how music notation is mathematic in nature.

Whoa, I don't agree with that (taking your words literally, but think I get why you put it that way).

Is there a direct correlation to the popularity of the 12-tone musical scale and the historic rise of calculus?

As the answer you mention implies, 12TET is a way to have a small number of notes and still be able to have reasonably consonant intervals (compatible with previous musical practice). The 12 tone part was achieved without all semitones being equal, and all of that happened before the 17th century.

The Ancient Greek tuning system was not based on 12 tones per octave. The dominant tuning system in the so called middle ages, Pythagorean tuning, was also not based on 12 tones (you can get 12 tones from it, but they weren't into that). Both of those systems are of the family of just intonations (mentioned in the AskScience answer), and medieval theory was influenced by the Ancient Greek ideas.

Things started to get complicated around the 12th-13th century, with what is called musica ficta. What happened? People started wanting to use notes that were not in the hexachordal system that was built on Pythagorean tuning. This becomes particularly problematic because polyphony was booming.

The thing with Pythagorean tuning is that you can get as many notes as you want from it, but if you want to keep having a 2:1 octave and put all other intervals inside, you are going to run into some unuseable sounds (at least unuseable for their musical practice).

Music slowly became more and more "chromatic" (using the notes that would be the black keys on a modern piano), this means more intervals were used and more notes were needed. Why? Because thirds became fashionable intervals. When the Pythagorean system was used, fifths were the bees-knees, but thirds were not so great. Pythagorean was super nice for solo melodies, and worked fine for simple polyphony. But people started to use more thirds and sixths in their polyphony. That is not a new problem, the "I want my thirds to sound good" thing had happened before (see Claudios Ptolemy), but that discussion had been kind of forgotten long ago... Not to point fingers but, the English were rocking the boat a little more vigorously than the other fellows, and a certain Walter Odington started giving people some crazy ideas in the 14th century.

So, what happened? Well, people started with a normal Pythagorean tuning and started tweaking here and there. People were still talking about a Pythagorean tuning in the 16th century (ok, even later), but everybody was making modifications to it (different modifications, some more radical than others). What kind of modifications? Well, Pythagorean tuning is about having 3:2 fifths but if we just keep some of those fifths and change the size of the others, we can get some nicer thirds/sixths here and there without getting into too much trouble. This tweaking was called "temperament," from the latin "temperare." You managed to have semitones, but they were not all equal (you could end up having two or more types of some other interval, too).

An Italian guy called Pietro Aaron proposed the next big tuning system in the 16th century. It allowed you to have nice thirds, decent fifths (that were NOT 3:2), and triads (chords made of thirds) worked fine (by then, those had been a thing for a while). This tuning system allowed people to have 12 notes per octave and all was fine. Of course, not all intervals were useable and you had to be careful with which you used to avoid those nasty wolves. What happened? People kept tweaking here and there to get the results they wanted. People ran into problems: "our bloody instruments aren't compatible! We just can't play together!" (mentioned by Artusi and Bottrigari), they were talking about fretted instruments (the violin family didn't have this problem). Temperaments would probably not be a thing without instruments with fixed tuning: organs, lutes, and later clavichords and harpsichords...

People used to talk about making all the semitones equal to get this one tuning system to rule them all. But that is easier said than done. They didn't have proper mathematical tools to figure out that kind of thing and were either trying to find intervals by ear, using geometrical/mechanical methods to work tunings with complicated ratios they could not manage analytically, or just working with approximations.

The first known correct ratios for equal temperament come from China, and were done by a Ming prince. The Chinese didn't really need this kind of thing for their musical practice at that time so, yeah...

Vincenzo Galilei, Galileo's father AND an important figure in the birth of opera, proposed an approximation that is still some times used by guitar builders to set frets (semitones): 18:17. Marin Mersenne proposed a more complicated one, among a lot of really important things.

Nicola Vicentino just said "fuck this" and decided to go for an instrument with a hell of a lot keys to be able to cope with all the tuning problems.

Logarithms were the great mathematical breakthrough that helped people work out the problem.

People said they were using equal semitones (equal temperament), but it was not true.

Variations of Mean-tone were used until about the 19th century. Circulating temperaments, those in which you can use ALL the intervals (and keys) without running into horrible dissonance, were used from the late 17th to the 20th century. True Equal temperament is more of a 20th century thing. Even when you KNOW what ET should exactly be like, it's very difficult to tune it in acoustical instruments (it's more like a model than a universal tangible reality).

Sources:

• Barbour, J. M. - Tuning and temperament; a historical survey

• Reitman, Boris. - History of mathematical approaches to western music.

• The Cambridge history of Western music theory.

• Benson, D. - Music: a Mathematical Offering

If you want to get your hands dirty, the primary sources for this kind of thing are super interesting, but things get complicated quite quickly. There are translations for some very relevant writings.

TL;DR

No.