The two diverging ideals of a theory

Like every theory, the theory of music moves between two extremes. On the one hand, a theory enables us to summarise quite different observations and explain them in a simple manner – the simpler, the better. On the other hand, we also want to apply this explanation, if possible, to everything that we observe. Thus a theory is good if it is as simple as possible but also explains as much as possible.

The challenge is to attain these two extreme objectives of every good theory at the same time.

What is typical is the moment when during the application of a theory, an observation suddenly crops up that is incompatible with the theory. Such observations plunge the theory into a crisis, as for instance when Max Planck noticed an inexplicable phenomenon in blackbody radiation, which gave rise to the quantum theory, or when Kurt Gödel’s observation of a gap in the logic of sets (incompleteness theory, 1931) plunged both set theory and classical logic into a serious crisis.

Every theory works well until it reaches its limits. Then, suddenly gaps emerge.

Is the pure temperament really pure?

Now, the crisis of which I am writing here is somewhat older than the ones triggered off by Max Planck and Kurt Gödel. Also, a practical solution to it was found long ago. It is a crisis in music theory, and the solution that was found is the equal temperament. This is the way in which we tune musical instruments today, but it isn’t a matter of course.

How did this come about? It had long been known that mathematical laws were behind the intervals which we perceive as euphonious. Musical scales with these intervals that are defined by simple fractions are considered to be pure; our major scale (Ionian) and all other ecclesiastical scales are perfectly pure, provided that the intervals are tuned in accordance with the simple fractions. Then they are “pure”.

However, this only works if we remain in the same tonality, i.e. if the music does not change its fundamental tone, i.e. does not modulate. In the Renaissance, however, composers increasingly felt a wind of change and started to modulate by changing the fundamental tone (the tonality) within the same piece of music. This made the limits of the pure (= Pythagorean) temperament evident.

The gap in the Pythagorean tonal system

When I first heard about the Pythagorean comma, I could hardly believe it. Like already in the Renaissance, our present music system consists of twelve halftones. If I ascend the ladder of halftones one by one, I will reach the fifth after seven halftones and the octave after twelve halftones. So when I ascend by twelve fifths (=12×7 halftones), I’m in the same place, mathematically speaking, as when I ascend by seven octaves (=7×12), right?

That’s as far as the mathematics is concerned, which made a great deal of sense to me when I was a child, and I was amazed that this should not be the case. In reality, we reach a slightly higher tone after twelve fifths than after seven octaves. In this case, 12×7 does not equal 7×12. This difference is the Pythagorean comma.

How come? As so often, the cause lies in unexpected exponential growth. In the post about the Pythagorean comma, I will explain how and why this gap occurs in the Pythagorean tonal system.

This is a post about the theory of the three worlds.