Metre

Groups that are not there

Play a completely uniform series of clicks. Same sound, same spacing, no accent anywhere. What people report hearing is not a uniform series of clicks. They hear them in groups, usually twos or fours, sometimes threes, and they will tell you which click is the first one.

Bolton reported this in 1894 and it has been reproduced ever since under the name subjective rhythmisation. The grouping is not weakly present in the sound, or present in some subtle statistical way. It is not present at all.

What that figure shows, and what it does not

With accent off, the grouping you select never reaches the audio. The click is scheduled from a single constant, the same one every time, and no part of the sounding code asks how many are in a group. So when you switch from twos to threes, the thing that changed was not the signal.

Then turn the accent on. Now the group is genuinely in the sound, and the contrast is the useful part: you can compare a grouping somebody gave you against one you were manufacturing yourself, and notice they feel much the same from the inside.

Being careful here, because this figure proves slightly less than it appears to. Bolton's subjects had nothing to look at and grouped the clicks spontaneously. This figure gives you a row of dots, which helps you switch deliberately, so what it demonstrates is that grouping can be imposed at will on an unchanged signal, not that it arises unprompted. The second claim is the stronger one and it is the experimental result; this is the half that fits in a browser. If you want the unprompted version, play it with the dots out of view and wait.

What a metre actually is

Once grouping exists, the positions inside a group stop being equivalent. Everyone learns that in 4/4 the first beat is strongest, the third is next, and the offbeats are weak. It is usually presented as a fact to memorise about that time signature.

It is not a fact about 4/4. It is what you get from one instruction: divide, then divide again. Halve the bar and you have two halves. Halve those and you have four beats. Halve again for eighths, again for sixteenths. Each division defines a level, each level marks certain positions as pulses, and the weight of a position is simply the number of levels that mark it.

Nothing in that says the downbeat is important. It comes out important because it is the only position that is a pulse at every level.

The shape that falls out

For 4/4 down to sixteenths the count returns 5 1 2 1 3 1 2 1 4 1 2 1 3 1 2 1. That is the familiar hierarchy, in order: the downbeat highest, the half-bar next, then beats two and four, then the eighths, then everything else. Nobody entered those numbers. They are a tally.

The sharpest demonstration is 3/4 against 6/8. Both have twelve positions, both run at the same step rate, so both bars are the same length and contain the same number of clicks. The only difference is the order of the divisions: three then two then two, against two then three then two. The weights differ at exactly three positions out of twelve, and the two do not sound like the same bar.

This is the object lesson 3 needs. Syncopation is not a vague quality of sounding off-balance; it is a measurable relationship between where the notes are and what these numbers say. Which means it cannot be a property of a rhythm on its own. The same pattern against a different metre is a different amount of syncopated, and the figures in the next lesson will show a pattern crossing from one to the other without a single note moving.

The divisive model has a real edge, worth marking before it gets over-applied. It builds metres by repeatedly cutting a whole, which handles 4/4, 3/4 and 6/8 comfortably and does not handle the additive metres of Balkan and Turkish music at all. A 7/8 grouped 2+2+3 is not the result of dividing anything by seven; it is three unequal beats laid end to end. That is a different construction, it is entirely regular on its own terms, and calling it irregular says more about the model than the music.