So, the problem was, what do you give up. You could tune the instrument so that melodies or harmonies played on one scale sounded good, but on some nearby scale they would sound weird or even horrible. Or did you tune so that things sounded somewhat bad on any scale. Throughout the 15-19the century this battle raged, producing a huge variety of tuning systems ( temperaments as they are called). Each tried to strike a different balance between having some of the keys (ie scales based on different base notes) sound good (ie the important intervals having whole number ratios) while other would sound weird.
Even by the 17th century it was known (published by Mersenne, a French musician and mathematician) that one could define a semi-tone so that exactly 12 fit into the octave. This would make the perfect fifth slightly sharp over the Pythagorean, the fourth slightly flat, and and the major third noticeably sharp in comparison with the just third, sharp enough to really bother musicians.
Bach's Well Tempered Claver was a series of pieces written for a harpsichord to show the range possible with a particular temperament which his friend devised ( called Well Tempering) This was not equal temperament. The purpose of these pieces was not, as many music teachers say, to show that each key sounded the same, but rather that each key in this well tempered system sounded different. This temperament tried to preserve the thirds in some of the keys, and the fifths in most, but as a result the thirds and fifths in some of the keys were decidedly out. Pieces written in the former keys tended to sound harmonious, giving a feeling of peace and harmony, while the pieces in the latter keys sounded weirder.
One of the musical tools available to the composers in the 17th to 19th centuries was precisely that the various keys, especially on keyboard instruments, sounded different. You got different moods depending on which keys you used. going to a different key was not simply a matter of starting higher or lower in pitch, the piece actually sounded different. A piece written in C felt different from one written say in C# ( a semi-tone higher).
Around the end of the 19th century, possibly driven by the mass manufacturing of musical instruments, and the unwillingness of people to re-tune their instruments (a constant feature of life before then), the equal temperament came into vogue, and has remained in vogue ever since. This temperament is the one in which all semi-tones are the same, a frequency ratio of about 1.059. In this temperament all keys are the same (except for the absolute pitch of the notes). There is no difference (except pitch) of a piece played in C or C# or F or B. The whole point of the Bach Well Tempered Clavier is lost. The additional emotional impact carried by the different relations between the notes in the different keys which composers relied on in the late 18th and early 19th centuries for music on keyboard instruments is lost. Thirds, again as in the early Middle Ages, have become almost dissonances, they are so far from the harmonious just thirds. Fifths are slightly less harmonious.
Of course, the development of the piano (as we shall see later) hid some of this loss. The different modes of the piano strings are not, as they are for the less stiff brass keys of a harpsichord or the forte-piano (the piano of Beethoven's time), in a nice harmonic relation. A piano note is naturally a harsher, and more dissonant note, all by itself. The frequencies of the modes are not harmonics of the fundamental (they almost are and the lowest modes have frequencies that are very close to being harmonics) and the ear, or rather the brain more or less presents them to us as a single pitch, but also notices that the relations are not harmonic and ``complains" by presenting the experience to us as harsher and more dissonant than for those other instruments. To some extent the harshness of the ``out of tune" fifths and thirds is hidden in the natural harshness of each individual note of the piano itself.
(Note that this is not to say the piano is worse or better- one may want that additional harshness, and may regard it as a feature of the music, rather than a detraction, but however one regards it, this difference is there).