I've been playing with DTMF frequencies and the Goertzel algorithm, and would like to know exactly HOW the DTMF frequencies were determined. I understand that they were chosen such that none of the frequencies would be equal to sum and difference frequencies, or intermodulation products, but I can't find any information on whether the frequencies were simply determined through experimentation, mathematical calculation, etc… does anyone know more about this?

Also, I would like to understand how to extend the two sets of frequencies to two sets of 8 frequencies, or 3 sets of 4 frequencies, such that it's possible to represent 64 keys. I'm not sure whether it would be easier to work with 3 sets of 4 versus 2 sets of 8 each. I guess the analysis would be easier if just using 2 sets of 8? There would be less intermodulation products to keep in mind …

Thanks! B

EDIT: if it's too difficult to track back how this was achieved, i would like to know what the best way is to find 2 sets of frequencies, giving 64 unique combinations, and such that if you pick 2 frequencies from each of the 2 sets randomly, and mix the sine waves together, and put them through some non linear distortion, they are still easily detectable using goertzel, and the other frequencies are not "hit" (their bins remain at a normal background noise level)

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    $\begingroup$ Perhaps someone who worked on the original Touch-Tone spec would know. Otherwise, you may need to find some obscure ATT memos or such. And there is info in other sources ( eg: repeater-builder.com/tech-info/dtmf/dtmf.html or: nemesis.lonestar.org/reference/telecom/signaling/dtmf.html ) $\endgroup$ Commented Dec 3, 2013 at 3:19
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    $\begingroup$ As noted in the first link above: 1) TouchTone was designed to fit an 8x8 matrix, and 2) the frequencies follow a 21/19 ratio. $\endgroup$ Commented Dec 4, 2013 at 4:24
  • $\begingroup$ @KevinMcGee I checked the above links but did not see an 8x8 matrix? $\endgroup$
    – b20000
    Commented Dec 4, 2013 at 5:13
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    $\begingroup$ The author states: "And 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 64-buttons into the voice band and only the top left corner was ever implemented." $\endgroup$ Commented Dec 4, 2013 at 17:08

3 Answers 3


From page 293 of Engineering and Operations in the Bell System (1982-1983) by AT&T Bell Labs:

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.

P.S. Yes, I know this answer is 6 years late, I just happened to be researching for the answer and I thought this could help.


See Bell System Technical Journal 39: 1. January 1960: Pushbutton Calling with a Two-Group Voice-Frequency Code. (Schenker, L.).


This is a question that has actually captivated me for some time.

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.

There are three mathematical models which describe the determination of the tones used for DTMF tones. Here's how one could predict the 5th-column DTMF tone from that:

  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

  1. 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.

  1. 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

A few phone collectors have actually tried extending WECo 2500 sets to do this. Experimental results come out to an average around ~1880 Hz or so. But the phones weren't designed for fifth-column DTMF, so it's possible that's why there's a mismatch between the theory and reality.

Here's a project you could try for yourself if you have a standard (model 2500) Touch-Tone phone:

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.

So, to answer your question, what would the extension to 8 x 8 look like? This is my best guess:

enter image description here

After exhausting the alphabet, it's anyone's guess what the remaining keys ought to be used for! There's exactly enough leftover to repeat the alphabet all over. I'll be the first to say it would be nice to be able spell things out using DTMF over the phone (e.g. "Please enter your name:" beep...)

Source - PhreakNet: Touch-Tone Odds and Ends


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