Why can we count?

Remember in 4th grade when your teacher tried valiantly to teach your whole class to play the recorder?  Learning the fingering was hard enough, but then you had to learn note values and how to count them, and halves of them, and quarters of them, and rests, and how to keep the beat and so forth.  This is one of the most frustrating aspects of learning music for just about any beginner.  But, by the time students actually enjoy playing their instruments, the whole issue of counting notes, and time between notes and all of that, has become second nature.  They've internalized something that the rest of us never really did.

Recorder; Wikipedia

Our son-in-law Niocla Barbieri, an Italian professional double bass player who plays baroque and classical music, put it this way when I asked him about the experience of keeping time in an ensemble, including about the rubato, an expressive stretching of the beat.  I quote him at length because it's such a beautiful and evocative description.

Staying together in synch is not usually a conscious thing and I would say that the more the ensemble we play with is good, or close, or “harmonious”, the less you have to think about that and the less it stays conscious… Or the less you have to want it, I would say, because it happens by “itself”… I wouldn’t say it is something automatic, either, because it is very far from any idea of being something stuck or rigid, but the feeling is more something towards a fluid idea of a continuous chain of perceiving and reacting, detecting and responding, more like a very relaxed dialogue… 

Sometimes, from the top of a bridge you overlook down into the water of the river and you see the flowing of it, the general flowing, you perceive a whole fluid movement of a big mass… But some other times you watch better and you can notice several small currents and streams that seem to have independent “will” from the main one… They seem to slow down and then accelerate and then move sideways, as if a part of the water is going to move away from the rest… That’s just an impression, because all the big mass is still traveling as one... 

This is the flowing of the time, in orchestra and this is the rubato, I would say… Of course in a good group, where nice things happens without effort and with a sort of natural relax (“sprezzatura", we used to call it in the Italian baroque era), like the one of the flowing of a river...

A new paper just published in the Journal of the Royal Society ("Optimal feedback correction in string quartet synchronization, Wing et al.) takes a look at how professional musicians correct lapses in synchronicity. Sensorimotor synchronization happens in many organisms (fireflies that synchronize their pulsing, e.g.), so synchronization is of long-standing interest for a variety of reasons.  Musicians do sometimes have lapses -- indeed, sometimes it's intentional -- but they also are adept at getting back in synch.  How?  What have they internalized?

Wing et al. analyzed two different professional string quartets playing the fourth movement of Haydn's  quartet Op. 74 no. 1.



The question was how the members of each quartet responded to expressive, unrehearsed variations in timing.  Generally, members of a string quartet follow the lead of the first violinist, although with more or less strict adherence to this rule, depending on interpersonal dynamics and the philosophy of the group and so on.  So, correcting timing may be a matter of getting back in synch with the first violinist, or it may be more fluid than that.  In any case, the authors propose a feedback mechanism to correct timing that gets off, a linear phase correction.

Time series analysis of successive tone onset asynchronies [that is, musicians who aren't in time with each other at the start of a tone] was used to estimate correction gains for all pairs of players. On average, both quartets exhibited near-optimal gain. However, individual gains revealed contrasting patterns of adjustment between some pairs of players. In one quartet, the first violinist exhibited less adjustment to the others compared with their adjustment to her. In the second quartet, the levels of correction by the first violinist matched those exhibited by the others. These correction patterns may be seen as reflecting contrasting strategies of first-violin-led autocracy versus democracy. The time series approach we propose affords a sensitive method for investigating subtle contrasts in music ensemble synchronization.

If musicians always played the music in front of them exactly as scored, it could be dull.  And, they aren't automatons -- they hear a piece in their own particular way and want to express what it means to them and they have plenty of freedom to do this, within limits.  We, the audience, give them that freedom, and also adjust whatever internal metronomes we listen to music with (even those of us who failed 4th grade music have developed a sense of timing) to go with the flow of the music we're listening to -- within limits.  This is the rubato.  But there are limits, and apparently they are internalized.

