TouchTone Odds and Ends
The DTMF dialing system traces its roots to a technique AT&T developed in the 1950s called MF (MultiFrequency) which was deployed within the AT&T telephone network to direct calls between switching facilities using inband signaling. In the early 1960s, a derivative technique was offered by AT&T through its Bell System telephone companies as a "modern" way for network customers to place calls. In AT&Ts Compatibility Bulletin No. 105, AT&T described the product as "a method for pushbutton signaling from customer stations using the voice transmission path." The consumer product was marketed by AT&T under the registered trade name TouchToneĀ®. Other vendors of compatible telephone equipment called this same system "Tone" dialing or "DTMF". — DTMF Reference
Most consumer telephones these days have 12 buttons: the numerals 1 through 0, plus star (*) and pound (#). However, 16button telephones were common in the AUTOVON military phone network, allowing for multilevel priority and preemption. But what about more than 16 buttons?
There is a "fifth column" DTMF. Whether the actual frequencies are specified in any Bell System Practice or Bell Labs Record article is uncertain (and probably unlikely), but the tones can be easily deduced mathematically.
Officially, this is the 16tone pattern that has been standardized for DTMF:
1209  1336  1477  1633  
Frequencies 
697  1 
2 
3 
A 

770  4 
5 
6 
B 

852  7 
8 
9 
C 

941  0 
D 
The tones have been carefully selected to minimize harmonic interference and the probability that a pair of high and low tones will be simulated by the human voice, thus protecting network control signaling. — Engineering and Operations in the Bell System, 2e. (1984) p. 276
Mathematics
There are three mathematical models which describe the determination of the tones used for DTMF tones. Below, we use this model to predict the "fifthcolumn" DTMF tone mathematically.
 This method is the one officially used by Bell Telephone Labs. The ratio used is approximately 21/19. This is about the ratio of adjacent consecutive tones in any row or column:
1336/1209 = 1.1050
1477/1336 = 1.1055
1633/1477 = 1.1056
Taking the average of those (3) slightly different multipliers comes to 1.1054
Then:
1209 x 1.1054^1 = 1336.43
1209 x 1.1054^2 = 1477.2
1209 x 1.1054^3 = 1632.8
1209 x 1.1054^4 = 1804.89  This method involves using the differences of differences between adjacent consecutive tones.
Jump 1: 1209 → 1336 = 127
Jump 2: 1336 → 1477 = 141
Jump 3: 1477 → 1633 = 156
So:
Jump 1 → Jump 2 = 14
Jump 2 → Jump 3 = 15
Well, it seems intuitive then that Jump 3 → Jump 4 ought to be 16, following this linear pattern.
In which case, the fifth column frequency ought to be 1633 + (156 + 16) = 1633 + 172 = 1805.  This method involves differences of differences again, but this time with square roots.
Taking square roots:
1209  34.770
1336  36.551
1477  38.431
1633  40.410
Differences are: 1.781,1.88,1.979
Differences of differences are: 0.099, 0.099
So, putting it back together:
1.979 + 0.099 = 2.078
40.410 + 2.078 = 42.488
42.488 squared = 1805.23 ~= 1805 Hz
Here are some comments from Chuck, who has been experimentally using 5thcolumn DTMF for two decades:
There have been some people who have stated that it's never listed in any of the BSPs nor W.E. nor Bell System records. But it works. So why not use it? It's an easy project. Just locate the rightmost column switch, clip it free with a small wire cutters. Then solder on some fine wires (I like to use #30 gauge wirewrap wire). Bring those wires out to (2) SPDT push buttons in series so that the 3rd column on the pad gets reassigned by the buttons to either the 4th or 5th column. — Chuck R.
And there you have it! Forget about four DTMF columns, let's have five!
There have been a few experimental attempts at using this fifth DTMF column. Frequency measurements have found this tone to be around 1871 Hz and 1890 Hz, well above the predicted 1805 Hz. What gives?
The deal is that if this had been planned as a tone tap, it could have been targeted at 1805 per the pattern. But it's just the end of the coil, so it makes a tone which is around 70 to 80 Hz greater than the math predicts. — Chuck
So, a supposedly almost unknown fifth DTMF column, although consumer telephones were not really designed to produce it. But why stop there?
It's not common knowledge that the tone matrix is larger than four rows and four columns. If you extend the tone mathematical sequence (it's a 21/19 ratio) it works out to an 8 by 8 matrix. Yes, the Bell System / Western Electric engineers designed the TouchTone matrix to fit 64buttons into the voice band and only the top left corner was ever implemented. — WA6ILQ
So, how to determine these 8 column tones and 8 row tones (4 of each of which we already know for certain)? Pick any mathematical method you like. We'll use the first one, since it's the "official" one. We simply start with 697 and 1209 Hz and keep multiplying by 21/19:
1209  1336.263158  1476.922438  1632.387957  1804.218269  1994.135981  2204.045032  2436.049772  
Frequencies 
697  
770.3684211  
851.4598338  
941.0871847  
1040.148994  
1149.638361  
1270.652926  
1404.405865 
Something's not quite right here, though. If you round the result for 851, you get 851 Hz, instead of the actual 852 Hz. Likewise, if you round the result for 1632, you get 1632 Hz, not 1633 Hz. So, what about multiplying instead by the ratios 1336/1209 and 770/697 instead? That actually leads to even more error.
For the row tones, if we benchmark at 852 Hz instead of at 697 Hz, 941 Hz rounds up to 942 Hz, which is still 1 cycle per second too high. In other words, using ratios of any adjacent tones introduces error. If we use 1.1054 for column tones, then, we're bound to get better results. And we do.
Doing the same for the row tones, we get 1.104735, 1.106494, and 1.10446. The average of these is 1.105229. If we then perform similar calculations as in #1 for the row tones, we get the following matrix:
1209  1336.4286  1477.288174  1632.994348  1805.111952  1995.370752  2205.682829  2438.1618  
Frequencies 
697  
770.344613  
851.4598338  
940.9999352  
1040.020417  
1149.460726  
1270.417329  
1404.102074 
851 is still wrong. It should be 852, not 851, as this method predicts. However, the other results are more accurate. In particular, 1633 and 941 Hz are almost exactly predicted. All the known column tones are correctly predicted. We know that 851 should be 852, and if we correct for this, we seem to have something that "ought" to be right. The column tones skew higher than before due to compounded error while the row tones skew lower than before due to compounded error.
What would such an 8column/8row keypad actually look like? This seems like the most logical approach:
1209  1336  1477  1633  1805  1995  2206  2438  
Frequencies 
1 
2 
3 
A 
E 
I 
M 
Q 

4 
5 
6 
B 
F 
J 
N 
R 

7 
8 
9 
C 
G 
K 
O 
S 

0 
D 
H 
L 
P 
T 

1040  U 
Y 

1149  V 
Z 

1270  W 

1404  X 
Other Odds & Ends
Western Electric 2500 keypads came in two main varieties. Some of them had buttons you could push in all the way while others only clicked and didn't go in. Many people didn't like the change and so WECo reversed course and went back to the old buttons at some point. At least, according to a discussion on the TCI list in 2017 or 2018 or on CRPF around this time which I can't locate anymore. If you have more details, please contact me!