Lab ii · the drum kit
A guitar string plays a tune; a drum can't — and that's physics. Skins, cymbals and snares each shake in a completely different way, which is exactly why a kit owns rhythm instead of melody.
A drum kit looks like one instrument. It is really five, bolted to the same rug — and they don't just sound different, they run on completely different mathematics. The kick is a proportion sum. The snare is a statistics lesson. A cymbal is a decay curve. The toms are a sequence. And the timpani is the exception that breaks all of it.
Five drums, five different sciences.
strike a piece of the kit — or pick one below
The timpani isn't on the rug — it lives in the orchestra pit, which turns out to be the whole point of section v.
There is one number that decides a drum's pitch before anyone touches it: how wide the skin is. Wider skin, slower wobble, lower note — and the two numbers multiply to a constant. This is inverse proportion, and a drum kit is a rack of it, sorted by size.
the same four drums, drawn honestly to scale
Every circle is drawn at its true relative size — thirteen pixels to the inch. Read left to right: the head shrinks and the number of hertz climbs. Nothing about the wood, the sticks or the player changes. Only the size of the skin.
diameter × frequency — is it constant?
| drum | d | f | d × f |
|---|---|---|---|
| Bass drum | 22" | 60 Hz | 1320 |
| Floor tom | 16" | 85 Hz | 1360 |
| Snare | 14" | 190 Hz | 2660 |
| Rack tom | 12" | 145 Hz | 1740 |
Three of them land between 1320 and 1740 — roughly the same number, however different the drums look. Multiply the size by the pitch and you keep getting about the same answer. That is what inverse proportion is: d × f ≈ constant, so f ≈ 1470 ÷ d.
The snare is the odd one out at 2660, and that is worth noticing rather than hiding. The rule says drums of the same type. A snare head is cranked far tighter than a tom's — and tension raises pitch too, so it sits high above the line.
Worth being straight about what kind of rule this is: 1470isn't a constant of nature, it is the average we just measured off these three drums. A real drum's pitch depends on how tight the head is and how heavy it is as well as how wide. Size is simply the one that dominates when the drums are built the same way — which is why the rule works down a rack of toms and falls apart the moment a snare joins in.
the reciprocal graph, in wood and metal
This is the y = k ÷ xcurve from your graphs book, with drums sitting on it. It never touches either axis: a drum can't be zero inches across, and however huge you build one it can never reach 0 Hz. Tap any dot to hear that drum.
hear the proportion
Play the 12" rack tom, then the 22" kick. The kick's head is 1.83× wider, and its note comes out 2.42× lower. Under perfect inverse proportion those two numbers would be identical — that is the whole claim, written as a check you can do. They come out the same size but not the same number, because a real kick also has a deeper shell, a slacker head and usually a pillow inside it. The proportion sets the ballpark; the drummer tunes the rest.
so what is a drum actually for?
Add it up. Four of the five have no usable pitch at all — you cannot write a tune for a snare, and no amount of practice will let you play a melody on a hi-hat. That sounds like a limitation. It is actually a job description.
Freed from carrying the notes, percussion takes the other half of music: time. And time in music is pure arithmetic — a bar cut into halves, quarters, eighths and triplets, counted out and stacked. When a drummer plays a groove they are adding fractions to exactly 1, several times a second, without writing any of it down.
For the classroom