The higher the variety of notes (out of the overall 12 sounds in an octave) in the song, the less this becomes possible.

If your song is really simple, e.g. only consists of the 3 notes that make up a major triad (root, third, fifth), then this is definitely possible and you can just use natural thirds and natural fifths.

But as you start adding more notes, more chords and perhaps change of keys etc, it starts to break down.

That's the reason why J. S. Bach wrote The Well-Tempered Clavier. It's a collection of 24 preludes and fugues, in each possible major and minor key.

The basic idea was that if every prelude and fugue sounded good on an instrument (organ, harpsichord etc.), than it meant that the instrument was "well-tempered".

Using natural tuning instead of 12-TET would have resulted in some pieces sounding very good and other sounding very bad.

Yes, people try this. Check out dynamic tonality. It doesn't necessarily need a system. Experienced guitar players often find themselves unconsciously making little microtonal adjustments through bends and other techniques when playing leads. I found myself doing this just because it sounded better to me. I didn't even notice there was a consistent pattern until I eventually learned the math. For example I'd always want to bend minor thirds slightly sharp and bend the neck to slightly detune major thirds.

https://en.wikipedia.org/wiki/Dynamic_tonality

Sure, that's basically just intonation (JI). You pick what key you want to play in and a scale, and then you build an instrument around that scale.

(Though something that happens in just intonation is that you often find out you need more notes than you might have originally thought, because JI makes distinctions between notes that are treated as the same in 12-TET. For instance, you might have 10/9 or 9/8 as your major second, or your minor seventh might be 9/5, 16/9, 7/4, or 12/7 depending on context.)

I don't think any just intonation guitar has been mass produced, but you can definitely build one or modify an existing guitar if you have the right tools and are willing to do a bunch of math and learn how to install frets.

This page is about a JI keyboard I built a while back, but there's also a few pictures of a couple old Harmony guitars I adapted to JI: https://jsnow.bootlegether.net/jik/keyboard.html

Here's a so-so performance of myself playing a Bach piece on a newer and vastly improved version of that just intonation keyboard: https://www.youtube.com/watch?v=rqbWnDhip0A

In 12-EDO the song has 11 distinct pitch classes. (Bach used the tritone, but not the minor second.) In my straightforward JI interpretation, I use 15 pitch classes. (I would have used 16, but my keyboard simply doesn't have a key for that note.)

> Can't we have a system that is optimized for the notes that are actually played in a song rather than the hypothetical set? And what if the optimization is done per note rather than over an entire song?

You can. It’s called adaptive tuning, or dynamic just intonation, and it happens naturally for singers with no accompanying instruments.

It’s impractical on a real instrument, but there’s a commercial synthesiser implementation called hermode tuning.

You’re trading one problem for another, though. No matter how you do this, you will either have occasional mis-tuning or else your notes will drift.

Actually Bach's Well Tempered Clavier IS a book written in a single set of tuning system that actually lost/forgotten. We still have discussions about how it's constructed. For more information google "Well Tempered Clavier interpretation"

You can listen to variations here: https://youtu.be/kRui9apjWAY?t=622

It doesn’t work per-song. Songs have multiple chords, some even with alterations. If you tune an E so that it is perfectly a major third above C, then that E won’t be a perfect fifth above an A note. The Am chord has the notes A, C and E, so Am has notes that all belong to C major.

Additionally, some songs even change keys, which makes “per-song” not enough of a constraint.

That's how it works when you sing! But if you have an instrument you need to tune it would be annoying if you had to retune it between every song.

I highly recommend the book “How Equal Temperament Ruined Harmony (and Why You Should Care)” if you are interested in this subject.

You may be interested in:

https://en.wikipedia.org/wiki/Just_intonation

https://en.wikipedia.org/wiki/Comma_pump

Logic Pro has Hermode tuning, which does this per chord: https://support.apple.com/guide/logicpro/hermode-tuning-lgcp...

You can with instruments without fixed pitches, like human voice and string instruments, in fact choirs and string quartets do play this way, adjusting each note.

But for instruments with fixed pitches, like guitar or pianos,12 equal temperament is the best compromise to be able to play in all keys.

Kyle Gann's Arithmetic of Listening goes deeply into this. Given an infinite number of ways of dividing the range from f to 2f, some other equal-division temperaments (31 or 53, for example) get closer than 12TET to maintaining low-integer ratios across key centers, but each additional pitch adds complexity. I'd recommend that book in particular. https://www.kylegann.com/Gannbooks.html

I think you can only be "perfectly" in tune for a single mode so a multi-modal song would become very difficult to play?

Not really, because notes do double duty.

You might play a G# note in the context of an E chord (where it's the third), and then you might play it in the context of a C# (where it's the fifth).

These are discernably different pitches, but the same "note", in the same key, in the same song!