Photo: Steve Taschuk (Flickr)
Iamblichus is known for having written books about the Pythagoreans–a mystical sect of early Greek philosophers who investigated numbers. They were very secretive, and without him we might not know how much they actually accomplished.
According to Iamblichus, the Pythagoreans noticed that the hammers being used by blacksmiths in town made a ringing sound when they were struck.
If the weight of two hammers bore a simple numerical relationship to each other–such as, say, one being twice as heavy, or three quarters as heavy, as another–their sounds when struck were in harmony. If no such relationship existed, the sounds they made were not harmonious.
So in modern terms we would say that musical tones are vibrations in the air, and that basic numerical relationships between two vibrations cause them to sound either harmonious or dissonant.