More and more I get the impression that the pythagorean tuning and Pythagoras have not so much to do with each other.
One can easily think that this tuning was discovered or invented by him. No, it is definitely not so.

Pythagoras discovered, most sources agree, above all the constant relationship between the length of the strings of a lyre and the basic chords of music (1:2 for the octave, 2:3 for the fifth and 3:4 for the fourth). These numerical ratios are also fundamental to pure tuning.
The tuning system with the Pythagorean circle of fifths does not originate from Pythagoras. The sources are scanty, it is not clear to me where the circle of fifths comes from.

In the Middle Ages, this tuning was the generally valid and used tuning. At the beginning of the 16th century, in addition to the octave and fifth, the major third was also intoned purely in chordal combinations, and for keyboard instruments pythagorean temperament was increasingly replaced by meantone temperament.

Pythagoras in the forge is an ancient legend describing how Pythagoras discovered in a forge the euphony of hammers sounding together, the weights of which were in certain integer ratios. This observation had formed the starting point for experiments, which became the basis for the music-theoretical description of intervals. With the knowledge gained in this way, Pythagoras had founded the theory of music. The legend had the consequence that Pythagoras was called the inventor of „music“ in the Roman Empire and in the Middle Ages, by which music theory was meant.