From: Engineers Garage
Dual Tone Multi-Frequency:
- $f_1$ = 697Hz, 770Hz, 852Hz, 941Hz.
- $f_2$ = 1209Hz, 1336Hz, 1477Hz, 1633Hz.
A key press on a DTMF keypad means a combination of $f_1$ and $f_2$ (Dual Tone) frequencies are heard at the same time. $f_1$ (rows) and $f_2$ (columns) have four frequencies (Multi-Frequency) associated with them, although 1633Hz was only used in military phones. $f_2$ frequencies are slightly louder than $f_1$ frequencies to compensate for high-frequency roll off of voice audio systems.
Frequencies were selected so harmonics would not be interpreted as a fundamental frequency. 2nd harmonic of 697Hz is 1394Hz, which is midway between 1336Hz and 1477Hz. Harmonics of $f_1$ can not be intrepreted as a fundamental $f_2$ frequency.
The combination of 941Hz and 1209Hz means that the sum (2150Hz) and difference (268Hz) are heard at the same time. 697Hz and 1477Hz means 780Hz and 2174Hz. None of these can be intrepreted as $f_1$ or $f_2$ frequencies.
Adjacent $f_1$ and $f_2$ frequencies were selected to have a 21/19 ratio (1.10), which is slightly less than a whole tone (musical reference Music and Noise) and can vary no more than ±1.5% (or ±1.8% depending on sources) from their nominal frequency.
The range of human hearing is 20Hz to 20kHz, most sensitive at 2 to 4kHz and the normal voice range is about 200Hz to 3.5kHz. Need 8 frequencies for 16 keys within 1950's phone's 4kHz bandwidth.
Human speech is unable to produce the combined tones, so you could implement modes, where users could talk and use touch tones to cause the system to react (operator).
As for the actual frequencies. From: Engineering and Operations in the Bell System, which I stole from how were DTMF frequencies determined exactly and how can I extend them?
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.
The nominal voiceband channel is defined as 4 kHz although the speech signal is
essentially bandlimited to between 200 Hz and 3.5 kHz. The additional bandwidth allows for a guard band on either side of the speech signal to lessen interference between channels.