### Ratios (WARNING: MAY CONTAIN MATH CONTENT)

#### by Jungreis

Earlier in the term, we discussed how the style of tuning used in Bach’s time was based on ratios from Pythagoras. Pythagoras did indeed like ratios. He believed that every number could be written as a ratio. He was wrong. The canonical example is the square root of 2. There is a simple proof of this, and anyone interested in seeing the proof should construct it, or you can let Vi Hart explain.

Back to music…

It makes no sense to me that musical tuning should be based on the incorrect theory of Pythagoras. Yes, there was that demonstration in class where an interval in Pythagorean tuning gave the pretty picture while the interval played in modern tuning gave the fuzzy image. What’s wrong with the fuzzy image? It sounded fine to me; it sounded like music (or at least the beginning of music).

The modern style of tuning ruins the idea of basing intervals on ratios. That mathematical idea is replaced by something much more beautiful in my mind: symmetry. The circle of fifths is a cyclic group! Starting on any tone, I can ascend by any interval and eventually arrive back at the original tone. (There will be an octave or multi-octave difference. There is a mathematical way of dealing with this, and I can get into that with anyone who is interested.)

Does anyone support Pythagorean tuning over equal-tempered tuning? Why? Keep in mind that Pythagoras was wrong about ratios! Most numbers can’t be written as ratios! (It is mathematically sound to say that almost all numbers are unable to be written as ratios of integers.)

We are now used to hearing the modern tuning, but I do think that something good might have come from Pythagorean tuning. Sure, the Circle of Fifths is a wonderful thing to have as a composer, but what if something neater and cleaner sounding could have come from Pythagorean tuning that would make our Circle of Fifths sound useless?

I think it’s evident that the circle of fifths isn’t useless.

Symmetry is pretty.