I'm not sure why I haven't been exposed to this earlier in my life, but it's definitely a clear example of brillance. (Reminder: My definition of brilliance is any solution to a problem that, once seen, can't be unseen. And it makes you wonder why you never thought of that yourself. Which isn't to say you would've, but that the solution fits that perfectly.) 

Let's cut to the chase: Studying analog computers today I ran into (via Wikipedia) an old Navy doc from 1944 describing an analog computer they used titled, Basic Fire Control Mechanisms, Ordnance Pamphlet (OP) 1140, which has been nicely scanned and presented in a pdf version.  Lookit how these things work. Simplicity itself, but I'd never thought of making something like it. 

Easy, right? If you want to multiply by 36, you have a gear where X teeth of movement (say just enough to move a dial so that what's in the output window changes from 1 to 2) turns a second shaft 36 times X teeth. Then you check the number that shows in the output window (also listed on a dial) that's regulated by that shaft.

In this case, the Navy wanted to set up a computer where sailors would enter the same number of specific variables for each calculation and have the machine compute the values needed to set and fire shells from their artillery.

That's pretty cool. No, more to the point, that's brilliant. No energy needed. Easy to repair, all things considered. Doesn't require any insanely specialized knowledge to work on or with. Not real flexible -- you're give or take doing the same calculation each time -- but in this case, who cases? As the OP says...

The basic mechanisms described in this book were especially developed over a period of about 30 years to do a highly specialized job. That job is to solve mechanically the mathematics for surface and anti-aircraft fire control. These basic mechanisms make the necessary computations to point the guns and set the fuzes to hit fast moving targets with shells fired from the deck of a ship which is moving, pitching, and rolling. To aim the guns correctly under these conditions about 25 things must be taken into account all at the same time. These include target speed, climb, and direction; target range, elevation, and bearing; pitch and roll; and initial shell velocity.

If the enemy were to announce six or eight hours beforehand just where the target would be at a particular instant and just how it would be moving, a lightning mathematician would be able to calculate where to point the guns to hit it at that one instant. But, the results would be good only for one instant.

Now I have my doubts about how "lightning" the mathematician would have to be, since we're just doing an easily, if tediously, delineated set of multiplication/division/logarithmic (?) operations each time (how many math questions can you answer in "six or eight hours"? Um, lots), but point taken. Pretty cool.

But why [do you care]?

Why did I run into this today? One pastime I've been pouring waaaay too much time into recently is the study of DIY headphone amps and cassette players. And one of the important parts of any (well, most any) headphone amp is its "op amp". The op amp is the piece that, when fed a little juice, makes the tiny electric current that's created by the magnets recorded into [sic] your tape as they pass your cassette player's read head loud enough to hear.

If we read our canonical work, Op Amps for Everyone, we learn that the name "pop amp" is short for "operational amplifier", and they're, in a sense, a new twist on the shaft-based analog computers we just saw the Navy used for aiming shells in the Forties.

The heart of the analog computer was a device called an operational amplifier because it could be configured to perform many mathematical operations such as multiplication, addition, subtraction, division, integration, and differentiation on the input signals. The name was shortened to the familiar op amp, as we have come to know and love them. The op amp used an amplifier with a large open loop gain, and when the loop was closed, the amplifier performed the mathematical operations dictated by the external passive components.

Does that make sense? You pass in a voltage and the operational amplifier -- which, it should be noted, requires a power source to perform its operation -- cuts it by two or multiplies by 7... or 36!... or whatever. It is, in effect, just a way of moving from one setting on one shaft through "electric gears" to another.

In an amplifier, we take in a current (?) and multiply it to produce more volume (or is that gain?). Different op amps, like different gears, multiply by different amounts. That multiplication here is measured in decibels, a logarithmic scale 

Here's a list of op amps that work reasonably well in a CMoy DIY headphone amp (for more on the CMoy, sort of "the" famous DIY amplifier, read the original post here, learn to build here, buy a kit here, or learn about alternatives). Note that each has some measurements of the power you've got to put into the op amp to get a corresponding decibel gain for your sound; one this page, he's listed Vmin, 0.5V into 33โ€‰ฮฉ and Vmin, 2.0V into 330โ€‰ฮฉ, so how much power (how many teeth in the gears?) is necessary for a specific decibel gain (multiplication factor)?

Anyhow, that's a long-winded way of saying that your old cassette player was very likely a computer. An analog computer capable of just one calculation, but that's all you needed! From the docs for the Elenco Electronic's AK-200 Cassette Player Kit (or the per-soldered AK-250), we see what the goal of your player's integrated circuit was...

To maintain a flat frequency response over the full audible range, both the high and low frequencies must be given a boost... It is therefore possible to boost the high frequencies during the recording process without saturating the tape. This is called pre-equalization.

It is not practical to fully boost the low frequencies during the recording process. It is therefore done by boosting the low frequency response of the playback amplifier. This called post-equalization. The National Association of Broadcasters (NAB) has set a standard response curve for playback amplifiers. In general, pre-recorded tapes are recorded so that the response is flat over the audible range when played back through an amplifier having this response.

...

As shown on the schematic diagram (Section 13), each amplifier consists of a pre-amplifier and driver with a volume control circuit between them. The playback signal from head A is input to the pre-amp on pin 3. The pre-amp has a gain of 30dB (about 32 times) at 1kHz. Resistors R2, R5 and R6 and capacitors C2 and C4 are placed in the feedback circuit of the pre-amp to provide the NAB standard frequency response.

Emphasis mine, as usual. 

Does that make sense? Because of limitations of the cassette tape medium, you need to boost some frequencies. That calculation is what all the innards of your player are for (aside from all the buttons and mechanics for engaging the capstan motor and all that): They're there to translate the feed from your tapes using the "NAB standard" formula for boosting the signal to a "flat frequency response".

Neat!

Some more cMoy stuff:

Btw, hummingbirds chirp, sometimes when feeding. TILx2.

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