PAUSE: Have an ice cream break from differential equations

So yesterday, an interesting question was posed concerning both music and math. I’m pretty sure she was joking (and trying to prove a point), but let us not miss a grand opportunity to combine music and math into one super-subject, even if it is for just one problem. (Granted, there is quite a bit of physics and math involved in the musical art, but that’s not the point).

This is a crescendo:

Here’s a website about crescendos if you don’t know what they are:  http://en.wikipedia.org/wiki/Dynamics_(music)#Gradual_changes

This is a funnel:

Here’s a website about funnels if you don’t know what they are: http://en.wikipedia.org/wiki/Funnel

My section leader in marching band said to imagine crescendos as funnels, because the music can move along a lot better this way. If we stress the crescendo on the last one or two beats of a measure, then our audience will hear a change in volume (versus spreading the crescendo over the four beats of the measure, to which our audience will hear the same volume).

My section leader also added that more ice cream can fit into a funnel instead of an “ice cream cone,” which is the real world equivalent to a crescendo (obviously).

Let’s find out for sure, shall we? I’m ahead on homework and time so we might as well. Also, differential equations can be a lot to swallow all of the time. Let’s fill our world with ice cream!

And puppies!

Since we need the math equivalent to a crescendo (i.e. our ice cream cone), let us denote one inch as one beat in a measure. We are in 4/4 time, which means there are four beats in a measure. Thus there are four inches in a math measure.

For musical nerds reading this, this means a whole note is four inches, a half note is two inches, a quarter note is one inch, an eighth note is half of an inch, a sixteenth note is a fourth of an inch, and so on.

Suppose we have a crescendo for one full measure. If we think of our crescendo as an ice cream cone, this is what our crescendo would look like:

If we were to correctly play our crescendo, we would play it like a funnel. Our funnel would look like this (I am starting the incline on the third beat/inch of our measure/line because this is the earliest we should start the actual crescendo):

Let’s think of the area first, considering these are two dimensional musical objects first, and then we’ll move onto the volume of the three dimensional funnel and ice cream cone.

Our ice cream cone crescendo is just a triangle, and the area of our triangle would be

The area of our funnel would be the area of the trapezoid and the rectangle. This would be

This makes sense, considering I made our crescendo triangle and funnel end at the same level (one inch above the starting point). This means if we were to cover these two dimensional objects with a thin layer of ice cream, they would have the same area the ice cream would cover.

Now, let’s think of these objects as three dimensional things and compute the volume.

Our ice cream cone would just be a cone (if you couldn’t conclude from the phrase “ice cream cone”) and therefore our volume is just the regular computation of the volume of a cone:

Now our funnel will be slightly more difficult to compute the volume of (but not that more difficult). Just like with area, we have two different things to compute: the volume of the cylinder and the volume of the partial cone. This would be

This means the volume of our funnel is more than our ice cream cone.

I have just mathematically proven why we should play a crescendo in the form of a funnel rather than an ice cream cone.

Also, I have mathematically proven why we should eat ice cream in funnels instead of ice cream cones.

Today has been momentous.