## How Many Grooves Are On A Record’s Side?

One. OK. We know that.

Every person who ever suffered through that innocent-looking problem where you’re given the size of a record and data about how wide the groove is and asked how many are on the side of the record and then after a lot of confused algebra handed in an answer and discovered it was a trick question has that burned into their brain, and maybe still resents the teacher or book of math puzzles that presented them with the challenge only to have the disappointing answer revealed.

This may be a generational frustration. I think but don’t know that compact discs and DVDs actually have concentric rings so that the how-many-grooves equivalent would be a meaningful, non-trick question; to check would require I make the slightest effort so I’ll just trust that if I’m wrong someone will complain. In another thirty years the word problem may have disappeared from the inventory. But it irritated me, and my Dearly Beloved, and I’m sure irritated other people too. And, yes, we’ve all heard of those novelty records where there’s two or three grooves on a side and you don’t know until fairly well into the performance which version you’re listening to, but I’ve never actually held one in my hand, and neither have you. For the sake of this discussion we may ignore them.

But the question we plunge into answering before we’ve noticed the trick is more like this: If we drew a line from the hole in the center straight out, a radial line if I want to make this sound mathematical, then it crosses some number of grooves; how many? Or maybe like this: how many times does the groove go around the center of the record? And that’s interesting. And I want to describe how I’d work out the problem — in fact, how I did work it out a few nights ago — including a major false start and how that got me to a satisfactory answer.

Since I’m pretty good at calculus, my first instinct is to try using calculus methods to solve the problem. How?

Well, imagine taking a little ring slice out of the record — geometers might call this an “annulus”; my father might call it a “washer” since it’s the shape of that thing you put under screw heads for some reason that my father knows. That segment will have some part of the record’s groove on it. If the inside and the outside rings are far enough apart the groove will wind its way around several times. If they’re close enough together the groove winds its way around not even a full loop. If they’re just the right distance apart, though, they’ll wind around exactly once.

Now here’s the calculus-approach thought to things: the area of the record — the playable area, anyway — is made up of some number of these annuluses of one loop each. It’s easy to figure out the area of an annulus (really: it’s the difference of the area of the outer rim’s circle and the inner rim’s circle; holding up a washer, it’s the amount of metal you need to make the whole disc, minus the amount you punch out to make the hole). So the number of grooves you cross in one line is the playable area (again, the area of an annulus) divided by the area of each one-loop annulus. If you follow the reasoning here then you understand integral calculus. (This part might need to be expanded on before you take an exam on it.)

This method is madness, and attempting to follow it quickly falls apart. There are several reasons for this. The most obvious is that the width of this one-loop annulus is really just how far any point on the groove is from its neighbors, going in toward the middle or out toward the edge. Yes, yes, it’s the same groove, but you know what I mean: if we imagined being tiny enough to stand in the groove on the radial line, and stared in the direction that the needle seems to go, there’s the point on the left where the groove crosses the radial line, and the point on the right where the groove crosses the radial line, and how far is it to that intersection? And I’ll call that the neighboring-groove distance even though it is really the same groove because you think of a way to say that which isn’t too cumbersome to deal with. Anyway, if we knew that neighboring-groove distance we wouldn’t need to fiddle with the rings any. We could just divide the diameter of the playable area by the neighboring-groove distance and be done with it.

Also, it’s conceivable that the neighboring-groove distance is different closer to the center of the record than it is far away. A record turns at a constant rotational speed, after all; the linear speed or the radial distance between one part of the groove and the next isn’t obviously important to the design. At the rim of the record, where one drops the needle before starting, and where it gets stuck at the end if the auto-return of the arm isn’t working, the neighboring-groove distance certainly does vary. I don’t believe that it varies too much in the main section of the record, apart from the little stretches most records have to mark where one song ends and the next begins. But obviously, I can’t say with certainty that it doesn’t vary enough. Assuming it doesn’t vary too much simplifies the problem, though, and it seems reasonable enough, but that doesn’t help me say what the distance is.

So calculus fails us, a refreshing reversal. But how can we still solve the problem?

That thinking about the neighboring-groove distance gave me an idea, though. Think about that radial line from the center of the disc to the edge. We want to know how many times the groove crosses that line. That’s the same number of times the record turns, if we play the complete side: on each complete turn the needle will have moved in just a little bit, and crosses the radial line at a different point. So if we know the number of rotations the disc makes to play one side, then we know the number of times the groove crosses that radial line.

