Why do push button telephones use dual-tone for signalling?
The two reasons are simple: 


Eight frequencies are easier to discriminate with simple analog electronics - banks of bandpass filters or even vibrating reeds - than sixteen frequencies.
The equally tempered scale is too close to the natural scale, which has simple fractional relationships between the frequencies.


Consider that a phone line may be highly distorted : the second harmonic is one octave above the fundamental, and the third harmonic of a note is an octave plus a fifth. If you used more musical intervals between dialing tones, harmonic distortion could result in dialing the wrong number.

The frequencies were chosen (citation needed, no doubt - here for example ) to reduce or eliminate the possibility of harmonic or intermodulation distortion between tones being mis-detected as the wrong number.


  The tone frequencies, as defined by the Precise Tone Plan, are selected such that harmonics and intermodulation products will not cause an unreliable signal. No frequency is a multiple of another, the difference between any two frequencies does not equal any of the frequencies, and the sum of any two frequencies does not equal any of the frequencies. The frequencies were initially designed with a ratio of 21/19, which is slightly less than a whole tone. The frequencies may not vary more than ±1.5% from their nominal frequency, or the switching center will ignore the signal.