This is a sequel to this post on Lerdahl’s GTTM.
Okay…apologies for the delay…I was busy, but also was uncretain whether I understood the material myself, that stopped me from saying more. Additional disclaimer: I’ve tried my best to pull out all the melody and counterpoint related content of the theory to leave things chordal. I’ve savaged the original theory in the process. Apologies to all affected by my act of gross violence.
Okay, so what is a prolongation tree? They look like this:
So basically it looks like a time-span reduction with three different types of vertices.
The lines represent chords, and the coloured vertices1 between them tell you roughly what the type of the progression is.
An aquamarine one means that there is no change between the chords represented by the two branches essentially repeats itself. This is called a strong prolongation. Given two chords that are the same repeated, the initial one is the dominant one, so aqua vertices almost always branch to the right (except sometimes in the case of up-beats and the like).
In the first diagram, we can see that chords 1 and 3 must be the same.
A blue vertex represents a chord progression where there’s only a little difference between the two, say a change in inversion, possibly the addition of a note to the chord2. This is called a weak prolongation.
In the diagram, we can see that chords 1 and 2 must be only slight modifications of eachother.
All other forms of chord progressions are just given boring old black vertex, and called progressions.
Interpreting these in terms of relaxion and tension-building, we have the following diagram
Well-formedness and Suggested Shapes
What sort of trees are allowed?
Firstly the trees must be planar, so shapes like the following are not allowed:
Normative Prolongational Structures
Lerdahl considers the following pattern to be more or less a good ‘top level’ for trees, and calls it his prolongation basic form (with the suggested chords at the bottom):
Note that this doesn’t mean that there’s a I-V-I progression in the piece necessarily, it means that at the highest level or oranisation, The piece is oriented around the sequence of chords I-V-I.
Lots of pieces he analyses have strong prolongations at the top vertex instead of weak ones: you can take your pick I guess.
Two slightly more elaborate forms he gives that might serve as useful skletons are:
Either of these can be called the normative prolongational structure. The first has a minimal tensing-relaxing pattern, the second has a repetition of the opening)
Before I describe his two other guidelines4, I should say that the context in which I’m presenting them is the one where you are generating a prolongation tree from the top down, and trying to decide what additional branches to add.
For instance, say I’ve decided already on the prolongation tree below in back, and am trying to decide where I should attach a new branch, c.
The green lines represent allowable attachments, the red lines ones that are simply not (because I’m trying to add branches in a hierarchical manner, I’ve already essentially decided that either b or d must dominate c).
The Balance Constraint
If you’re adding brances that are framed3 by a weak prolongation, prefer to have the same number attached to each branch.
The Recursion Constraint
This is a rule advocating alternate left/right branching of progressions as opposed to repeated branchings in the same direction, though allowing for repeated branching in the same direction provided progressions alternate with prolongations.
In the following diagram, green branches are good, red are bad.
Here’s an illustration from the book
Say I am looking to add a branch at c to the following
and to attach it like this:
The recursion constraint says that if the chord I put at c is the same as the one at a then I must rearrange the tree and put in an explicit strong prolongation:
This is the only way that adding a new branch is allowed to interfere with the existing branches of the tree. Though it’s called the recursion constraint, it isn’t to be applied recursively.
I guess, if you were trying to add chords to given tree (which is the application I have in my mind here: a program generates a tree, then generates a chord sequences from that tree, then generates some music with that chord sequence), you want to avoid this happening.
It’s not necessary (or necessarily advisable) that one construct a single tree encompassing an entire piece, one could have a group of them, one for each phrase, or section. If your pieces doesn’t have harmonic changes every beat, however, a single tree could suffice for, say, a 16-bar section.
1The blue vertices should be notated with black circles, and the aquamarine ones with similarly-sized black dots, but I can’t easily draw them at the moment for some stupid reason.
2In the original a weak prolongation is used when the chord stays the same but either the melody or the bass change a note. It might be worth testing my suggestion though.
3Note the word ‘framed’, the balance constraint doesn’t say anything about branches that aren’t directly enclosed by the prolongation.
4These are not strict rules, but I think for a basic generative model it’s appropriate to enforce them strictly.
Okay, I’m tired again. Have to stop. I’ve hope given enough, and in unambiguous enough a manner, that you should at be able to generate grammatically correct prolongation trees (there’s still going to be a lot of choice/randomness involved in the generation, it’s something that should be possible to refine easily enough if you find it lacking). Filling them out shouldn’t be too hard in principle…I’ve sort of hinted at how to do it already…but this is enough for one post…
Duplicated from here. The content diverges somewhat from Lerdahl.