Touch-Tone Odds and Ends

The DTMF dialing system traces its roots to a technique AT&T developed in the 1950s called MF (Multi-Frequency) which was deployed within the AT&T telephone network to direct calls between switching facilities using in-band 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 Touch-Tone®. 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, 16-button telephones were common in the AUTOVON military phone network, allowing for multi-level 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 16-tone pattern that has been standardized for DTMF:

Frequencies (Hz)
High-Group Frequencies
1209 1336 1477 1633
Low-Group
Frequencies
 697
 
1
ABC
2
DEF
3
FO
A
770
GHI
4
JKL
5
MNO
6
F
B
852
PRS
7
TUV
8
WXY
9
I
C
941
*
OPER
0
#
P
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 "fifth-column" DTMF tone mathematically.

  1. 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
  2. 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.
  3. 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 5th-column 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 right-most 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 re-assigned 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 Touch-Tone matrix to fit 64-buttons 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:

Frequencies (Hz)
High-Group Frequencies
1209 1336.263158 1476.922438 1632.387957 1804.218269 1994.135981 2204.045032 2436.049772
Low-Group
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:

Frequencies (Hz)
High-Group Frequencies
1209 1336.4286 1477.288174 1632.994348 1805.111952 1995.370752 2205.682829 2438.1618
Low-Group
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 8-column/8-row keypad actually look like? This seems like the most logical approach:

Frequencies (Hz)
High-Group Frequencies
1209 1336 1477 1633 1805 1995 2206 2438
Low-Group
Frequencies
697
 
1
ABC
2
DEF
3
FO
A

E

I

M

Q
770
GHI
4
JKL
5
MNO
6
F
B

F

J

N

R
852
PRS
7
TUV
8
WXY
9
I
C

G

K

O

S
941
*
OPER
0
#
P
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!