The Matrix Tooth Pattern as a Binary Number

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1. The Linotype as a Computing Machine

In addition to being a metal foundry, a Linotype or Intertype is also a computer (albeit a special-purpose computer). Try that on your laptop!

The double-wedge spaceband justifying mechanism is an example of rather sophisticated analog computation. (I'll look into spacebands and justification in other Notebooks, not this one.)

The matrices and their distributor are examples of digital computation. The short explanation of this, in computer jargon, is that the matrix tooth pattern and therefore also the distributor bar rail pattern are a 7-bit big-endian binary numbers, and the Distributor executes a logical comparison operation upon them.

This of course makes no sense at all if you don't have a computer background, so the long explanation, without computer jargon, follows below.

2. How to Count on Your Fingers in Binary

This part is fun. At least I thought it was fun when I first figured it out in junior high school. It will be old stuff to computer programmers, but may be new in this form to Linotype machinists.

Remember back to school math, when they talked about numbers as having a "one's place," a "ten's place," a "hundred's place," and so forth? So that in the number:


There were 4 hundreds, 5 tens, and 1 one, giving 400 + 50 + 1 = 451.

By itself, that is profound but not necessarily useful to the student who isn't going to become a mathematician. ("Why are they teaching us this stuff?" - Well, because someday perhaps you'll be rebuilding Linotypes? Probably not.) But as a foundation, it is a good basis for extrapolating this idea of a number to an area which is very useful indeed to the computer programmer: binary numbers. Computers deal exclusively with binary numbers, which is to say numbers which have only two individual digit values (normally thought of as 0 and 1). Partly this is because it's easy to build a machine to distinguish just two things (vs. 10 things), and partly it's because all of the logical operations of a computer work much better with two-valued logic.

In any case, this means that to understand computers at a low level you need to learn how to count in binary. This is an easy extrapolation from the decimal (10-ary) "451" example, above. In a decimal number, the "places" are really not just "one's," "ten's," etc. but rather powers of 10. So the 1's place is 10^0 (ten to the power of 0, which is not really a sensible number but which by convention is taken to be 1). The 10's place is 10^1 (ten to the 1st, which is 10). The 100's place is 10^2 (10 * 10 = 10^2 = 100). The 1000's place is 10^3 (10 * 10 * 10 = 10^3 = 1000). And so forth.

Now for a binary (2-ary) number, the base number (which was 10 in base 10) becomes instead the number 2. So the rightmost digit of a binary number is the 1's place (because 2^0 = 1, by convention, just as 10^0 = 1 by the same convention). The next digit to its left is the 2's place (2^1 = 2). One more to the left is the 4's place (2^2 = 2 * 2 = 4), and then of course the 8's place (2^3 = 2 * 2 * 2 = 8). So reading from left to right, a four-digit long binary number has an 8's place, a 4's place, a 2's place, and a 1's place.

So the binary number 1 (or "0001") is just 1 (decimal). The binary number 10, though (or "0010") is 2 (decimal) because it has the 2's place occupied. The binary number 11 ("0011") is 3: (2 * 1) + (1 * 1) = 3 (decimal). The smallest number that you can represent in this scheme is 0 (of course), and the biggest is 1111. Binary 1111 works out, in decimal, to (8 * 1) + (4 * 1) + (2 * 1) + (1 * 1) = 8 + 4 + 2 + 1 = 15. So in four binary digits, you can count from 0 to 15 (decimal).

I find this counting most instructive to do on my fingers, because it shows very clearly the distinctive pattern of binary counting. This pattern will be important in understanding Linotype matrix teeth.

Take your left hand, and use just your four fingers (not the thumb). These four fingers represent four binary digits. The index finger is the 1's place, the middle finger the 2's place, the ring finger the 4's place, and the pinkie the 8's place. Adopt a convention for what finger position means zero and what means one; for example, finger-down is 0, and finger-up is 1. Now count.

Start with all four fingers down. That's zero:


Now lift your index finger. That's 1:


How can you represent 2? (This part is apparently trivial, but was of mind-blowing importance to me when figuring this out as a kid.) You cannot represent 2 using just the index finger. The index-finger position is "full"; it's represented all the numbers it can; you can't fit another one in there, no matter how hard you wish.

