There is a lot of bad information floating around out there.
The age-old myth about Pythagoras and inventing the circle of 5ths comes from a old story about him hearing blacksmith’s hammers producing harmonic frequency ratios in proportion to their masses, which is wrong, because hammer heads vibrate irregularly, unlike strings which vibrate regularly( at integer ratios). He may discovered the phenomenon with strings, however.
The thing to note about Pythagoras is that the earliest mention of him is by Cicero over 500 years after Pythagoras’ death. That would be like me being the first authoritative source on something from the early 1500’s, and being cited for it in the year 4017.
The origins of the diatonic scale stretch back into prehistory. Most cultures’s music incorporate something like it or the pentatonic scale. These distance between the steps in these scales come directly from the harmonic series, and it is very natural if you are inventing music for the first time to sing something like one of these scales. Some animals (bird songs for example) will make noises at intervals of octaves 5ths or major 3rds.
The diatonic scale existed without the other 5 notes or any other notion of keys up all through ancient Greece and Rome and all through the Medieval period. Most of their music had simple simple harmony and they either sang or played solo, or when they made music together they sang in unison or octaves, and eventually 5ths.
But as somewhere after the 11th century when music notation began to take shape it became possible for people to play and sing in large groups and to have different parts with complex harmonies and counterpoint. Instrumental groups had to tune their instruments to one another and singers had defined vocal ranges. As they started building pipe organs pitches became quasi fixed. Until the Renaissance transposing music into a different key had the singular function of fitting to a singer’s vocal range.
With no notion of accidentals yet, musicians used modes instead of keys for tonal-like effects. There was an authentic mode and a plagal mode for C through A (no locrian or hypolocrian because tritone) which determined whether you ended on I or IV.
Eventually near the beginning of the renaissance some clever folks figured our that you could transpose these modes to star on any note, if you just added half steps in between the existing 7 notes. There’s a lovly treatise on it called Dodecachordon written in 1547 by Heinrich Glarean id you feel like reading a butt-load of Latin. Eventually they learned that playing a melody in a mode, and then playing that melody on the note a 5th above the original mode’s tonic sounded good and strong so they called it modulating to the dominant key, which was now possible because of the addition of those 5 other notes.
The actual circle of 5ths didn’t show up until the 1670s in a treatise called Grammatika by Nikolai Diletskii and from here on music theory begins to take on familiar form, although things like functional harmony don’t show up like we know them today, until the 18th and 19th centuries with people like Jean-Philippe Rameau and later Hugo Riemann.
Until the mid 19th century all tuning of all instruments in practice was done by ear. There was no objective was to measure pitch until after the industrial revolution. The cent is directly related to the system of equal temperement and is a 1/100th of a perfectly equally tempered semitone. Equal temperment is a compromise in tuning that makes it possible to play in all the keys without re-tuning keyboard instrument. Cents do not exist until the 1830s.
There is a great video on early music temperaments here by Early Music Sources: https://www.youtube.com/watch?v=R75unSXKJXQ
This is the type of instrument they were building to get around the comma issue. https://www.youtube.com/watch?v=0akGtDPVRxk
On this keyboard the circle of 5ths wraps around 3 times before coming back to the original note. But the 3rds are more in tune in all keys(and you have 8 different types of 3rds including major and minor).
Maybe I’ll make some videos on all of this some day.