Plus Blog

May 17, 2006
Wednesday, May 17, 2006

Maths maps musical emotion

Music makes emotion and maths underpins music. But can the emotional part of music be captured by maths? To find the answer to this question Elaine Chew, assistant professor of industrial and systems engineering at the USC Viterbi School of Engineering, set up a graduate course called "Computational Modelling of Expressive Performance" to look for answers. The aim of the course is to teach students how to use tools from engineering and computation to understand our emotional responses to music.

Chew pursues a successful career as a pianist along side her research, and so is perfectly qualified. But as she admits, teaching the course is a challenge, as students need expertise from a huge range of different disciplines. They need to use tools from music theory, cognitive science, artificial intelligence, experimental psychology, signal processing and neuroscience, to name but a few. Not many students have the necessary pre-requisites, but some of those that do have now put an exhibition of their projects online.

One pair of students have created "emotiongrams" that map musical characteristics to colour patterns to capture their emotional impact. Another project involves analysing scores of silent movies that have been specifically created to evoke certain emotions, and yet another one involves a system that allows you to add extra rules to a musical score and observe whether this changes the impact on your feelings.

You can read about all the projects on the course website. Plus has a number of articles on the mathematics of music, which you can find in our archive under keyword or topic "maths and music".

posted by Plus @ 1:19 PM

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April 26, 2006
Wednesday, April 26, 2006

29 steps to universal (26/04/2006)

Think of a number, any number, and see if you can write it as the sum of square numbers: 13 = 22 + 32, 271 = 12 + 12 + 102 + 132, 4897582 = 62 + 952 + 22112...

In 1770, Lagrange proved that every positive integer, no matter how large, can be written as a sum of at most four squares, x2 + y2 + z2 + t2. In the centuries since, mathematicians searched for other universal quadratic forms which could represent all the positive integers. Another 53 expressions, including 1x2 + 2y2 + 3z2 + 5t2, were found by Ramanujan in 1916. So many such universal forms exist, but how can you predict if a particular quadratic form is universal?

Now the brilliant young mathematician Manjul Bhargava and his colleague Jonathan Hanke have found a surprisingly simple result that completely solves the problem of finding and understanding universal quadratic forms. They found a shortcut to deciding if a quadratic form is universal — to check if the form represents every single positive integer, you only have to check it represents a mere 29 particular integers, the largest of which is 290. Bharghava and Hanke then went on to find every universal quadratic form (with four variables), all 6,436 of them. You can read about this surprising result in Ivars Peterson's excellent article in Science News.

Apart from Bhargava's brilliance as a mathematician (he was one of the youngest people to be made a full professor at just 28), he is also an accomplished musician. Both number theory and tabla playing may be viewed as the study of patterns, Bhargava told Peterson. "The goal of every number theorist and every tabla player," he explains in the article, "is to combine these patterns, carefully and creatively, so that they flow as a sequence of ideas, tell a story, and form a complete and beautiful piece."

If you're struggling with a sum of squares, try out this useful applet by Dario Alpern.

posted by Plus @ 2:15 PM

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March 29, 2006
Wednesday, March 29, 2006

Frequently flipping fields (29/3/2006)

In the past news story, Core business, we reported on research that the Earth's magnetic field may be in the early stages of a polarity reversal. These geomagnetic flips have occurred hundreds of times in the last 160 million years, and as the last one was 780,000 years ago many researchers think we are overdue for the next reversal. Until recently it was thought that the frequency of these polarity reversals was random, following a Poisson distribution. However Italian physicists have found that the process is better described by a Levy distribution — a distribution used for processes that exhibit "memory" of past events. So the pole reversals are not independent events, and this information should inform models of the magnetic dynamo process believed responsible. But as for when the next flip is due, that still remains a mystery.

You can read about the strange magnetic fields of Uranus and Neptune in Untangling a magnetic mystery, and about the important role of the magnetic field in the Sun in The dynamic sun. You can also find out more about the Poisson distribution in On the ball.

posted by Plus @ 2:13 PM

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March 14, 2006
Tuesday, March 14, 2006

Happy Pi Day to you! (14/3/2006)

Write the date March 14th in the date format used in the US and you get 3.14 — which makes it Pi Day! Plus is celebrating by reflecting on the ubiquitous usefulness of this number to mathematics, pondering its many unsolved mysteries, and of course eating an appropriately shaped pi pie (pie are squared you know). How will you be celebrating? If reciting digits of pi is to be your pi party trick, then make sure you read Remembrance of numbers past from issue 31.

"If a billion decimals of pi were printed in ordinary type, they would stretch from New York City to the middle of Kansas. Only fortyseven decimal places of pi would be sufficiently precise to inscribe a circle around the visible universe that didn't deviate from perfect circularity by more than the distance across a single proton."
- from The Riciculously Enhanced Pi Page

posted by Plus @ 2:13 PM

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February 16, 2006
Thursday, February 16, 2006

Gravity kills dark energy? (16/02/2006)

In 1998, scientists found that the rate at which the universe expands is accelerating. This was extremely puzzling, as none of the standard theories could account for the acceleration. So scientists decided that something unknown and mysterious must be responsible, and they called it dark energy. But now three scientists claim that we may not need dark energy after all, but that the acceleration can be explained by a modified model of gravity. They altered the equations from the theory of general relativity in a way prescribed by cosmological theories based on a universe with more than four dimensions. When applied to short distances (on the cosmological scale), the differences between the old and new models are not noticeable, however, on ultra large distances the new equations predict the accelerated expansion. Moreover, the researchers say, the predictions match observations extremely well and don't conflict with other observations.

You can find out more about the new model for gravity on Physicsweb, and more about dark energy in Plus article New light shed on dark energy.

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January 27, 2006
Friday, January 27, 2006

Small, cool planet found (27/01/2006)

Scientists have found the smallest extrasolar planet yet. The inspiringly named OGLE-2005-BLG-390Lb orbits a red dwarf about 22,000 light years away, close to the centre of the Milky Way. Its mass is between 2.8 and 11 times that of the Earth's mass, possibly making it the least massive planet that has ever been found outside our own solar system. It's about 2.6 times as far away from its star than the Earth is from the Sun. This distance, together with the fact that the star is a lot smaller and less bright than our Sun, means that temperatures on this icy, rocky planet are somewhere around -220° — too cold for life as we know it.

The planet was discovered using a new technique called gravitational microlensing. Until now, planets were spotted by the way their gravitational pull makes their host star wobble, a technique which uses the Doppler effect (see Plus article Brave young worlds). But for this to work, the planet has to be massive enough or close enough to its star to exert the necessary pull. Consequently, all planets found so far are much more massive than OGLE-2005-BLG-390Lb.

Gravitational microlensing uses a different approach: as a star passes in front of a more distant reference object, its gravity acts like a lense and increases the object's brightness for a period of a few weeks. If the moving star has a companion planet, then this gives the brightness an extra spark. Although this is the third exoplanet discovered using gravitational microlensing, it's the first one of such low mass. Scientists think that these small worlds are much more common in the universe than was previously assumed. To find out more and see some pictures read this article from the European Southern Observatory.

posted by Plus @ 2:11 PM

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