The Physics of Music: Subdivision

Part I: Subdivision

This is the first in a series exploring the physics of music. They are geared toward musicians seeking a better understanding of the basics of music. If nothing else, they're food for thought.

I'll address subdivision in front two angles: first by establishing it's importance in keeping accurate time and secondly, why it works.

First Experiment

A bell tower strikes 12 times indicating the time is noon. Suppose the bell rings every 4 seconds. The experiment is simple: clap at the same time the bell rings.

Interpreting the music above: the bell rings on downbeats, subdivide according to the notation (1 beat per ring, then 2, then 4, then 8). Be careful to not subconciously subdive the 1 or 2 beats -- try to "feel" the time.

When I have tried this I found that it is difficult to match the ring with 1 or 2 beats. With 4 beats per bar I do a pretty good job and with 8, even better. The takeaway is simple: subdividing beats is an effective way of keeping steady time.

Second Experiment

Imagine yourself with a monk (high production values on this blog...).


The rules of this are the same as the first: clap at the same time the monk hits the gong. What makes this difficult is that you are not allowed to use any external or internal "clock" -- don't look at a watch, count numbers, or even notice the rhythm of your breating. Focus on "feeling" the time. The idea is that in real life we're very good at syncronizing with external clocks, but how good is our internal, natural, clock?

 

Here is what we might expect if our target is 15 minutes. We'll probably do a poor job but, on average, we might hope to be on target. What's important is the relative error, for example, if you were off and clapped after 14 minutes (an error of 1 minute) your relative error is 7%. 

Now imagine you do this again for a baseline of 1 minute. To get that same relative error you need to be within 4 seconds. Finally, imagine the baseline is our 4 seconds from the 1st experiment. To achieve the same relative error as before you need to be within 0.3seconds!

Hypothesis

Our ability to divide intervals of time does not improve as the intervals get shorter. In other words, whether one is counting in minutes, seconds or milliseconds the relative error is the same.

Why subdividing works

If smaller subdivided intervals are still error-prone, then how can it be more accurate? The answer is that as we add up many small intervals the errors cancel out.

As a more concrete example imagine we have two options rushing or dragging, each count can be ahead or behind by some random amount. Whether we are ahead or behind on any given count amounts to a coin flip -- pure chance. As we add up many coin flips the result is 50% heads (ahead) and 50% tails (behind) -- right on time!  [Fun math note: there are an infinite ways to be behind or ahead of the beat but only one way to be exactly on time -- in practice you'll always be either ahead or behind, if only very slightly!]

Conclusion

Scientists create phonemically accurate atomic clocks using the same principle -- adding up many finely spaced "ticks" -- the only difference that their clocks finely divide nanoseconds (0.000000001sec) instead of minutes.

It is imperative that long held notes are subdivided to ensure subsequent entrances are played in time. For example, rushing a 4 beat whole note by 7% leads to an early entrance by a over an sixteenth note [at M.M. =60: 4s*.07= 0.28s while a sixteenth note is 4s/(4*4) = 0.25s].

We all have natural limit to how quickly we can count. For myself, in moderate tempi, I find subdividing eighth notes to be satisfactory; beyond that I get diminishing returns for my efforts. After all, there's more to music than keeping precise time. 

Appendix: Some math

A fair coin is equally likely to come up heads or tails, and is an example of the more general Bernoulli_trial in probability theory. There is a proof than may physicists encounter while studying Condensed Matter Physics or Thermodynamics showing that for a large number of coin flips, the probability distribution is a Gaussian (Bernoulli Process).

The takeaway is that flipping a coin many times produces a "sharply peaked" Gaussian centered on a probability of 1/2 (heads comes up half the time). What's interesting is that there are no other options, in other words, for a large enough number of coin flips you will always get 50% heads!