You are right that there is an unavoidable mathematical compromise. A pure 5th would be a ratio of 3:2 or 1.5:1. But in equal temperament, semitones are 2^(1/12

You are right that there is an unavoidable mathematical compromise. A pure 5th would be a ratio of 3:2 or 1.5:1. But in equal temperament, semitones are 2^(1/12), so a 5th is 2^(7/12), which is about 1.4983:1, so just slightly off from 1.5:1. This issue applies to all instruments. BUT, while all of that is true, it isn't the reason a piano is tuned how it is. It's because the strings don't behave like the mathematical ideal of a string. The harmonics aren't at exact integer multiples of the fundamental frequency. My source for this is asking a piano technician a bunch of annoying questions while he tuned my piano. He helpfully responded to by explaining the reasons behind stretch tuning, and then showing me the process of using his Sanderson Accu-Tuner to measure the inharmonicity of my piano and compute an individualized stretch tuning for it. Another source: https://en.wikipedia.org/wiki/Inharmonicity#Inharmonicity_leads_to_stretched_tuning https://en.wikipedia.org/wiki/Inharmonicity#Inharmonicity_le...