It was mentioned on the YMMV page, but Andrew's fanhood of Buddy Rich is supremely ironic, considering the importance that Andrew (and Fletcher) place on being able to properly read sheet music: Buddy Rich was one of the greatest musicians of all time, and perhaps the greatest "big band" drummer in history. He also couldn't read a lick of sheet music. Thus, everything that he was capable of was organic, spontaneous, and free-form…precisely the opposite of Fletcher's rigid adherence to, well, rigidity.
Furthermore, some of the greatest musicians of all time (or some of the most popular, at least) couldn't read sheet music. Rich, Phil Collins, and even Paul McCartney couldn't read sheet music, for example. It's less about convincing Andrew that "good job" are the worst two words in the English language and more about Fletcher finding an opportunity to be a complete and total asshole with the victims buying into it.
Buddy Rich was also known to be a legendary jerk, alienating everyone he worked with (including being slapped by an irate Dusty Springfield on one occasion) despite his supreme talent. Sound like Andrew at all by the end of the film?
To that end, Rich also was a band leader cut from a very similar cloth as Terence Fletcher. He demanded perfection from his band and would get monstrous towards them if he felt like the band was not stepping up. There are an infamous series of recordings known as "the Bus Tapes", where Rich excoriates his band in tirades that sound eerily similar to Fletcher's.
However, Rich was also known for not being a very hard worker off the clock or practicing very much when he wasn't playing, believing it didn't hold much value and real experience and improvement only came from regular performances, the opposite of Fletcher's demand that students put music above everything else and literally work themselves bloody. Rich also claimed he didn't think of music in terms of who was the greatest, once famously claiming "there's probably a kid in Idaho no one's ever heard of who's better than all of us", putting him in opposition to both Fletcher and Andrew's views of music as a brutal, cutthroat competition and desire for greatness, no matter the cost.
One of Andrew's Jerk Jock cousins makes a surprisingly pertinent point in wondering how bands win competitions when "isn't it all subjective?" Andrew wants "greatness" but that is a fluid, totally subjective concept. What he really wants is Fletcher's approval, which he has at the end, at the cost of everything else. Victory over Fletcher would be realizing his opinion means nothing, and telling what to do with his damned tempo.
The titular piece is in 7/8, a time signature that is very unsettling due to its irregularity and 'incompleteness' (in the sense of it being ever so slightly off 'common time'). This is perhaps reflective not only of Fletcher's volatile character, but of Andrew striving to gain Fletcher's approval - he never seems to fully get it, although he gets incredibly close.
Fletcher is quite fond of the story about Jo Jones throwing a cymbal at Charlie Parker's head and Parker becoming determined to be better as a result. As detailed on the main page, the actual story is that Jones threw it at Parker's feet which is a small but notable difference that sums up Fletcher's personality. Throwing a cymbal at someone's feet may leave them shaken but ultimately doesn't hurt them while throwing one at their head could cause serious harm. Fletcher is someone who repeatedly shows he's completely indifferent to any damage left in his wake if there's even a slight possibility that it will benefit jazz music, up to casually physically assaulting students for minor mistakes.
The context of the incident is also left out. Parker was not playing the charts note for note but jumping off them, modulating into strange keys, and apparently at one point, losing his pitch. Jones throwing the cymbal at his foot was to alert him to leave the stage for this reason. It was not a personal attack and witnesses even described it as playful with Jones not intending to humiliate Parker. On the other hand, Fletcher repeatedly uses personal insults in order to further his musical agenda and his humiliation of his students is entirely intentional.
Similarly, Fletcher frames this as a pivotal moment when Parker committed himself to becoming the legend everyone would know. In reality, he didn't play again for months and even then, only for enjoyment and as an art, the polar opposite of Fletcher's intense rigidity and the "life or death" importance he places on music. Fletcher's own telling of the story reflects how he sees his training methods, as a harsh but necessary way to spark greatness, rather than an outlet for his anger and sadism. It shows how much Fletcher has deluded himself about the kind of person he is. As an added bonus, Parker was notorious for losing his sax or pawning them to feed his drug habit while losing equipment or sheet music is one of the many things that piss Fletcher off.
During the famous "Not my Tempo" moment, Andrew actually is on tempo, but Fletcher attacks him anyway. Showing that he isn't interested in quality music but psychologically breaking vulnerable people.
Fletcher repeatedly accuses his students of intentionally playing badly to sabotage him, which initially comes off as him just spouting off deliberately absurd accusations in order to humiliate them further - but considering the kind of ruthlessly competitive environment Fletcher tries to cultivate, and the fact that Fletcher himself later sabotages his own concert just to spite Andrew, its possible he really does believe his students would sabotage him on purpose, because in his mind that's a normal thing for a musician to do.
If it wasn't obvious already, the fact that Fletcher's treatment of Andrew is a trademark example of emotional abuse and that Andrew never gets out of the cycle.
Screenplay-only fridge horror: In the screenplay at one point Fletcher uses as support for his unique educational style an anecdote about a father enlisting his children for an educational experiment on whether exceptional ability in a particular arena (in that case, chess) could be created through rigorous instruction — a clear reference to the real-life Polgár family. Fletcher is also shown to have had a wife/girlfriend and young daughter from whom he now appears to be separated. In the case of the Polgár daughters, while their training was indeed rigorous it hardly seems to have been on a level with the hellish verbal abuse Fletcher lavishes on his students, but Fletcher clearly sees the two as acceptably equivalent. What kind of father was Fletcher?
It's never commented upon or explained, but none of Fletcher's band are women. It's not as if he seems obviously misogynistic, as he encourages a friend's daughter to be a musician, listens to the female players in Andrew's first class, and he has female players in his final show. Maybe he uses sexist insults as an easy tool, which drives out any female students who get into his band.
In the first informal-audition scene, he faux-affably asks a female student to play for him with a quip about seeing whether she got her position in the band just for being "cute". When she doesn't meet his standards, he dismisses her. Misogyny is definitely as much part of Fletcher's toolset as are homophobia, antisemitism, and even racism against the Irish. (Strangely, racism against his one African American student, credited as "Bassist", does not seem to be in his repertoire — that might be too much even for him, given that his chosen genre is jazz.) Fletcher is not just some frothing bigot — he's clearly calculating what he thinks he can get away with.
This could count as Fridge Brilliance depending on your opinion of Andrew at the end of the film, but his final performance may not have the effect he wanted. While technically masterful, it's also clearly presented as showboating. The jazz critics may be impressed with his skill, but would likely be turned off by a player who selfishly upstages his own bandmates in attempt to call attention to himself.
One has to wonder just how many students Fletcher has zeroed in on and tried to "mold" into "greatness" throughout his teaching career. The film itself only gives the example of a former student named Sean Casey, who killed himself possibly (and partially) because of Fletcher's abuse, but there may be others out there, and it's likely that, although they might not be dying by suicide, they're still deeply traumatized by what he put them through.
", where Rich excoriates his band in tirades that sound eerily similar to Fletcher's.