@mrlyonsmaths Another interesting thing: the ear tends to like very musical intervals whose frequencies are in simple rational relationships (just intonation), which often sound "consonant." Since we can compose two intervals by stacking them on top of one another to form another one, the resulting structure is isomorphic to the multiplicative group of strictly positive rationals.

But what if we want the most "dissonant" intervals as well? Then in some sense we want to add "maximally irrational" numbers. Which numbers are these?

Well, there are a few ways to do it, but one reasonable candidate for this is to look at those numbers whose continued fraction ends in a tail of all 1's (noble numbers). These have a few number-theoretic properties that give them a pretty solid claim to being the "most irrational" numbers. Also, in a musical setting, these tend to be pretty close to the perceptual "most dissonant" interval point in between two neighboring consonant intervals (given a few extra criteria I'm handwaving here), at least approximately as measured by models such as harmonic entropy. So, we may just say they're good enough as a basic idealized model to get started.

The resulting structure we get is the set of positive elements from the quadratic number field Q[√5]. It makes perfect sense, but I still think it's somewhat magical and bizarre. You want the most consonant intervals and you get Q, and if you want to add in the most dissonant as well you get Q[√5].