JULY 3RD, 2017
J/S:
J: Ringo says, “What’s this? Dartboard paper?”
S: I don’t know, Ringsy… I’ve never seen anything like this before, though it’s probably some type of graph, since we’re doing polar functions.
J: That’s right. And a polar function is a function expressed in terms of a radius r and an angle Θ (theta), as opposed to x and y.
S: So, ONLY r and Θ? No numbers?
J: Oh! You can have anything you want, but the point is that x and y are rectangular coordinates-
S: And this is a circular graph!
J: -whereas r and Θ are polar coordinates.
S: Polar, in this case, means that the graph is circular.
J: Ringo says, “Hmm… So, if I lay this graph on a snare, then I can see, in terms of r (the radius) and Θ (the angle), where I’m hitting the drum and note how the sound varies.”
S: But then, if anyone discovers the equations, they would know how to play in the elusive, top-secret Beatles percussion style.
J: I suppose you could write it in secret code, or, as da Vinci did, by writing things backwards in different languages.
S: I don’t think Ringo is bilingual.
J: Well, they all knew Liverpudlian, right?
S: *rolls eyes* WELLLLLLL, I guess Liverpudlian does count because of all the words they use that we don’t use in America (or in the rest of Britain, for that matter)… or they could use Paul and John and Ivan’s secret language! All I know of the secret language is “Cranlock naval, Cranlock pie”. No one knows what it means, but it’s from the poem that Paul wrote for Ivan when he died… in 1993… of Parkinson’s disease… I know way too much…
J: Hm! You know a lot.
S: So, Ringo, we will attach this polar graph to your snare drum and see the angle and radius at which you hit it.
J: John is writing down something. Hey! It’s a polar function! Looks like r = Θ/60°.
S: Ahhhhh… That’s a nice, simple function. But where ARE the Beatles, just so that I can picture this more clearly?
J: They could be in a secret lab at Apple.
S: So, it’s 1968? ‘Cause it must be.
J: Why must it be?
S: In ’67, they didn’t spend much time at Apple and it hadn’t really got going, and by ’69, it was falling apart…
J: “Hey, Ringo! Play THIS!” says John, handing him the paper with the equation on it.
S: Have we already put the graph on the drum?
J: You can see it there, on the drum!
S: Oh, now I see. Ringo thinks that it probably has something to do with the 60° mark on the circle.
J: Θ represents all the points going around the circle – it’s an angle. So, if the point that Ringo hits has radius 0.25 and an angle of 15°, how would we graph that?
S: We would graph it by first finding 15° on the Θ line and then starting from the origin (the center of the circle) and going up the 15° line (which is a radial, not to be confused with a radian) 0.25 units.
J: And we get there because of John’s equation: 15 ÷ 60 = 0.25.
S: Oh! So, should I graph it?
J: Absolutely!
S: *graphs it in red*
J: And now Ringo has hit a point on the 30° radial!
S: What IS the point? Oh! I know! Paul and George are the point! PUT THEM IN HINT HINT
J: “We’re trying to find out where the sweet sounds are on a snare,” George says.
“And whether it’s worth it to compose for snare in terms of r and Θ,” says Paul.
S: So, they’re all in their secret lab in the basement of Apple with a sterile table and-
J: Why does it have to be sterile?
S: It’s a lab!
J: It could be a loose R&D lab!
S: Whatever. I just meant that it’s cleaned regularly. Anyway, as I was saying, there are four plastic chairs in their favorite colors, and Ringo’s drum kit (the biggest one, of the three he had), and a tiny window near the ceiling that no one can see through so that their experiments are kept secret. The sterile table is made out of marble. They bought it back when they didn’t know that Apple was bankrupt and broke and in-debt and moneyless.
J: Goodness!
S: No, really – it was. Also, John, Paul, and George are sitting at the table and Ringo is at his drum kit. So, you said that Ringo hit a spot on the 30° radial?
J: Mm-hmm. Can you keep up with him?
S: Are we still following John’s equation?
J: Yes.
S: 30 ÷ 60 = 0.5, so I’ll graph that… *graphs that in blue*
J: BAM!!! He hit the 45° radial! Looks like he’s hitting one radial every second, going around the paper from one radial to the next!
S: Snare drums don’t go “BAM”; they go “TSCHHHHHH”. But, anyway, 45 ÷ 60 = 0.75, so… *graphs 0.75 on the 45° radial in purple* Hey! It’s going to make a circle, isn’t it?