So to count to 2, I must move over one finger and raise my middle finger. I must also lower my index finger back to its "zero" position, because I want just 2, not 2 + 1 = 3. So 2 in binary is:


To get to 3, I've got a bit of breathing room. I can just raise my index finger, and add another. So 3 in binary is:


But to get to 4 I'm stuck again. I simply cannot fit another number in the right two fingers; I've used all of their possible combinations (00, 01, 10, 11). So to get to 4, I must move over one place. So I raise my ring finger and lower the other two:


Then 5 is easy:


And so forth:

0110 = 6 
0111 = 7 

(Now all of the right three positions have been used entirely, so I go another digit to the left: )

1000 = 8 
1001 = 9 
1010 = 10 
1011 = 11 
1100 = 12 
1101 = 13 
1110 = 14 
1111 = 15 

Note the absolutely regular pattern. It works visually, and it works when counting it out on your fingers. It's going to work on a Linotype 90-channel distributor bar, too. Every time you are forced to move left one position, you repeat the entire previous pattern. So to count from 0 to 3, you do this:

0 = 00 
1 = 01 
2 = 10 
3 = 11 

and then to count further from 4 to 7 you put a "1" in the 4's place and repeat the previous pattern:

4 = 100 
5 = 101 
6 = 110 
7 = 111 

3. Counting from Left to Right

So far, I've been counting from right to left, which is the way we ordinarily write numbers. So, numbering the places, counting in binary starts out like this:

8 4 2 1 
0 0 0 0 
0 0 0 1 
0 0 1 0 
0 0 1 1 

Of course, the order is just a convention. So long as we know what the order is, it can just as easily be left-to-right. So the following is an equally valid way of counting in binary:

1 2 4 8 
0 0 0 0 
1 0 0 0 
0 1 0 0 
1 1 0 0 

As it happens, while all computers treat numbers correctly as numbers, some of them lay out the binary digits of these numbers, in storage, from left to right while others lay them out from right to left. In what was originally a joke with reference to Jonathan Swift, the right-to-left layout is called "little-endian" (because the "little" end is on the right), while the left-to-right layout is called "big-endian."

4. Matrix Teeth (and the "90-Channel" Magazine)

Now, if I pull out a copy of Linotype's Useful Matrix Information (in any of various editions), somewhere in it will be a set of tables entitled "Tooth Combination Chart." (In the October 1937 edition, it's on page 16. In the undated but probably 1950s edition which is black with a gold matrix partial profile on the cover, it's page 18. There are equivalent charts in Linotype Machine Principles (p. 274) and elsewhere.) The first of these tables lists the channels of the standard "90-channel" Linotype magazine (which actually has 91 channels, numbered 0 through 90). It lists the channel number, the physical width of the channel, the character which normally is to be found in the channel, and finally the "Teeth in Combination."

This listing of the tooth combination for each channel position is at first confusing. It starts out:


Clearly, there is some pattern - but what is it?

The numbers, 1, 2, 3, 4, etc. refer to the tooth numbers. Linotype numbers the teeth on a matrix from 1 through 7 (there are 7 teeth on either side of each matrix, but since the left and right teeth of any pair are always the same this works out to 7, not 14, teeth per matrix; you just count one side).

(From Useful Matrix Information. (Brooklyn, NY: Mergenthaler Linotype Company, October 1937.) p. 14.)

So what they're saying is that the matrix teeth for the tooth positions identified are those which are present. Let's call that state "1". Matrix teeth which are not identified are absent; let's call that "0". So for a start, let's lay out all of the teeth and see which ones are present or absent. It starts out like this:

1 2 3 4 5 6 7 
0 1 0 0 0 0 0 magazine channel 0 = tooth combination "2" 
1 1 0 0 0 0 0 magazine channel 1 = tooth combination "1-2" 
0 0 1 0 0 0 0 magazine channel 2 = tooth combination "3" 
1 0 1 0 0 0 0 magazine channel 3 = tooth combination "1-3" 
0 1 1 0 0 0 0 magazine channel 3 = tooth combination "2-3" 
1 1 1 0 0 0 0 magazine channel 4 = tooth combination "1-2-3" 
0 0 0 1 0 0 0 magazine channel 5 = tooth combination "4" 

This starts to look very much like binary counting from left to right ("big-endian"), but there are some complications.

