Need help understanding the difference between diatonic and chromatic semitones?

Question

I am writing an essay on musical temperaments for which I was doing something extensive research on the history of tuning and how it evolved over time. I understand the Pythagorean tuning system quite well now but I am confused about a few things and would love an explanation for them. \n\n1) What is the difference between a diatonic and chromatic semitone in reference to Pythagorean tuning? \n\n2) Why were these semitones used differently and notated differently? \n\n3)Did they use it in one chromatic scale itself like we do on the keyboard instruments today or were diatonic and chromatic tunings entirely different and never combined into one scale? So for example you could either have a C to C# tuned as a limma or tuned as a apotome but not together on a fixed pitch instrument?\n\n4) During the time of Pythagoras and several centuries after him, did they only use the diatonic and pentatonic scales? Didn't they use smaller intervals back then?

Answer 1

The difference between a diatonic and a chromatic semitone is the spelling of the notes. A semitone, as you probably already know, is a fancy way for half-step. Each note is a half-step interval apart. For example; C to C# would be diatonic; but C to Db (the same note) would be chromatic. The most natural diatonic notes within a scale are always mi to fa and ti to do. Chromatic semitone would be a secondary dominant chord modulating. These semitones were thought the same way but written differently. Music was used to worship the gods and really nothing else. They used modes not scales. Theoretically, they know about the chromatic scale mathematically, but never used it musically.