Ancient Greece had only four musical instruments that got much air play:  the aulos, which was a V-shaped double flute, pan pipes, the kithara and the lyre.  MP3 players were thin on the ground back then, and music notation was primitive and not much written, so we have almost no idea what the music sounded like.  (Though there've been attempts to reconstruct based on the few surviving lines of notation).

But weirdly, we have a very precise knowledge of how the instruments were tuned.

That's because one day, a lad by the name of Pythagoras noticed a few things about how the kithara and lyre players tuned their strings.  This is the same Pythagoras who came up with the theorem you learned in school about triangles, so it will come as no surprise that he did a minute analysis.

What Pythagoras noticed was firstly, that a string twice as long as another produced the same note but one octave down.  Ditto that a string exactly half the length was precisely one octave up.  This is a piece of pure physics that anyone could spot.

The second thing he noticed, and this was hugely important, was that musicians consistently went for a note in between the octave ends that divided the string in the ratio 3:2.

Musicians will be nodding their heads knowingly, because these days we'd call that ratio a perfect fifth.  Ancient Greek music was based entirely on perfect fifth intervals.  What's more, the interval from the perfect fifth to the next octave up is what we'd call a fourth.  We almost have enough to play the blues.

With no electronic tuners and no way to delicately adjust string tension, probably the best they could manage was to get every string to the same tension and vary the lengths.

So here's how they tuned:  start with a note X, with a string that I'll call a length of 1.  (X has some arbitrary frequency that the musician's picked by ear.)

Now the next octave up is a string of length 2:1 compared to the first string.

The perfect fifth between those octave notes is at 3:2.   That's the first perfect fifth.

Remember in this system, we get each successive note in the scale by going up a perfect fifth from the last.  So to get the perfect fifth up from one at 3/2, we have to multiply the length again by 3/2.  That gives us 9/4. The only problem is, 9/4 is more than 2:1.  We've fallen off the end of the scale!

Not to worry, just drop that note down an octave.  Which we do by halving the string.  That note becomes 9:8.

Our scale now has notes at:  1, 9:8,  3:2, and 2:1

We now go up a perfect fifth from the 9:8 note.  Which we do by multiplying it once again by 3/2.  That gives us a note at 27:16.  Our scale now has notes:  1, 9:8,  3:2, 27:16, 2:1

I won't leave you in suspense.  Here's the final Pythagorean Scale:

1:1   9:8   81:64   4:3   3:2   27:16   243:128   2:1

It's doubtful that Pythagoras invented this.  It's much more likely that he formalized a system that was already in place.  But I'd be willing to bet anything he was the first to work out the ratios.  Musicians were probably tuning up perfect fifths by ear.

You could, in theory, continue adding notes forever, but Pythagoras stopped at 8, presumably because of the practical difficulty of adding more strings.

Pythagorean Tuning was unbelievably successful.  The Greeks used it.  The Romans used it.  It survived to be used in mediaeval times.  In fact it survived until another genius by the name of Johann Sebastian Bach finally killed it off when he wrote a work called The Well Tempered Klavier in the 1700s.

Believe it or not, there are still a few instruments that use Pythagorean Tuning.  This isn't ancient Greek music, but it's using their scale: