Friday, June 16, 2017

Zeno's Quarter Note

I present for your enjoyment: Zeno's Quarter Note! It's impossible to get to the end of this note, but it'll will nonetheless be over in two beats.

"What the heck is this?" you may ask. Here's a quick explanation.

The Greek philosopher Zeno presented a set of paradoxes, the most famous being the Dichotomy Paradox.

If you wish to travel from point A to point B, you must first travel half the distance. Then from there you must travel half the distance again, and half again, and so on. Thus, you would have to pass through an infinite number of points, which is impossible.

In music, we can emulate this with time and rhythm. In 4/4 time, a quarter note lasts 1 beat. If you add one dot to the note, you must hold it for 1.5 beats. The dot adds half the time value of the item right before it. A second dot adds another half the value of the first dot, bringing the total to 1.75 beats. A third dot would bring us up to 1.875 beats. Four dots gives us 1.9375 beats, and so on.

In practice, one dot shows up in nearly all pieces of music. Two dots are seen occasionally, and three dots are very rare. Beyond that, I haven't come across four or more dots. Though, theoretically, you can add as many dots as needed, ad infinitum, and that's exactly what I've done here.

No matter how many dots you add, you get closer and closer to 2 beats, but you can never reach it nor pass it. Then again, this is only a thought experiment, applying Zeno's paradox to time instead of space, and it makes for a great math-meets-music joke.

A couple of weeks ago, I submitted this gem to a prominent music engraving page on Facebook, which received 340 reactions (to this date), and scores of funny replies. Here are my favorites:

[It] takes an infinitely long piece of paper to notate.
Unless you make each dot half the distance to the barline.... 
Fortunately, on my saxophone, the A can be fingered using the left hand only. This leaves the right hand free for the infinite number of page turns. However they come faster, one after the other, as one plays more and more of the dots.
I did the math! The note pictures has 19 dots, which would give it a value of one and 524287/524288 beats.
Sisyphus would be proud.
An infinite series is usually denoted by three dots. Hmm... now I'm extra confused!
a metaphor to life?  so long but so short

And finally, Vili Robert Ollila offers alternate notation (the second note in the picture being the very fast note you must play to finish out what's left of the 2 beats):

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