Harmony 'driven by mathematical ratios'

It was apparent to mathematicians going back to Ancient Greece that musical harmony was fundamentally driven by mathematical ratios.

The public are generally unaware that the "three chords trick" which is the basis of all popular music is an idea going back to the father of geometry Pythagoras.

The Greeks established that the first note in the scale and the fourth and fifth are the most harmonious together and they are governed by ratios.

For instance a standard A has a pitch of 440hz, the fourth note a D has a pitch which is three-quarters of that at 330hz and the E note or fifth is two-thirds of the pitch of the A at 293.33hz. The higher A at the end of the octave is at 220hz half that of the lower A.

The ratios remain as important today as they were in Greek times.

How mathematics provides the fundamental basis of music was the subject of a talk by two mathematicians Professor Robin Wilson from Oxford and Professor Ehrhard Behrends from Freie Universität Berlin at the Euroscience Open Forum (ESOF) yesterday.

Prof Wilson, a son of the former British Prime Minister Harold Wilson, said the Greeks regarded such ratios as providing a key to the structure of the whole universe as they were supposed to relate to the ratios of distance between the planets.

He explained though that our current 12 note scale where all the notes are of equal temperament (equal distances between them) does not exactly correspond with the Pythagorian ideal that notes should be based on fractions of pitches between each other.

Therefore, he said, the vast majority of pianos are slightly out of tune.

He went on to note that many composers over the century have used mathematical constructs to produce complex pieces of music citing the example of Bach's Crab canon from his Musical Offering where the first part of the piece is the same as the second part backwards and vice versa.

There is also a huge amount of mathematics in jazz and also in The Beatles' music, he added.

Professor Behrends spoke about the mathematical conundrum posed by an American mathematician Mark Kac in 1966. "Can one hear the shape of a drum?" It was a conundrum which took three decades to solve.

He explained that in one dimension such as a guitar string you can tell the length of the string by the sound at certain intervals on the string, but you cannot do the same with a two dimensional shape such as a round or square drum for reasons which take a very complicated formula to explain.