of which 63% falls within national parks or other protected areas. Hopefully that is enough to stop this beautiful amphibian from becoming extinct, although it continues to face threats from urban sprawl, feral cats, and vegetation changes.