One can only conclude that the identity of a "piece of music" was a far more fluid concept for the Notre Dame cantors than it is for us. An organum as actually performed was essentially a patchwork created more or less on the spot, or after a brief consultation, from the many available parts in the manuscripts we have...He is talking about the fact that there are many different versions of the pieces in different manuscripts and also that each has a multitude of interchangeable sections. To a somewhat lesser extent, this is true of all music prior to the invention of recording technology and is still true of all live performances of music. [UPDATE: this overstates things. In the Notre Dame repertoire it is the actual text, the notes, that has multiple variants. In music since about 1750, the text is stable, but the variations are on the level of expression and interpretation--which I think is also very important.] Even with simple pieces, every time you sit down to play or sing them, there are real differences in the pacing, expression, dynamics, tone-color and so on. I regard this as one of the great strengths of music.
Compare to some other art forms: differences in lighting and environment aside, all paintings and sculptures are essentially the same over time. There are no 'performance' differences. But in the performing arts, such as theater and music, each performance is different. What strikes me is the analogue with persons. Every time you meet someone and have a conversation with them, there are differences. A different side of them appears--and a different side of you as well. Proust mused that we only really come to know ourselves as we are reflected in others. A piece of music only seems to come fully to 'life' in a performance before an audience.
How great are these differences? Take a famous piece like the Chaconne to the D minor violin partita by Bach. There a many different ways of playing this. Here are a few of them:
Each on a different instrument! All the same piece. But the 'same' in a way akin to the way a person is the same dressed for a formal occasion, or dressed to play soccer, or in an extrovert mood, or subdued and so on. Different facets of a piece are revealed in different performances by different people, even if on the same instruments, similar to the way different facets of a person emerge depending on the situation: at a party, in a job interview and so on. You can only stretch the analogy a little ways, of course. But it is easy to see why a piece of music is a bit like a person: because people are involved in at least three ways. First, the music is composed by a person and due to its unique fluidity and expressive qualities, some of the personhood of the composer is transmitted. Then it is performed by a different person who takes what is there and contributes yet another level of personhood with their own expression. Thirdly, there is an audience receiving this complex of expression and they receive it with their own personhood.
Music is a medium for the transmission of some aspects of being a person. It is the fluidity, the undefined aspects of music that lend it the malleability that is so important.
Now think about recording for a moment. I had to use recordings to give you examples, but I would have much preferred not to. If this blog post were a talk instead, to a live audience, I could have played the beginning of the chaconne in several different ways as an example. Myself on guitar, perhaps some friends on violin or piano. Then the point would be even stronger, because even when the same person plays the same piece on the same instrument, it is different on different days. Just as you yourself are different on different days, in a different mood, and this is amplified when you interact with another person, who is also in different moods on different days.
Recordings are fixed, transportable and saleable, which has made them so hugely important. But they eliminate the aspect of music that I have been talking about: the fluidity of music, the different-each-time-ness of music. A recording is music frozen. Interestingly enough, this aspect is least evident the first time you listen, but with each subsequent hearing of the same recording, the frozenness of it becomes more evident.
Even though I have avoided using the word, this post is really about the ontology of music...
UPDATE: I just remembered an excellent example of this. During Expo 86 in Vancouver, the great Croatian pianist Ivo Pogorelić was a guest artist performing, if I recall correctly, a Tchaikovsky piano concerto with the Vancouver Symphony. There were two evening performances scheduled and a review appeared in the morning paper after the first of them. Vancouver being just a bit of a musical backwater, but somewhat belligerent about it, the critic was careful to find some negatives to mention in the review. I wasn't actually there, a friend of mine, another music critic, related to me the story. In any case, in the first night's performance the critic tweaked Pogorelić for being too something: too subdued or too boistrous, too subtle or too crude. I don't recall which! But let's say, too subdued. Obviously he read the paper the next morning so for that evening's performance he played the piece in exactly the opposite way! This is what we often forget: a great artist can play a piece of music in fifty different ways...