And then one of the bits thought up during my false start comes into play: the record turns at a constant rotational rate per minute. For an LP, that’s 33 1/3 revolutions per minute. How many minutes does a record take to play? (This same sort of reasoning can be put to 45’s or 78’s or even freaky other records you might find.)

Well, that’s different for every record, of course. But one side takes usually something like 20 minutes, which can be surprisingly short if you’ve got so used to CDs or MP3 players or Internet radio where music just seems to sprawl out without limit. 33 1/3 is an awfully suggestive number, though: it’s 100 divided by three. Let’s say that a record takes 18 or 21 minutes on the side, because those are multiples of three. Then an 18-minute record side would turn 600 times over the course of playing; a 21-minute record side would turn 700 times. So I would call it something like six to seven hundred times the groove crosses the radial line.

That sounds satisfying as an answer. It just leaves one nagging question: am I right?

Solving a problem is fine, but it’s not complete. We have to verify the solution somehow.

A quick search of the Internet reveals that the question about grooves on a record side are overwhelmed by people answering “one”. Thanks heaps, guys. A few sources will give the specified dimensions of a record, by the surprisingly large number of standards out there. Groove width, for mono or stereo, is pretty well specified, but sources get vaguer about the average separation between adjacent grooves.

There are some fair reasons for this. The width of a groove itself depends on peculiarities of the recording, such as the volume of the sound and the noise compression at play. How close together grooves can be depends on the quality of the material. The record player is very forgiving of groove distance, so, perhaps it’s better not specifying too tightly what the average should be. Let the recording staff figure that out.

It’s easier to find information about the gross physical dimensions of a record. Taking the playable outside of a record to be 11.5 inches in diameter, and the inside playing surface 4.75 inches in diameter, suggests it’s 3.375 inches from the outer surface to the innermost. That seems to be about the right size if I actually hold up and look at a record, although I would probably have rounded that to around three and a half inches. Fitting 600 to 700 grooves in that … call it about 200 per inch … and I can’t find a source that says what the typical number of grooves per inch should be. 200 per inch sounds plausible, but I can’t really tell that by eyeball, and I’m fresh out of diffraction gratings to check against the records I can lay my hands on.

Let’s plunge into the mess of people saying things on the Internet. The Genesis Of Vinyl Stereo Record page, from Badenhausen.com, looks neatly authoritative what with appearing in both English and German or maybe Dutch, and asserts “the radial distance from groove to groove will be about 120 micrometers at an average signal length for both channels” on a stereo record. 120 micrometers from groove to groove, if I’m doing division correctly, implies just over 200 grooves to the inch, so I seem to be pretty dead on, given how few significant digits I have in play.

Since every web site has an equal and opposite web site, though, here’s a comment thread from a web site I never heard of before where a guy I don’t know anything about cites two web pages that don’t exist anymore to say a long-playing record averages about 225 lines per inch, but could be as high as 300. Well, the lower end of 225 is close enough to my figure that I can’t argue it too much.

And in something I thought would never happen, I ran across a Yahoo! Answer page with a clear, direct and useful answer to the question, “what is the standard groove spacing on a CD-R and LP record?” (It was Yahoo! Canada Answers; perhaps that makes the difference.) Canadian answerer billrussell42 claimed the groove spacing was variable but about 150 micrometers, which works out to 170 grooves per inch. I’m still comfortable with that answer and delighted to find an actual answer on a Yahoo! Answer page. For the curious, billrussell42 also says the spacing on a CD is 1.6 micrometers, on a DVD 0.74 micrometers, and on BluRay 0.32 micrometers.

I’m not very happy that I can’t verify my answer with a direct measurement. I have some authorities who may well be qualified endorsing the answer I reasoned out, but part of why I find them believable is that they do agree with my answer. But my reasoning is pretty straightforward and the assumptions which go into it seem easy to check. If I have it very wrong, I’d like to know just where I go wrong.

So that’s my answer: the groove on an LP loops around the center something like 700 times, or around 200 times per inch.

Those disputing my reasons are welcome to say so, and I will respond with an essay on what the Muzak corporation did to help win World War II.