Looking back: 1982

In 1982 (35 years ago!) I finished my basic undergraduate degree, majoring in Mathematics and Computer Science (after sniffing around the job market, I continued my studies for an honours year). This was the year that the compact disc and the Commodore 64 computer came out:

Also that year, Stephen Cook won the Turing award for his work on computational complexity theory. The then Soviet Union landed two spacecraft in the hellish inferno that is Venus, and took photographs:

It was also a year of conflict – Argentina started a war with the UK over the Falklands Islands, and Israel invaded Lebanon. On a more positive note, there were several movies which became cult classics, such as Tron, E.T., and Conan the Barbarian. The superb science fiction movie Blade Runner stood out from the crowd (even with the flaws in the original cinema release):

In literature, Isabel Allende published her debut novel, as did Kazuo Ishiguro. In music, The Alan Parsons Project released their album Eye in the Sky and Australian band Icehouse (originally Flowers) released their classic single “Great Southern Land”:

Overall, it was a great year (apart from the wars).


Looking back: 1978

In 1978 I started senior high school (year 11 and 12). That was a year of terrorism – a bomb was exploded outside the Sydney Hilton Hotel by the Ananda Marga group (apparently in an attempt to kill Indian prime minister Morarji Desai), and former Italian prime minister Aldo Moro (below) was kidnapped and murdered by the Red Brigades. On a more positive note, John Paul II became the first Polish pope, and helped to chip away at the power of the Soviet Union.

That year also marked the debut of the soap opera Dallas and the comic strip Garfield. In science, James Christy at the United States Naval Observatory discovered Pluto’s moon Charon. We finally got a good look at it in 2015:

In computing, the Turing Award went to Robert Floyd, for his work in programming languages and algorithms. Intel introduced the 8086, the first of the x86 microprocessors which are still the most common CPUs in personal computers and laptops today. The game Space Invaders also had its debut:

The year 1978 also saw the release of the unsatisfactory animated version of The Lord of the Rings, and a number of interesting albums, including The Kick Inside by Kate Bush, Pyramid by The Alan Parsons Project, Dire Straits by the band of the same name, the electronic Équinoxe by Jean Michel Jarre, and Jeff Wayne’s Musical Version of The War of the Worlds:

Of the books published that year, The Road Less Traveled by M. Scott Peck, the exceedingly dark The House of God by Samuel Shem, and A Swiftly Tilting Planet by Madeleine L’Engle (below) stand out.


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).


Looking back: 1980

In 1980 I began my undergraduate university studies (in Chemistry, Botany, Mathematics, and Computer Programming). That year, Iran seemed to be constantly in the news. In April, the US tried and failed to rescue the embassy staff taken hostage in Tehran. Some days later, the British SAS rescued Iranian embassy staff held hostage in London. In September, Iraq began a war with Iran that lasted until 1988 (and which essentially continues today as the Sunni–Shiite conflict in Iraq and Syria). In Australia, the death of Azaria Chamberlain caused great controversy. Natural disasters that year included the eruption of Mount St. Helens (which was to inspire the 1997 film Dante’s Peak):

On a more positive note, the World Health Assembly declared on 8 May 1980 “that the world and its peoples have won freedom from smallpox, which was a most devastating disease sweeping in epidemic form through many countries since earliest time, leaving death, blindness and disfigurement in its wake and which only a decade ago was rampant in Africa, Asia and South America” (Resolution WHA 33.3). The Olympics were held in Moscow (though with only 80 nations) and in Lake Placid. The space probe Voyager 1 made its closest approach to Saturn, taking spectacular photographs of the ringed planet and its moons (such as Rhea, below):

Ronald Reagan was elected President of the United States, and Tony Hoare won a well-deserved Turing Award. Usenet (a precursor to Internet discussion forums) was launched. The Rubik’s Cube was licensed to a US toy company and became a craze. The arcade game Pac-Man was released, and it became a craze as well:

Two much-loved films, The Empire Strikes Back and The Blues Brothers, also appeared. An obscure Italian book called Il nome della rosa had yet to make an impact, although The Clan of the Cave Bear was undeservedly trendy. Music of the time was on vinyl or the now-forgotten cassette tapes. During the year, Stevie Wonder released his album Hotter than July, Supertramp released their live album Paris, Flowers released Icehouse, the Alan Parsons Project released their superb The Turn of a Friendly Card, and Kate Bush released this very strange single: