Well, though I'm not a medievalist and medieval music is not my job, I’ve done some work in early music and some of the same questions apply. I will try to be useful.

Generally speaking, the further back in time you travel through music manuscripts, the less about the music is notated. All the small comments disappear ( like Satie's "Play this in the shape of a pear") then all indications for instruments ( like up or down bowing for violins) then all the dynamic and tempo markings ( like Presto), as well as all the marks for ornaments, then the indications for instrumentation, then even indications of meter. Back in the 9th c., notation could just be squiggly lines over the text to indicate the pitch rising, falling, turning- but no definite indication of notes. Look at a piece by Vivaldi, and you know a lot about what you're supposed to do, what instrument you should be playing, etc. New editions of transcribed Troubador and Minnesanger songs just give you blobs on a staff. There is no “codex”, as you put it, stating how things had to be played. There’s much more known about later music of the 15th c. than there is about that of the 12th. But generally speaking, the further back you go, the more choices you have to make.

This is somewhat parallel to regular history: further back you go, the fewer the records, the more detective work is needed. Scholars need to get more familiar with, think real hard about, their fewer sources, and so tend to be a special bunch with special knowledge. I like reading it, but I would never dare do it. The publishing industry known as David McCullough is quite happy to generate books on a lot of American history but he hasn’t set foot in the medieval world, and that’s wise. Barbara Tuchman tried it once and got little love among medieval scholars for A Distant Mirror. And medieval scholars themselves have to specialize. The period is almost a thousand years, there are a lot of languages and cultures. Someone who knows a lot about 13th c. France might not want to say anything about 11th c. Wales. And as Tuchman herself confessed, it can seem an odd place, with people saying and doing unfamiliar things. So, when you ask how these people performing medieval music can know how to play it, you could also ask questions about what’s known of medieval history, generally. And the answer to those questions are, very often, going to be hypothetical.

There are medieval writings on music, but mostly on theory. Though more on performance would be nice, just Knowing how they tended to think about music is useful for arranging, improvising a part. Some music is easier than others. Most all medieval music surviving in manuscripts is vocal, which means it’s got words below the notes. Those words give very important clues as to the meter, the rhythm…even the tempo. And sometimes you know something about the situation of the composer, and who’s singing it, and that helps. Hildegard was writing songs for nuns, was already singing the offices of the mass with them, and so we can plausibly say that her songs had some of the same quality as Gregorian chant. If a good women’s choir – like Sequentia, or Anonymous 4- sings them as Hildegard notated them we could say it’s likely to be close.

Instrumental music is far harder. There are mountains of sacred vocal, In Nomine and Agnus Dei vocal pieces, written for the mass, but just a handful of instrumental dances. For the modern performer of medieval music, that’s a real problem, because instrumental secular dance music is what’s most popular for audiences, now, and audiences want to hear instrumental music. But vocal pieces can provide examples for arrangements of instrumental ones- like possible harmonies, and ornamental flourishes. And some songs have catchy-enough tunes in their own right to work as dances or melodies- lots of the Cantigas de Santa Maria hare been used this way, as instrumentals.

Almost no medieval instruments survive, to give us an idea of what they sounded like. A big resource here is the iconography. Paintings of angel bands serenading the Virgin and Child, carvings of musicians in corners of cathedrals, doodles of musicians in corners of manuscripts…they give an idea of the construction of instruments, but also how they were played and what combinations- like, sackbut and shawms together. But a very important development in the past hundred years of the revival of medieval music performance is the acknowledgement that most medieval musicians worked by ear and with memory, with hand gestures and not with a piece of sheet music in front of them, telling them what to play. Musicians play differently, when they are working by ear- they listen differently to others around them, react better, improvise. In other words, more like a bluegrass jam, less like a symphony orchestra. There’s also more been more interest, more acknowledgement, of eastern influences. Modern western music has tended to add stress and drama with more and more complex harmonies ( like those 8-note jazz chords) Arabic and Persian music adds stress and drama with microtones and ornaments, not harmony, and earlier medieval music likely operated more in the same way.

Though performance of the stuff involves educated guessing, that’s not entirely a bad thing. As Nicholas Kenyon shrewdly observed, every time a scholar puts forward a new interpretation, it tends to fit in with the current taste in music generally; and the lack of medieval sources frees up those possibilities of interpretation. French court music of the late 17th. c. is a bit mannered for modern ears, but there are very good period sources for performance and they quite limit how it can be played , so there's not much choice, there. On the other hand, Stella Splendens, from the 14th c. Catalan Llibre Vermell, is just melody, in two parts. It feels like a two-part pilgrim song, sounds like a two-part pilgrim song, and mostly it is performed at a walking pace. But that’s still just a feeling…you can also perform Stella Splendens very freely, without the marching feeling, and it gets very expressive and is still quite nice. And why not? We know that medieval music could have lots of expression ( all those troubadour and minnesinger songs) . But then, in the other direction, some have suggested it's a flagellant song, and that works, too. It's the 14th. c, plague hits, people begin to march in processions, beating themselves with whips. You can play it with that in mind and the tenth beat does, actually, seem as though it's the right moment for a blow to the back. But that's not more than a hypothesis. Any of these ways is right. Better to play it, than not play it.

And in that regard, though there is plenty of medieval music performed by historically-minded ensembles, there are others who just like it because it’s cool. The German medieval band Corvus Corax seem more inspired by Goth fashion and 19th. c. Romantic paintings of barbarian hordes. They claim to have some justification for their musical interpretations. But whether it’s historically informed performance is perhaps beside the point, because piles of bagpipes and drums onstage are, really, just fun to hear.

TL:DR We don't know exactly, but we have some useful indications.

Nicholas Kenyon: Authenticity and Early Music