The Maths of Tango

Turning a mathematical theorem and proof into a musical

How do you make a musical about a bunch of dead mathematicians and one very alive, very famous, Princeton math professor?

Andrew Wiles, the Eugene Higgins professor of math, gained worldwide fame for his 1993 solution to Fermat’s last theorem, which dates to 1637. The theorem states that for the equation xn+yn=zn there are no positive whole numbers that solve this when “n” is greater than 2.

French mathematician Pierre de Fermat had noted the theorem in the margin of a book and wrote that

A mathematician’s guide to mating

by John Billingham

When you were small, you probably heard the fairytale The Frog Prince. The original version of the story is rather more complicated.

The Frog Prince: original version

Once upon a time, a princess was walking through a forest and stumbled across a pond. Out of the pond rose a witch, who cackled, “Stop! I have turned a handsome prince into a frog and cast him into my pond to live with 99 other frogs. Each frog has a different number on his back. The prince has the largest number on his back, and this is your only way of spotting him. You must find him and kiss him if you want to leave my enchanted forest. The frogs will jump from the pond one by one. When each frog appears, you must decide whether to kiss him or throw him back in, never to be seen again. If you kiss a real frog, or don’t kiss any of the 100 frogs, you will never leave the forest, and the prince will remain in the pond.” And with a suitably evil laugh, the witch sank back into her murky pond. Fortunately, the princess was very good at maths, and knew the best strategy for deciding which frog to kiss.
Frog 1 hopped out of the pond with the number 2 on its back. The princess kicked it back into the pond. Frog 2 had the number 12 on its back. Better, but it received the same treatment from the princess. Frog 3 was number −6 (Did I say the numbers had to be positive?) and was duly dispatched by the princess. She repeated this for the first 37 frogs, and noted that the highest number she had seen so far was 23.2 (Did I say they had to be whole numbers?). She then waited until she saw a frog numbered higher than 23.2 and kissed it. (Of course the frog numbered 23.2 might have been the prince, and she’d then have missed him.) The frog disappeared in a puff of smoke to be replaced by a handsome prince, and they lived happily ever after (or, if you prefer unhappy endings: the witch rose from the pond laughing as the frog remained unmoved by the princess’s attentions. Oh dear! There was a higher numbered frog still to come).

The magical mathematics of music

by Jeffrey S. Rosenthal

The astronomer Galileo Galilei observed in 1623 that the entire universe “is written in the language of mathematics”, and indeed it is remarkable the extent to which science and society are governed by mathematical ideas. It is perhaps even more surprising that music, with all its passion and emotion, is also based upon mathematical relationships. Such musical notions as octaves, chords, scales, and keys can all be demystified and understood logically using simple mathematics.

Where Math meets Music – Joseph Heimiller

Where Math meets Music

Ever wonder why some note combinations sound pleasing to our ears, while others make us cringe? To understand the answer to this question, you’ll first need to understand the wave patterns created by a musical instrument. When you pluck a string on a guitar, it vibrates back and forth. This causes mechanical energy to travel through the air, in waves. The number of times per second these waves hit our ear is called the ‘frequency’. This is measured in Hertz (abbreviated Hz). The more waves per second the higher the pitch. For instance, the A note below middle C is at 220 Hz. Middle C is at about 262 Hz.

Now, to understand why some note combinations sound better, let’s first look at the wave patterns of 2 notes that sound good together. Let’s use middle C and the G just above it as an example:

Now let’s look at two notes that sound terrible together, C and F#:

Do you notice the difference between these two? Why is the first ‘consonant’ and the second ‘dissonant’? Notice how in the first graphic there is a repeating pattern: every 3^rd wave of the G matches up with every 2^nd wave of the C (and in the second graphic how there is no pattern). This is the secret for creating pleasing sounding note combinations: Frequencies that match up at regular intervals