Rational and Irrational Beauty

Much is happening in my life – I’ll tell more in another blog in the near future – but here’s an intellectual-ish problem that’s been executing ill-choreographed dances in my mind for the past several months. We all know that ‘beauty’, in all the many and varied ways in which this elusive abstraction is manifest, has much to do with balance and proportion: mathematical relationships, in other words. But are the proportions inherent in ‘beauty’ rational or irrational, in the mathematical sense of those adjectives?

A rational number is a number that can be expressed as a ratio of two integers: 2/3, 5/8, 99/101, and so on. Positive integers are themselves rational; the number 5, for example, can be written 5/1. Pythagoras and his followers from the 6th century BC onwards taught that the universe was constructed in accordance with rational numbers, and that beauty (particularly in music) consists in simple numerical relationships – simple rational numbers. Some Pythagoreans doubted the reality of irrational numbers.

An irrational number cannot be expressed as a ratio of two integers. An example is the square root of 2, which is approximately 1.414… but the row of digits after the decimal point extends ad infinitum. The ratio of the lengths of the diagonal of a square to a side of that square is the square root of 2. Pi, the ratio of the circumference of a circle to its radius, is also irrational. One irrational number that has captured the imaginations of philosophers of aesthetics is the so-called ‘golden ratio’, phi. If you divide a straight line into two unequal parts such that the ratio of lengths of the smaller part to the greater is equal to the ratio of lengths of the greater part to the whole line, then the ratio in question is phi, approximately 0.618…

The architectural proportions of classical temples are claimed to be phi. Leonardo da Vinci and his contemporaries discovered phi in many of the proportions of the human body. So contrary to what the Pythagoreans taught, beauty (at least in these manifestations) is irrational. But is it? The rational number 8/13 is slightly more than 0.615, which is within 0.8% of phi. Can the human eye tell the difference between phi and a proportion that is less than 1% below (or above) phi? Were the measurements of Greek temples so amazingly accurate? And don’t human bodies vary in their proportions by more than 1%?

(There’s one sequence of rational numbers that gets nearer and nearer to phi as you go along. The classical Fibonacci sequence is made by adding its latest two numbers together to make the next one – 1, 1, 2, 3, 5, 8, 13, 21, 34… – and it is the ratios between successive numbers that approximate more and more closely to phi: 1/1, 1/2, 2/3, 3/5 (=0.6), 5/8 (=0.625), 8/13 (=0.615), 13/21 (=0.619)… This connection between phi and the Fibonacci sequence was discovered by the astronomer Kepler early in the 17th century.)

Harmonic relationships in music are, in principle, rational numbers. In the diatonic scale that has dominated most Western music since the 15th century, a major second has the frequency ratio 9/8, a major third 5/4, a perfect fifth 3/2, a major seventh 15/8, and so on. This created difficulties for the designers of keyboard instruments. For example, if you start in the key of C, the frequency ratio of G sharp to the tonic C is 25/16 (=1.5625), but the frequency ratio of A flat to C is 8/5 (=1.60), a very noticeable difference. This is why the G sharp/A flat key on many early 17th century keyboard instruments is split; the player can press either the G sharp of the A flat part of the key and slightly different notes emerge. When composers wanted to write in different keys, headaches resulted. That is why, at the time of J.S. Bach, “equal temperament” was developed, a device by which each semitone interval on the keyboard had a frequency ratio of the twelfth root of two (= 1.0594… – an irrational number). In equal temperament, the major second ratio is 1.1224…, very close to the diatonic 1.125; the major third is 1.2598, very close to the diatonic 1.25; and the troublesome augmented fifth/minor sixth (G sharp/A flat) is 1.5871…, a reasonable compromise between the conflicting diatonic ratios of 1.5625 and 1.60.

The prevailing, though often implicit, belief is that although the frequency ratios between the notes issuing from the instrument when it is played are irrational, the listener ‘hears’ them as rational; the brain ‘adjusts’ them to the appropriate diatonic intervals. This hypothesis – conjecture, rather, because I have no idea how it could be tested critically – is brother and sister to the conjecture that the proportions in Classical temples and human bodies, which are claimed to be (or to be perceived as) phi, are actually perceived by the viewer’s eye and brain as rational numbers taken from an early part of the Fibonacci sequence. Pythagoras still rules, okay.

But the truth could be the opposite. Perhaps, when we listen to music, we don’t ‘hear’ diatonic intervals – even when they are generated exactly by, for example, electronic oscillators. Perhaps what our ears and brains perceive is an equally tempered interval based on the twelfth root of two. And perhaps the proportions in classical temples and human bodies are perceived as phi, even when they are not, in objective external reality, exactly equal to phi. (The great 14th century polymath Nicole d’Oresme proposed something akin to equal temperament in the perception of music, which verged on heresy at a time when the Pythagorean account of musical and heavenly harmony was accepted without question. Oresme is well worth reading for those who’re interested in such matters.)

The beautiful cathedrals of medieval Europe have many proportions based on the square root of 2, so even in that generally Pythagorean era, the master masons built with ‘irrational beauty’ in mind.

On the other hand, some parts of the natural world that are generally considered beautiful seem to have rational numerical proportions; arrangements on leaves and flowers on many plants, for example, accord precisely with Fibonacci numbers. Nature provides us with examples of ‘rational beauty’.

At risk of an accusation of sexism, I should also comment on the subject of feminine beauty. The proportions considered ‘beautiful’ in adult human females have changed over time. Today’s slim-line cat-walk models have a height to waist ratio of about three to one; the so-called Venus de Milo (a statue of Aphrodite prejudicing the Judgment of Paris) has a height to waist ratio of about two to one. The modern ‘beauty’ has more or less equal bust and hip measurements, but the Venus de Milo has a bust to hip ratio of (approximately) the fourth root of 2. In other words, the Venus de Milo is rather pear-shaped. So, we would infer, was Helen of Troy. The only ‘ideal’ major ratio in female bodies that seems to have remained constant between Classical times and the present is hip-to-waist: approximately the square root of 2. Past or present, therefore, we must conclude that feminine beauty is apparently irrational.

At the end of these reflections I have no clear idea whether ‘beauty’ is rational or irrational. I have no idea whether the Pythagoreans or Leonardo got it right. Maybe the answer is ‘both’, because there is no such single abstract entity as ‘beauty’. Maybe, mathematically speaking, beauty is different in its different manifestations. And perhaps, after all, whether it’s rational or irrational is a matter that lies in the eye, the ear, and the brain of the beholder.


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