According to Wing et al., musicians use linear phase correction to regain synchronicity with either other musicians or with the tick of a metronome.  That is, they know when their count is off, because it has been set previously, by the relationship between note values and time between notes, and they are able to tell when they're off, and adjust to get back into the beat.  Musicians learn their skill by spending tens of thousands of hours counting notes; that they can internalize it quickly, and correct it when it's off is no surprise.

It is often said that musicians are good mathematicians and vice versa.  The earliest formal musical theory was by the Pythagorean school of mathematics, in ancient Greece.  They figured out the mathematics of basic harmonies.  But many musicians didn't take, or hardly squeaked by, in mathematics.  So are they doing implicit time-series math in their heads implicitly?  How?

And try this on for size:

This is an excerpt of an ensemble playing John Adams' "Shaker Loops", with its fiendishly minimalist, relentlessly repetitive and syncopated measures (with endless slight deviations) that goes on for around 25 minutes.  We heard an ensemble play this a few years ago at Ithaca College, and the violist who was a friend told us the stress and horrors of trying to stay in synch.  But they did, beautifully, and in that instance (unlike the YouTube) they had no conductor!

In this general vein, what explains this curious phenomenon that I've observed numerous times, after decades of knitting?  Just last night I was casting on stitches to make a scarf.  The pattern called for 85 stitches.  There are too many distractions for me to be able to count stitches as I cast on, so I just take time out to count them a few times as I go along.  Remarkably -- at least I think so -- last night when I stopped to count I had cast on exactly 85 stitches.  But I've had this happen when the pattern called for 285 stitches too.  No phase correcting there, do I have an internal counter that turns itself on as I start to cast on, and then alerts me when I've met the target number?  If so, it seems like a rather frivolous way to spend brain cells though, even if useful.

Casting on; Wikipedia

I remember once being at my daughter's youth orchestra rehearsal when one of the violin teachers told me to watch the conductor when he stopped to talk to the players.  "He'll pick up the beat right where he left off," she told me, and indeed he did.  After years of conducting, he had developed an internal metronome that kept on ticking even when he wasn't waving his baton.

And then there's the internal alarm clock that always goes off 2 minutes before the alarm we'd set.  I don't remember the last time I've heard an alarm -- except when I couldn't figure out how to turn the bloody thing off on my phone.

So, this mathematical explanation for musicians correcting themselves when they get off the beat.  It might well be a good description of what happens, but it's not an explanation of what they are doing or feeling. To us, the deeper question is why we're able to do all this counting and time keeping anyway.

It's said that infants can count.  Or at least have numerical awareness.  As do non-human primates, and even dogs. Crows can count, parrots can count to six, and even have a concept of zero, according to at least one source.  Apparently even frogs can count.



So the ability to count must have evolved long before humans. But why?  Or better put, what kind of ability is it really?  It's easy to imagine adaptive scenarios (the duck had to be able to count her ducklings, to shepherd any stragglers away from predators, the wolf had to know its pack was intact, and so forth), but like most such stories, impossible to test them.

But maybe it's not an adaptation at all, really.  Maybe it's just one of the many ways we take in information about our surroundings, a by-product of there being more than one of any of us, or of any food, or of anything else in our environment.  Maybe recognizing that there are two lions over the crest of the hill is exactly the same as noting their color or even just that they are there.  It's just another observation; when we turn it into a number, that's when it becomes higher math.

Many pre-agricultural cultural groups are reported only to have numbers one, two, and many in their language, which may be consistent with the idea that formal 'math' and counting are recent cultural add-ons.  It is interesting that math and music were seen by evolution's co-discoverer Alfred Wallace as attributes that could not have evolved because, after all (he argued), our primitive ancestors didn't need them and so could not have been selected for them.   He used this as a reason to invoke the existence of God and human exceptionalism.

We wouldn't go that far, and instead are grateful for music, which, if it is but an evolutionary spandrel, is one of the more beautiful things we do with our ability to count.