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:


Harp History

After some feedback on my harp twins post, I thought I’d say something about the history of the harp. It’s one of the oldest musical instruments (following the flute and the drum). Harps are known to go back to 3500 BC, in Ur. Harp design has varied considerably over the 5500 years since then.


Harpist depicted on the Standard of Ur, c. 2500 BC

Later harps were of particular importance to the Celtic people, and the harp is still a symbol of Ireland today.


The medieval Queen Mary harp, c. 1400s (photo: David Monniaux)

A limitation of harps has been that the strings correspond only to the white keys on the piano. A significant improvement was the pedal harp – initially the single-action version, and from 1810 the double-action version. The double-action pedal harp is typically tuned to C major, the key of 7 flats. There are 7 pedals, with e.g. the C pedal connecting to all the C strings. Using the pedal can effectively shorten all the strings in this group to give either C or C (and the same for other groups of notes).

Child prodigy Alisa Sadikova playing the pedal harp (at age 9)

The pedal harp is the main concert instrument today. Garrison Keillor once described the instrument as “an instrument for a saint” because “it takes fourteen hours to tune a harp, which remains in tune for about twenty minutes, or until somebody opens the door.”


A modern electric lever harp (photo:Athy)

Smaller harps (including modern electric harps, like the one above) use levers to modify individual strings (which makes key changes much more difficult than with the pedal harp). Electric harps weighing up to 8 kg are described as “wearable,” which reminds me a little of this 11 kg grand-daddy of the laptop.

Camille and Kennerly Kitt playing “wearable” electric harps

The harp is often seen as a stereotypically feminine instrument – when I look at American harpists on Wikipedia, I count 10 men and 60 women. There are, however, exceptions.

Jakez François (president of French company Camac Harps) playing jazz


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).


Congratulations, Toronto!


Toronto’s solar car Horizon comes 12th in the 2015 World Solar Challenge (my photo)

The Blue Sky Solar Racing team from the University of Toronto ( ) had their 20th anniversary on November 18th last year. The video below celebrates their two decades of solar car racing. Congratulations, guys!


The magnificent Doble steam car

The video above shows the beautiful 1920s Doble steam car owned by Jay Leno (see this article). This magnificent vehicle represents the pinnacle of a technology that was already dead when it was built. A front-mounted boiler powers four cylinders at the rear, which drive the back wheels via spur gears (see below). There is no traditional gearbox or transmission. The steam is condensed and recycled, so that water does not have to be constantly replenished. All very efficient.

Leno says that “The last days of an old technology are almost always better than the first days of a new technology,” and aesthetically (in spite of my love of solar cars) he is probably right. Something similar can be said about the ultimate examples of castle-building, which occurred when castles were already obsolete (see below). So watch the video of this wonderful vintage car!


Church and Turing


Alonzo Church (left) and Alan Turing (right)

Alan Turing is no doubt the most well-known of all computer scientists. His Turing machine is justly famous, and the video below shows a really cool Lego version of it.

However, it is not widely known that the fundamental result in Turing’s 1936 paper (“On Computable Numbers, with an Application to the Entscheidungsproblem” – with errors later corrected in 1937: i, ii, iii) was not actually new. This is revealed on the second page of Turing’s paper, where Turing admits “In a recent paper Alonzo Church has introduced an idea of ‘effective calculability,’ which is equivalent to my ‘computability,’ but is very differently defined. Church also reaches similar conclusions about the Entscheidungsproblem . The proof of equivalence between ‘computability’ and ‘effective calculability’ is outlined in an appendix to the present paper.

It seems that Max Newman encouraged Turing to explain how his work related to the result that Alonzo Church had already obtained, lobbied the Secretary of the London Mathematical Society to still permit publication of Turing’s paper, and arranged for Turing to go to Princeton, where Church would be his PhD supervisor (Church also supervised 34 other students). It’s a terrible feeling for a student to be “gazumped” like that, and invaluable to have helpful people backing you up. Turing went on to have a colourful, but relatively short, life. Church lived to 92, but was staid in comparison.

Church’s notation, the lambda calculus, has been quite influential, spawning programming languages such as Lisp and Haskell, as well as more theoretical work. The fundamental feature of lambda calculus, the anonymous function written function (x) { e }, has been incorporated into many other programming languages as well. Consider the following code in R, for example:

i <- function (x) { x }
k <- function (x) { function (y) { x } }
s <- function (x) { function (y) { function (z) { x(z)(y(z)) } } }
y <- function (f) { (function (x) { f(x(x)) }) (function (x) { f(x(x)) }) }

Notice that functions can themselves be arguments to other functions, and in x(x), a function is applied to itself.

The object i is the identity function, so that i(3) reduces to 3. The expressions k(3)(4) and s(k)(k)(3) also reduce to 3. The object y is the Y combinator, with y(k(3)) reducing to 3, and y(i) generating infinite recursion. The following code uses y to generate the factorial function:

fact <- y (function (g) { function (x) { if (x == 0) 1 else x * g (x - 1) } })
fact (3)

The appendix that Turing was forced to write was perhaps the most significant aspect of his 1936 paper. Showing that his definition of computability was equivalent to that of Church led to the Church–Turing thesis – the idea that that is the only kind of computability (at least on numbers). Independently of Turing, Emil Post produced another equivalent form of computability in 1936. Yet another form came out of Russia. Even Conway’s Game of Life (below) turns out to capture the same kind of computability, though in an inherently parallel form:

See more on the Church–Turing thesis at plato.stanford.edu. There was also a series of interesting talks presented on Church and Turing in 2012, including this one by Philip Wadler:

While the lambda calculus was probably the version of computability with the largest impact on programming languages, the more practical Von Neumann architecture probably had the largest impact on hardware design. It is difficult to imagine what the world would look like without those two concepts.