Musical Braids

We're all familiar with braiding hair. The classic hair braid is an interesting pattern that can be represented thus:

Let's say that a braid consists of some number of strands that are 'woven' together. So our classic hair braid example has three strands.

Braids are more common than one might think. Arguably, the DNA in each human hair root cell is also a braid — a two-strand braid, the famous 'double helix':

Unsurprisingly, mathematicians have studied braids in depth.1 We're going to ignore all of that and investigate artistic, specifically musical uses of braids. After all, weaving patterns are definitely an art form in their own right.

Graphically, we could musically represent our classic hair braid (Example 1) like this:

Strands Crossing

The reader may have noticed in Example 1 that strands cross over and under other strands. Example 3 ignores that detail. How might we mimic that musically?

Here the red strand crosses the blue one:

In this image we've represented the blue strand going under the red one by having the red strand pause, go silent. We can certainly do that with music. Here's a two-strand musical braid, with the strands going silent when they go 'under' the other strand:

We don't have to represent 'over' and 'under' if we don't want to — it's just a musical option.

Some Observations

So far all of our examples have been on one chord/harmony. There's no reason we can't change chords/harmonies:

Also, so far in all of our examples, the musical strands have been identical —just time shifted. Same melody, starting at different times.

But it needn't be thus. For example:

Pure Braids

So far in our examples, each strand ends up where it started, something like this:

Mathematical Braid Theory calls this a 'pure' braid, so we'll go with that name. Meanwhile, generally speaking, each strand doesn't have to stay in its own lane, it can migrate to another:

Each strand, however, does have to end up in a lane by itself — two strands can't end up in the same lane!

Also, it's worth noting that if you repeat a 'general braid' enough times, every strand will end up in the lane it started in.2

'Proper' vs. 'Improper' Braids

We've been discussing only 'standard braids' so far — the subject of mathematical Braid Theory where at any given step, there's only one strand per lane. This amounts to permuting the available strands at each step. This is what Braid Theory considers.

But if 'permutation' is 'random selection without replacement', there is also the option of 'random selection with replacement. Which means that, at any given step, more than one strand could occupy the same lane.

If generating a braid step using permutations results in a 'proper' braid, we'll say that assigning strands to lanes using combinations — random selection with replacement — results in an 'improper' braids. With its own musical possibilities

Conclusion

This doesn't begin to enumerate the possibilities. The examples above used the simplest of musical materials to demonstrate some basic ideas about 'Musical Braids'.3

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