Musical Numerology #3 ~ Harmonia Universalis vs. the Pythagorean comma

The Claim: simple whole number ratios are the building blocks of nature. In music these ratios produce the best, most stable, most beautiful harmonies. The movements of the planets and the sun produce a humming sound; the Harmony of the Spheres. This deep cosmic chord is also based on simple whole number ratios.

The evidence: musical arguments

The ancient Greek cult leader Pythagoras was taking a walk when he passed a blacksmith who was hammering away on his anvil. The ringing of the metal sounded quite harmonious to Pythagoras, and upon closer inspection he was able to determine that the weight of the hammers used by the blacksmith was responsible for the frequencies (or pitches) of the various metallic sounds.

Pythagoras was struck by his discovery, and it inspired him to do some serious research. He built lots of musical instruments, he hammered on them and then he would measure them very precisely. This is what he found: the musical intervals that please our ears the most are the ones based on the simplest whole number ratios,

octave 1:2
fifth 2:3
fourth 3:4

Deeply impressed by these new discoveries, Pythagoras concluded - a bit hastily perhaps - that all nature consists of harmony arising from number. Then he got into some very serious trouble. Pythagoras tried to construct a musical scale using the 2:3 ratio (producing a cycle of fifths) but he ended up with an imperfection, which is called the Pythagorean comma (it's size approximately 74:73). The exact method used by Pythagoras to create scales, and the comma troubles caused by his method, are explained in the two videos below.

The bottom line: all this harmonious interplay of whole number ratios suggests that there has to be some perfect and simple system behind it, but in reality this 'beautiful' perfect system doesn't exist. There's a fly in the ointment; it's called the Pythagorean comma. This little comma gave Pythagoras a massive headache.

The evidence: numerological arguments
Pythagoras really, really loved numbers. Here's what Aristotle had to say about the Pythagorean number fetish: "The Pythagoreans were the first to take up mathematics and thought its principles were the principles of all things. Since of these principles, numbers are the first, in numbers they seemed to see many resemblances to things that exist. More than just air, fire and earth and water, but things such as justice, soul, reason, opportunity."

The numbers which impressed Pythagoreans the most were the ones found in musical ratios. Here's another quote from Aristotle: "The Pythagoreans saw that the ratios of musical scales were expressible in numbers and that all things seemed to be modeled on numbers, and numbers seemed to be the first things in the whole of nature. They supposed the elements of number to be the elements of all things, and the whole heaven to be a musical scale and a number."

So Pythagoras liked numbers, and he also liked simplicity, as in simple whole numbers and simple ratios. Everything nice and clean. Pythagoras didn't know about negative numbers or zero and we know that he hated the idea of irrational numbers, like pi. He preferred to deal with 1, 2, 3 and 4, slowly working his way up to 10 and beyond.

Pythagoras isn't the only scientist who loves both numbers and simplicity. Most mathematicians and physicists prefer simple theories and simple equations, the most famous example being E=mc2. Scientists usually refer to simple theories as 'elegant' or 'beautiful' although elegance and beauty are not the same thing as simplicity.

In reality, the beauty and elegance of nature are not simple. Just look at the theory of quantum physics, which is a mess. Ironically, the strangest concept of quantum physics, the Uncertainty Principle, even seems to suggest that nature's deepest secrets are unknowable, and that nature likes to play hide and seek. Insofar as mathematicians and physicists accept this idea, they become mystics (Einstein, Bohr, Pauli, Buckminster Fuller). The rest of the field keeps chasing the latest, greatest, most beautiful and elegant pipe dreams. The recent death of supersymmetry theory (blame the Large Hadron Collider) is just another example.

The verdict
Nature likes to play hide and seek. A simple, harmonious theory of nature may seem 'beautiful' but that doesn't make it true. Musica Universalis is a great metaphor but as a scientific theory it's quite worthless. There are many ways to create scales, tunings and instruments. Music doesn't always have to be perfectly in tune to sound good: extremely stable, pure harmonies can also sound tame and uninspiring.

1. Try to think of 5 reasons why scientists might prefer 'elegant' or 'beautiful' theories over 'simple' theories.
2. Imagine yourself in the Head Space of a holistic whole number fetishist. Analyze the Pythagorean comma as a conspiracy to compromise the purity of the Universe and all its precious fluids.
3. Create a Theory of Everything based on the numbers in your date of birth.
4. Write a simple, solid tune using only octaves, fifths and fourths.