First, the tooth combination "0 0 0 0 0 0 0" (that is, no teeth at all) isn't indicated. Matrices with this tooth combination simply fall into the "quad box" (the Linotype term; Intertype calls it the "transfer channel quad box" or "quad tumbling box" {Sinclair. The Intertype. (1929) p. 238.}) This does, therefore, really count as a matrix combination (Sinclair indicates that it is used for logotypes, thin spaces, and other special characters). It doesn't show up in the "Tooth Combination Chart" tables because these matrices drop out before distribution, so they're never involved in getting put into a magazine channel.

Second, the tooth combination "1 0 0 0 0 0 0" simply isn't used.

Third, Ottmar Mergenthaler was a machinist, not a computer programmer. So he didn't indicate binary digit value; he just counted teeth. But this doesn't matter, since what is important is the pattern, and its recognition by the distributor bar - not the number it represents.

Putting it all together:

1 2 3 4 5 6 7 tooth number
1 2 4 8 16 32 64 binary digit placevalue (in decimal)
0 0 0 0 0 0 0 no teeth, falls in quad tray
1 0 0 0 0 0 0 tooth combination not used
0 1 0 0 0 0 0 magazine channel 0 (combination: "2")
1 1 0 0 0 0 0 magazine channel 1 (combination "1-2")
0 0 1 0 0 0 0 magazine channel 2 (combination "3")
1 0 1 0 0 0 0 magazine channel 3 (combination "1-3")
0 1 1 0 0 0 0 magazine channel 4 (combination "2-3")
1 1 1 0 0 0 0 magazine channel 5 (combination "1-2-3")
0 0 0 1 0 0 0 magazine channel 6 (combination "4")

So there it is. The tooth combinations are just counting in binary, from right to left.

That's of course a lovely pattern, but what does it do? The Distributor Bar answers that. Here's an illustration of the beginning of a standard 90-channel Distributor Bar.

(From Linotype Machine Principles. (Brooklyn, NY: Mergenthaler Linotype Company, 1940.) p. 270.)

Black lines indicate that a "rail" is present in the distributor bar to engage a matrix tooth, white space indicates that the rail is absent. The first part to the very left, which shows all seven rails, should be ignored - it just carries any matrix onto the Distributor Bar. The first detecting position is that marked "e 0" (magazine channel 0, which usually holds the extra 'e' matrices). If a matrix passing this position has only the "2" tooth present, it drops off. All other matrices which have reached this point have at least one other tooth present, so they continue on. In the next position ("e 1") matrices with only teeth 1 and 2 drop off; all others continue. This pattern continues through the end of the "90-channel" magazine (combination 3-4-5-7, which is "0011101" (left to right), which (again, reading left-to-right) is 92 (decimal). So the 91 channels of a "90-channel" magazine, numbered channel 0 through channel 90, correspond to tooth combinations from 2 through 92.

Note that this mechanism does also imply an order to the detection. Consider the fairly easy position for channel 2 ('t'), which has teeth with only tooth 3 present. In all positions previous to this which have in fact been encountered, there has been a rail for this tooth, so this matrix hasn't yet dropped off. But in fact this matrix will drop off in any rail pattern which is missing rail 3. So if, for example, the Distributor Bar order were somehow reversed to "e e a t", then this matrix would drop off at "a" (which would be wrong). The binary counting pattern assures that this cannot occur (which is why the exceptions to binary counting which will come up later are all the more interesting).

In case the whole concept of matrices dropping off the bar at certain rail/tooth combinations is still a bit murky, here are some diagrams to help. The first is Linotype's version, showing a matric for a lowercase 'l' as it encounters position 11 on the Distributor Bar (the diagram of the Bar, above, shows this location with an arrow):

(From Linotype Machine Principles. (Brooklyn, NY: Mergenthaler Linotype Company, 1940.) p. 270.)

