We musicians are always boasting about how diverse music is. It’s art, it’s science, it’s mathematics. A lot of mathematics. Sadly, no one believes us – after all, how mathematical can music get? Aside from pitch frequencies, we never have to deal with numbers. Right?
Of course not! Music is littered with numbers, whether you see them or not. Today I want to explore some fascinating mathematical patterns in music – specifically, the appearance of Phi, also known as the golden ratio, divine proportion, or golden mean.
Phi, as some of you may already know, is an irrational number 1.6180339887… which continues on forever. You can represent Phi in several different ways. One way to think about it is as a sequence, called Fibonacci’s sequence. The sequence goes thus: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34,… forever. It consists of adding the previous two numbers to produce the third. (0+1=1, 1+1=2, 1+2=3, 2+3=5, 3+5=8, and so on.) As you continue the sequence, the ratio of one number in the sequence to the number preceding it converges upon Phi.
1/1 = 1
2/1 = 2
3/2 = 1.5
5/3 = 1.6666…
8/5 = 1.6
13/8 = 1.625
21/13 = 1.61538…
34/21 = 1.61904…
Slowly, the sequence is working towards Phi, though because Phi and the sequence are infinite, the sequence will never converge exactly upon Phi.
Probably the easiest way to visualize Phi—and this representation is important, as you’ll find out later—is as a rectangle, nicknamed the golden rectangle. Imagine a rectangle which has a length which relates to the height in the same way that the whole (length + height) relates to the length. The ratio of that triangle’s length to height or whole to length equals Phi.
Here’s the golden rectangle:
Pretty normal looking rectangle, right? But it has some interesting properties, one of them being that if you section off a perfect square from the golden rectangle, the resulting smaller section is another golden rectangle. So you can continue to section off perfect squares and yield more golden rectangles forever, getting an unending spiral.
Now, I don’t expect you to be blown away by Phi’s mathematical properties, though they are interesting. What is really amazing is its appearance in day-to-day life. To give a few examples, the golden ratio crops up in its different forms in nature, architecture, art, and—as you’ve probably guessed—music.
Let’s begin with some examples of the application of the golden rectangle in architecture. The best one would probably be the Parthenon.
You can see that its proportions are based on the golden rectangle. The same goes for the Taj Mahal, Notre Dame, and even the Great Pyramid of Giza.
Now let’s hop on over to take a look at some art – in particular, Leonardo da Vinci’s Mona Lisa. There are plenty of other examples (you can take a look at them here), but I thought the Mona Lisa would work best because that’s a familiar painting to us all.
As you can see, there are golden rectangles at certain focal points in the painting, such as the top of her head and the corner of her eye.
But now let’s move on to something even more interesting – the golden ratio in nature. There’s no problem in using the golden ratio in art and architecture, but why do artists do so? Because the golden rectangle, based on the golden ratio, abounds in the real world. Artists aren’t using the golden rectangle for no reason – they are, consciously or unconsciously, mimicking the world around them. The golden rectangle is a very basic, aesthetically pleasing shape. The reason artists use the golden ratio so frequently is because it appears naturally in the real world all the time.
There are innumerable examples I could show you, from insects to DNA to the human face, but I’ll just show you one striking example: a galaxy.
Phi in Music
Now we finally arrive to the golden ratio in music. Prepare to be surprised! Plenty of contemporary composers have written music based on the golden ratio, but did you know that even Mozart and Beethoven are suspected of having used it?
Not only that, but our scales might be founded on the Fibonacci sequence. For instance, an octave is comprised of 8 notes – a number in the Fibonacci sequence. There are 8 white keys and 5 black keys (another Fibonacci number) in a chromatic scale which spans an octave – 13 keys in total. (Fibonacci again.) Some have even suggested that Phi factors into the relationships between pitch frequencies, but that’s getting into some pretty technical stuff, so I won’t elaborate on all that. (If you do want to read up on it, though, you can do so here.) Others have used Phi in building recording studios, due to its ideal acoustic properties.
Now, on to Mozart and Beethoven.
Well, I’m sorry to disappoint you after getting your hopes up, but there is scant evidence of Beethoven having used it. Derek Haylock decided to analyze Beethoven’s 5th Symphony in search of Phi. He claimed that Beethoven’s opening motif was repeated at .618 measures through the piece, and again at .618 measures from the end of the piece. However, Haylock had to manipulate the measure numbers to arrive at these calculations, so his conclusions are invalidated.
As for Mozart, John Putz analyzed his piano sonatas in search of Phi. He found that the exposition of Mozart’s Sonata No. 1 in C Major consisted of 38 measures while the development and recapitulation together consisted of 62 measures. He reported, “This is a perfect division according to the golden section in the following sense: A 100-measure movement could not be divided any closer (in natural numbers) to the golden section than 38 and 62.” This means that the ratio of the development + recapitulation to the exposition = Phi, just as the whole movement to the development + recapitulation = Phi. Still, after exploring deeper into Mozart’s sonatas, Putz had to conclude that, considering the large deviation from Phi in many movements, there was little evidence that Mozart used the golden ratio. It’s more likely that there was a tendency for Phi to appear in compositions due to the structure of sonata form. There is also the possibility that, because Phi has such natural aesthetic beauty, Mozart could have unintentionally used it, albeit rather loosely, in his music. Sever Tipei, a professor from the University of Illinois, wrote: “The golden mean ratio can be found in many compositions mainly because it is a “natural” way of dealing with divisions of time… It is a question if it was used in a deliberate way or just intuitively (probably intuitively). On the other hand, composers like Debussy and Bartok have made a conscious attempt to use this ratio and the Fibonacci series of numbers, which produces a similar effect.”
Yes, Bartok, Satie, Debussy, and Schubert all used the golden ratio. For example, Peter Smith—author of The Dynamics of Delight: Architecture and Aesthetics—notes that “In Debussy’s ‘Image, Reflections in Water’ the sequence of keys is marked out by the intervals 34, 21, 13, and 8, and the main climax sits at the phi position.” (Note that phi is the counterpart of Phi – it is just 0.618… not 1.618…) Bach, too, used the golden ratio very subtly in his music. You can read about it . So though it’s doubtful that Beethoven or Mozart used the golden ratio, a few great composers certainly did.
Lastly, I’d like to talk about the most surprising appearance of Phi – in the violin.
Yes, the dimensions of the violin were founded using Phi. As you can see, the ratios of certain sections of the violin equal Phi. Antonio Stradivari himself used the golden ratio to determine the placement of his F-holes. Also, although not shown in the chart above, the scroll of the violin is believed to be loosely based on the golden rectangle spiral.
So, Phi is everywhere, isn’t it? Especially in music. It’s interesting to consider how many other composers used Phi in their works, or what other instruments featured Phi in their dimensions. And we’re hardly done analyzing music and making new instruments; Phi is bound to be used for a long time to come.
“The most distinct and beautiful statement of any truth (as of music) must take at last the mathematical form.” – Henry Thoreau