From what I've gathered, tuning in the Baroque period was more "free" in the sense that the reference pitch would vary considerably between performances. So, how did music historians choose the frequency of 415 Hz? (Or, if historians didn't arrive at this number, why is this number circulated so frequently?)
Even if there was a single standard tuning pitch back then, how would it be possible to measure it hundreds of years later?
Short answer: Old Tuning Forks.
Since the invention of the Tuning Fork in 1711 (late Baroque period) by trumpeter John Shore there has been an easy reference tool for checking and setting pitches. By accurately analysing the frequency given off by different tuning forks over time we can build up a pretty good idea of how tunings have changed.
In 1880 Alexander Ellis wrote an important essay on the history of musical pitch for the Society of Arts in London ^([1].) The essay contains a list of old historical tuning forks and their frequency value measured with accurate equipment. Here is an extract taking us up to the end of the Baroque period:
YEAR, PITCH, PLACE & SOURCE
c. 1715 A= 419.9, England. Crude tenor fork, possibly made by John Shore, the inventor of the tuning fork.
c.1740-1812 A= 424.1, Eutin, Germany. Tuning fork owned by Franz Anton von Weber, father of Carl Maria von Weber.
c. 1750 A= 424.3, London. "Common music shop fork."
1751 A=422.5, London. Handel's tuning fork. The box which contains the fork bears the inscription: "This pitchfork was the property of the Immortal Handel and left by him at the Foundling Hospital, when the Messiah was performed in 1751."
c.1754 A= 422.6, Lille, France. Tuning fork found in the workshop of M. Francois, musical instrument maker.
1754 A=415, Dresden. Fork used to tune the catholic church organ built by G. Silbermanmn.
1776 A= 414.4, Breslau. Marpurg's pitch for clavichord tuning.
As you can see from the above, during this period the pitch seemed to hover around 422Hz - so why adopt 415Hz?
(1) We only have tuning forks from 1715 but if we look at the whole story we can see that pitches in general have been rising through the C18th and C19th - a phenomenon known as "pitch inflation". So, it is reasonable to assume that prior to 1715 (early Baroque and Renaissance) they would have been lower than the A=419.9 listed.
(2) The 1764 measurement (A=415) used by Silberman for his 50 or so European organs is critical. If organs were tuned to this pitch it is reasonable to assume that other instruments playing with the organ would have to have been tuned this way. Similarly choirs would have sung at this pitch.
(3) Practicality: A=415 is a semitone down from the modern standard A=440 - so it is easier for modern players and instruments to adapt to this pitch than something that is outside of our current pitch system
Finally, anyone who has played on a Baroque instrument will confirm that at this lower pitch, instruments sound better. Gut strings give a better sound at this pitch and tend to be a bit screechy when tightened up to 440Hz. It is hard to imagine a musician of the time doing this.
Edit: Just to add that although A=440Hz is an Internationally agreed standard (It is standardized by the International Organization for Standardization as ISO 16 ), as far as I know 415Hz as a "Baroque Standard" is still a matter of consensus. Most orchestras that specialise in Baroque music have agreed amongst themselves to adopt this but I don't believe it has been defined anywhere as "official".