Unfortunately, the drawing above, while accurate, isn't particularly clear. The equivalent Intertype drawing, reproduced below, is much clearer. It shows a matrix held on the distributor bar rails by its teeth, on the left, and the same matrix released at the appropriate location along the bar, on the right. (The matrix shown is a different one, 'v' rather than 'l', but the principle should be clear enough.)

(From The Intertype. (Brooklyn, NY: Intertype Corporation, 1943.) p. 272.)

For a partial reality check, here are images of actual matrices for "e t a o i n":

(These are straight out of a Linotype/Intertype Model X 90-Channel magazine as it ran at a small newspaper in rural Wisconsin. If you've never seen a Linotype mat before, I should perhaps note the obvious: that the white printing on them in the first image is something I did in the picture; it isn't on the mats themselves. The less vivid printing on the "narrow" side of the matrix, in the second image (the "reference" side), is what the operator would see. The distributor, of course, "sees" only the teeth. The study of the notches in this font of mats is a lesson in keeping a small newspaper running in adverse conditions, but that's a topic for a different Notebook.)

Below, I'll try to bring this together for an entire 90-channel magazine, in a single diagram using only ASCII characters (which should be cut-and-pastable anywhere, without losing formatting). In the text/diagram below, the first two lines are the tooth combination interpreted as a binary number (reported in decimal). Read the numbers vertically: 00, 01, 02, 03, ... Similarly, two lines of numbers near the bottom represent channel numbers in a "90-channel" magazine. These start out with "q" (for "quad tray"), which isn't in the magazine, and "x" (for not-used), which also isn't in the magazine. At the very bottom, I've shown some of the characters commonly (but not always) associated with this channel (there are variations not shown here, and I've used "." to represent untypable characters, but this should give a general sense of place relative, at least, to etaoin shrdlu). In the middle are the tooth combinations, with "T" repesenting the presence of a tooth, and "." representing its absense. Note how the pattern is pure binary counting, starting at 2.

0000000000111111 1111222222222233 3333333344444444 4455555555556666 tooth combo 
0123456789012345 6789012345678901 2345678901234567 8901234567890123 as a number 

xx.T.T.T.T.T.T.T .T.T.T.T.T.T.T.T .T.T.T.T.T.T.T.T .T.T.T.T.T.T.T.T 
xxTT..TT..TT..TT ..TT..TT..TT..TT ..TT..TT..TT..TT ..TT..TT..TT..TT 
xx..TTTT....TTTT ....TTTT....TTTT ....TTTT....TTTT ....TTTT....TTTT 
xx......TTTTTTTT ........TTTTTTTT ........TTTTTTTT ........TTTTTTTT 
xx.............. TTTTTTTTTTTTTTTT ................ TTTTTTTTTTTTTTTT 
xx.............. ................ TTTTTTTTTTTTTTTT TTTTTTTTTTTTTTTT 
xx.............. ................ ................ ................ 

qx00000000001111 1111112222222222 3333333333444444 4444555555555566 90-channel 
qx01234567890123 4567890123456789 0123456789012345 6789012345678901 magazine 
..eetaoinshrdluc mfwypvbgkqjxz... ...,.:;?.(|`!-.) .'*1234567890$.E 

6666667777777777 8888888888999 tooth combo 
4567890123456789 0123456789012 as a number 

.T.T.T.T.T.T.T.T .T.T.T.T.T.T. 
..TT..TT..TT..TT ..TT..TT..TT. 
....TTTT....TTTT ....TTTT....T 
........TTTTTTTT ........TTTTT 
................ TTTTTTTTTTTTT 
................ ............. 

6666666677777777 7788888888889 90-channel 
2345678901234567 8901234567890 magazine 

5. The 72-Channel Magazines, and Beyond

With the 72 channel magazine, the logical counting pattern breaks down and irregularities begin to appear. I'll stop the present Notebook here, then (because it's about counting in binary, really, not about distributor/magazine arrangements). In the next Notebook, I'll repeat the 90-channel section above, and continue with the 72-channel and other magazines.

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