Wednesday, 13 August 2014

Golden Ratio and Beethoven's 5th

I had an email yesterday from a student in Spain that began:

'I am currently carrying out a Research Project about the Golden Ratio. In fact, what I am particularly interested in is its appearance in classical music and its use by some musicians in their compositions. It has been brought to my attention that you have been studying this matter. If it is not too much to ask, given your expertise, it would be of great help to me if you could share some of your thoughts about this particular topic ...'

In September 1978 (yes, 36 years ago!) I wrote a brief one-off article that was published in Mathematics Teaching (volume 84) on this subject, mainly focussing on some interesting occurrences of the 'golden section' in the structure of the first movement of Beethoven's 5th symphony. My suggestion was that this might make an interesting investigation for arts sixth-formers doing general studies courses. I have never returned to this subject and not written any more about it. But that little article – from my perspective probably the least significant piece I have ever had published – continues to get cited and references to it turn up often in internet searches. So, 36 years later once again I get someone contacting me as though I am an expert in this field! Which I am not.

However, this is what the article was about.

First some revision. A 'golden rectangle' is one with the lengths of the sides in a particular proportion: approximately 1.618 (or, precisely, half of '1 plus the square root of 5').  So, for example, imagine a rectangle with sides 1 cm and 1.618 cm, a shown. This is a golden rectangle. If you cut off a 1-cm square, as shown, the piece that you are left with has sides of 1 cm and 0.618 cm. The ratio of the lengths of these sides is 1 ÷ 0.618 which, remarkably, equals 1.618! So this smaller rectangle is also a golden rectangle! 
This is the defining property of a golden rectangle: that if you cut off a square, as shown, you are left with a rectangle with sides in the same ratio as the original rectangle. From this, with a little bit of algebraic manipulation, you can work out that the ratio of the sides is 1.618 to three decimal places. Dividing the rectangle like this - or the division of any quantity into two bits that are in this ratio - is also called a 'golden section'. 

For example, if I had 1618 counters and put them into two piles with 1000 in one pile and 618 in another, then I would have applied a golden section to the set of counters. If I were then to apply a golden section to the 1000 counters, I would get 618 in one pile and 382 in the other. And so on.

If the smaller golden rectangle in the diagram is divided into a square and an even smaller golden rectangle, the 1 cm length is divided into 0.618 cm and 0.382 cm. So a golden section can be thought of as dividing 1 unit into two parts in the ratio 0.618:0.382 (or, to two decimal places, 0.62:0.38).

This is a magical ratio that keeps turning up in all kinds of situations, in mathematical contexts, but also in art, architecture, the natural world, psychology, and, indeed, music. Bartok, for example, deliberately incorporated the golden ratio into his musical composition.

My discovery was that there are a number of occurrences of this golden ratio in the first movement of Beethoven's fifth symphony – that's the one that starts with the famous motto theme: 'da, da, da, daah'! In Beethoven's original score there are 600 bars before the final statement of the opening motto. But a statement of the opening motto also appears at bar 372. So, we have this structure for the three main statements of the motto: motto starts ... 372 bars ... motto starts... 228 bars ... motto starts. 

Now what do we get if we split the 600 bars into two sections using the approximation to the golden section 0.62:0.38? Well, amazingly, 0.62 of 600 is 372, and 0.38 of 600 is 228. So the three statements of the opening motto theme begin at points that divide the score using a golden section! 

Is this just something that happened by accident, or did Beethoven do it deliberately? We shall probably never know. I have no view on this. But there are other instances in this movement. I will mention just two.

1. The 'exposition' in the first movement (the statements of the two main musical subjects) is in three sections: 24 bars of the first subject rounded off with a version of the motto; then 38 bars with an extended restatement of the first subject rounded off with a version of the motto; then a further 62 bars that begin with the statement of the second subject. 24:38 is a golden section of the first 62 bars; and 38:62 is a golden section of the last 100 bars!

2. The movement has the most extraordinary 'coda'. A coda is usually just a few bars of music to round off a movement. In this case Beethoven gives us a coda that is 129 bars long! Divide this coda up using the golden section and you get 49 bars and 80 bars. So what happens after 49 bars of the coda? This is the point where Beethoven actually introduces a completely new tune that has not appeared in the movement so far. Before Beethoven no-one would ever introduce new material in a coda! So this is a very significant point in the coda. Is Beethoven signalling his piece of radical creativity in this very long coda by linking it with the golden section? Again, we shall never know. But either way, it is interesting.

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