Tuning pitches to one another is wildly complicated. The rabbit hole goes unfathomably deeper than this introductory stuff on ratios and equal temperament. High

Tuning pitches to one another is wildly complicated. The rabbit hole goes unfathomably deeper than this introductory stuff on ratios and equal temperament. High schools physics introduces basic concepts in a way that works in certain situations and can be confirmed somewhat through experimentation. However, when you go deeper into any subject, fluid mechanics and material tolerances come to mind, those simple introductions cease to resemble the complexity of the systems. Music is no exception. I remember being surprised by the many ways this platonic explanation of pitches breaks down when confronted by real instruments in real air heard by real ears. Welcome to beautiful complexity of music! Example 1: Pianos are tuned with stretch tuning, where the high notes are tuned higher than their theoretical pitches, and vice versa for low notes. That's because of a thing called inharmonicity, where the overtones of short piano strings don't form perfect ratios with the fundamental. Maybe that has something to do with short strings vibrating more like metal cylinders than rods? Example 2: French horns have a lot of tubing! The valves let you include different lengths of tube into the air column. With a single valve combination, the player can hit the fundamental and all the partials up until their lips can no longer take it. To play a scale or melody, they switch between different horn lengths, choosing the partial that will sound closest to the pitch they're going for. Really, all brass instruments function this way, I just mention French horns because I like them. Example 3: String quartets tune their chords with each other on the fly. If you listen to a string quartet practice, they'll spend a good amount of time holding and adjusting intonation to each other. So unlike in the equal tempered world of keyboard tuning, strings and most instrumental ensembles are constantly adjusting to make their harmonies sound sweeter. Equal temperament doesn't come into play, unless you add a piano, which sort of forces everybody to uncomfortably conform to the limitations of set it and live with it keyboard tuning.