What is the relationship between a spell's incantation and effect?

Question

This requires a little bit of context, so please bear with me.

Ron Weasley: "Wingardium Leviosa!"
Hermione: "You're saying it wrong. It's Wing-gar-dium Levi-o-sa, make the 'gar' nice and long."

— Hermione Granger being condescending toward Ronald Weasley

In linguistics, one of the first things you're taught is that with a few exceptions (e.g. onomatopoeia, sound symbolism), the mapping between words and their meanings is completely arbitrary. To quote Shakespeare, "What's in a name? that which we call a rose by any other name would smell as sweet;". Nothing about the rose changes no matter what sequence of sounds you use to refer to it, and inversely, nothing about the sound sequence "r", "o" and "z" will tell you anything about the flower. Furthermore, contrary to popular belief, the specific signs of sign languages used by the deaf are just as arbitrary as spoken and written languages in their mapping between sound and meaning. In Harry Potter, however, the form taken on by spells does in fact seem to depend on the specific sounds of the incantation (as well as wand/hand movements). While spells do tend to have an etymology somewhat related to their function in other languages, Latin and Greek are no less arbitrary than English or any other language. What about wizards in other parts of the world who speak languages not rooted in a classical occidental language? Do Chinese wizards use the same incantations as English wizards? Superficially, it would make sense if the incantations and wand movements used are just traditional, the important part being a mental association between these actions and the effect that results. If this were the case, a young wizard could be taught to cast a levitation charm with the incantation Avada Kedavra, or Lumos, or Banana, or anything else.

"Lapses in concentration while charming can result in painful side effects — remember Wizard Baruffio, who said 's' instead of 'f' and found himself lying on the floor with a buffalo on his chest."

—The Standard Book of Spells, Grade 1

So what is it, the pronunciation or the concentration? Do the sounds spoken somehow influence or shape the magical "field" that is emitted? What about non-verbal spells? Has the wizard gotten skilled enough to mentally emulate whatever effect the sounds have?

One guess I have is the use of incantations as a means of "communicating" with a wand, which is shown in the last book to be something comparable to conscious and intelligent. Even then though, if the incantation were being used as sort of a code-word or instruction to the wand, whether wands come "preprogrammed" or learn the incantation concurrently with the wizard, this should be just as arbitrary unless specific sounds or thoughts of sounds are what create different types of magical effects.

Examples abound, the most relevant in this case is Sectumsempra. How is it possible that Harry could cast this spell correctly without knowing what it does, just from knowing the sound and wand movements involved? I vaguely recall something about wand movements next to it. The shape of a knife is not arbitrary, and used against someone will cause very specific, obvious effects. It's possible that the wand movements somehow directed the magic into a cutting "shape", but the incantation was still necessary, and it seems unlikely that if it were not arbitrary, Harry would have had to have had at least some idea as to the spell's effect.

The time Harry learned "Episkey" from Tonks brings to mind another possibility. I can't remember the exact quote, but he mentions that it's similar enough to other spells he already knows, and was later able to use it on one of the Gryffindor chasers. It seems that witnessing the spell was enough to learn it, but only because Harry had sufficient background knowledge. Possibly relevant,

"J. K. Rowling writes in Harry Potter and the Half-Blood Prince that Harry's knowledge tells him this spell could belong to a variety of healing spells, in the same way a species of plants belongs to a larger genus."

One thing from the books that seems very clear is that we know precious little about how magic actually works. It could very well be that there is a great deal more going on underneath the surface than what we can glean about the science of magic, but working with what is known, I've come up short trying to find an internally consistent hypothesis concerning the form-function relationship of incantations and spells.

Answer 1

While the HP universe doesn't go very deep into the "metaphysics of magic", there is a layer of consistancy in the way that magic works. Specifically, magic involves what you think and what you do, across the board. Wand movements act as physical focus points for the 'magical energy', while the incantation acts as a mental focus point. We see spells in both English & Latin, but never hear what the Beaubaxons, the Durmstrangs, or those from the Salem school, use for their spells. They may be the same, they may be different, and either way, the incantations still carry the same purpose - focusing the mind of someone who has the magical ability.

• When HP enters the Floo Network, thinks about going to Diagon Alley, but says "diagonally", He shows up in a fireplace in Knockturn Alley, which is diagonal from Diagon Alley.
• When Crouch-as-Moody is teaching the Defense vs Dark Arts, he mentions that the students "get your wands out and point them at me and say the words, and I doubt I'd get so much as a nosebleed" in reference to 'Avada Kedavra'... because the will to kill isn't there; Similarly, when the Moodyganger is showed using the Cruciatus curse on the spider later in the book (or when HP uses it on a Death eater in the final book) it is both through the words and a torturous intent.
• Also in Goblet o'Fire, when HP uses Accio Firebolt, it does not bring any nearby Firebolt, nor does it bring every Firebolt nearby, but specifically Harry's personal Firebolt that he was concentrating on.
• When HP uses Sectumsempra, he knows the exteremly simple jist of the spell - It's for Enemies. Being able to concentrate, using the wand as a focus, HP directs the spell towards his enemy, not knowing what will happen. It's the magical version of "Stick 'em with the pointy end".
• The closest we get to magic metaphysics is the discussion surrounding the Patronus charm. Lupin says directly, "With an incantation, which will work only if you are concentrating, with all your might, on a single, very happy memory". There are direct discussion referencing how it has to do with mental state, showing HP eventually succumbing after a dementor assault; it also may have something to do with inner morality, as the only Death Eater to show the use of one is Snape. The incantation for the patronas may be Incorporeal or corporeal, but it has to do with the mental state, not the actual incantation.

We also see that the incantations don't need to be spoken, which is shown through both Dumbledore, Snape and Voldemort using silent magic. The Trio also does this in the movies, with Stupefy, but I cannot recall them using silent magic in the books offhand. Dumbledore, Lily, & Snape are shown to use it and most cases of underage magic happen due to wandless magic. On one hand, those with training and discipline can direct the "magic energies" without needing a verbal or material focus; while those with the magical gift can do "things" without any control whatsoever.

All of that to say... there is a relationship between effects and incantations. There also may not be one. It very well may be like the transferrence of ideas via text - There are things that you cannot possibly know or experience, that you could learn once you understand a language. Once you can understand a language, you are no longer limited to just your own experiences, but to those who have written in that language. Later on, once you really internalize it, you can start thinking in that language for some or all things. Someone can sing "Voulez-vous coucher avec moi?", but not understand what they're saying. Later they can learn to read French, so they understand what they sung, but not converse in French in an useful manner, or without a dictionary in hand to interpret. Later still they can learn the full language, can think and converse in that language, understand completely foreign viewpoints and experiences.

Answer 2

If you don't speak clearly when asking Siri to look something up, you get unintended affects.

I don't think wizards taught in drumstrang or beaubatons use the same incantations as those from Hogwarts for exactly the reasons you said, but a sloppy incantation will give you sloppy results just like sloppy questions for Siri will get you sloppy responses.