[veli_joza]

I've been down that rabbit hole. Math and music aren't really in such harmony as advertised. Yeah, equal tuning gives your instrument ability to play in any key, but it will always be slightly off.

For more enjoyable tuning, your frequency ratios should actually be fractions of small integers. For example, note E to note C ratio should be 5/4. This is called "just intonation", you can hear some examples on youtube when compared to equal temperament described in article. It sounds much better to trained ear, but doesn't work for changing keys.

It would be nice for your digital instrument to be aware of key you are in (much harder than it sounds) and to re-tune all notes into just intonation. This would give you best of both tunings.

[TheOtherHobbes]

It's easy to set up just intonation digitally.

The problem is that most classical music modulates to other keys. So why not just set up some switches or programmed changes?

Because as you modulate there's a grey area in which you're not fully in one key or the other. If you interpolate the intervals as you go through this area, it sounds wrong. If you switch to a new tuning when you land in the new key, that sounds wrong too.

Equal temperament solves the problem by being a good-enough compromise. All the intervals are slightly off, but they're off by a consistent amount, so - paradoxically - key changes become smoother.

[isolli]

Note that violinists (and cellists, and others) are not constrained by their instrument and can play in different temperaments depending on the situation.

[derriz]

Temperament is unrelated/orthogonal to inharmonicity. Temperament deals with picking frequencies for the notes that make up the scale within an octave.

Inharmonicity affects how you tune the octaves themselves. The reason a simple doubling rule doesn't work is a result of the properties of real physical strings which are not perfectly elastic causing the harmonics of a single string vibrating to be slightly sharp. To avoid "beating", the ratio of frequencies between two piano notes an octave apart needs to be slightly greater than 2.

This article is half of what a "what every musician should know about piano tuning" article should contain - it's the "set-up" part where you derive a nice simple rule that is widely known. The second part would then deal with temperament (first) and then inharmonicity.