J: It’s going to do something, but I don’t think it’s a circle.
S: Okay. What point on the snare does Ringo’s drumstick hit next?
J: Well, Ringo’s hitting the next radial on the graph paper every second.
S: So (assuming we’re in 4/4), he’s playing quarter notes at 60BPM or half notes at 120BPM or whole notes at 240BPM or eighth notes at 30BPM or sixteenth notes at 15BPM or triplets at… what would that be?
J: Use a calculator.
S: *uses calculator* 20BPM. So, Ringo’s next point would be on the 60° radial and at 1, since the pattern increases by 0.25 every time he hits the drum, and 60 ÷ 60 = 1. *graphs that in green* Now I’m just going to graph all the points for this equation that are on the radials. *graphs up to radial 225°* Oh, you’re right! It’s not a circle… Maybe it’s a heart! No, that would be a badly-shaped heart…
J: A cardioid.
S: *finishes graphing* It’s a spiral!
J: Zzzzzzzhooooop! Nice spiral!
S: Ritchie is an artist. Actually, John is an artist because the spiral was his idea. I really feel like Paul and Geo are being left out, even though they’re sitting there in the lab.
J: Paul says, “Okay, I like the sound of that spiral, but can we try a circle that is off to the side?”
S: Okay, Paulie! But first, Dad, remember that the point of this is to see if it’s worth it to to them to do all the math, so keep that in mind as we continue… but it depends on their definition of “worth it”…
J: Well, if it’s a drag, they won’t do it again!
S: Yeah, because, since most bands don’t use precalculus to figure out drumbeats, it’s not a well-known drag.
J: *grimaces* Do you want to go to a drag show?
S: No! It’s from A Hard Day’s Night! You know – ‘she’s a well-known drag’ – Susan, the TV lady trendsetter!
J: Well, that’s what ‘a well-known drag’ means!
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S: OH! NOW I know why Geo was drawn in drag in that weird comic that Strabius on DeviantArt did! Because of ‘a well-known drag’!!!
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J: Paul says, “John, what’s the equation for a circle?”
S: I don’t understand why John is always the one with the equations! He skipped class, never did homework, hated school, and thought math was stupid.
J: John (looking up from a math reference book) says to you, “I’m just a newborn genius!”
S: I still think that if a Beatle had to do the equations, it would be Paul! He was a model student… until he met John.
J: Paul has trained John to look things up in his old CRC handbook.
“CRC?” George asks.
Ringo reads from the spine of John’s book, “Chemical Rubber Company.”
S: Is that really what it stands for?
J: Yes. *gets copy of CRC Standard Mathematical Tables from 1961*
S: *looks in book* WOW, this is complicated.
J: If you didn’t have a calculator, this is where you’d look.
S: But then how did the authors of the book figure out all this stuff?
J: By hand.
S: Ewww… Gruesome…
J: John speaks the function: “r = 4 cos Θ + 6 sin Θ.” And now we graph it as Ringo starts to play it, still tapping one radial per second.
S: Starting at 0°… r = 4 cos 0 + 6 sin 0. The cosine of 0 is 1, and the sine of 0 is 0. 4 · 1 = 4 and 6 · 0 = 0, so we’re left with r = 4. So, I’m counting four units from the origin on the 0° radial… *graphs that* Now I’ll do the others until I get back to 0°. *finishes graphing*
J: “Nice!” says Paul. “That circle sounds nice.”
S: I’m so glad you think so! 🤣 But Georgie’s still left out…
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S: …because his mouth is full of gummy bears!
J: Really? I was going to have him ask for a clover leaf or something with heart.
S: He tries to through the gummy bears.
J: George says, “How about something with heart: r = 1 + cos Θ.” And I have marked 1 where 4 would normally be, since the equation fits nicely that way. Now, this graph is in radians, so make sure that your calculator is set to radians instead of degrees when you do cosine.
S: *graphs equation*
It’s a sideways heart! We should note also that the black lines that connect the colored points are just approximations of the function – if we were to plot EVERY point in the function, it would be more curved.
OMG, Dad, I sound like you!
J: George Martin pops in and says, “Say, is that a spec for a cardioid microphone?”
S: So, what happens then?
J: I guess it’s just left there, hanging…
S: …on the clothesline stretched across the lab for no apparent reason that Ringo crashes into every day.
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S: Okay, that was random. Bye!
Beatles Math 