Analogies and Interference

When I try to explain something to others, (and to myself as well) I make extensive use of images and analogies. There are a few reasons for taking this approach, with my lack of real mathematical expertice as a front-runner. But using analogies has some serious drawbacks: the listener might be tempted to understand the analogy, but miss the underlying abstract concept being described. Similarly I do run into issues where my mental model looks like it fully captures a concept, but put to stress it turns out to be just an analogy.

One important analogy I've been working with lately is the concept of interference, specifically the interaction between (sound) waves which leads to new, different patterns. These patterns can't be fully described from the originating waves themselves individually. For a simple pattern of two constant waves, the mathematics describing the higher level patterns is easy enough. But as soon there are many more changing waves, the mathematical description of the patterns quickly becomes too complex. The resulting patterns quickly become chaos.

But precisely these interferences lead to music. Although it's theoretically possible to describe a piece of music in a complete mathematical fashion, only using the underlying soundwaves as the input, there is no musician who uses such tools. (Computers did slightly alter this statement, I know) No, for musicians a language on a more abstract level has been defined to describe music, based on specific notes, octaves, etc.

In some, hard to understand manner, the interferences between separate entities (waves) has lead to higher level structure (music). I've been using this analogy to describe the concept of emergence of complex structures from separate entities (or agents). To me it works to explain how complexity can exist. This leads me to using statements like: "The cell is the result of interference between molecules, which are nothing more then an interference between atoms".

I'm not sure yet if this analogy usage is actually helping in explaining (and qualitative proving) certain ideas about the world outside. As I'm not equiped to mathematically proof these statements and ideas, this is probably the best approach available to me. If any of you has found a better way, please tell me:)