Inspired by Henry Baker's musing about modem tones: Each Touch Tone button on a telephone sends two simultaneous audio frequencies through the phone line, one associated with the row it's on, and one associated with the column it's on. There are four rows and four columns (though only the first three columns exist on most telephones) for a total of eight frequencies. The frequencies were chosen such that the sums or differences of any two of the eight frequencies are as far as possible from any of the eight frequencies. Similarly with whole-number multiples ("harmonics") of the eight frequencies or of their sums or differences. This is to prevent the phone company equipment from getting confused, since non-linearities in the circuits could result in such sum and difference frequencies appearing. This is probably why Touch Tones are so unmusical. Music relies on frequencies with small whole-number ratios, almost the exact opposite of the Touch Tone criteria. (Or did they only concern themselves with sums or differences of row and column frequencies? Two different row frequencies, or two different column frequencies, should never appear together.) Also, the frequencies should be between 400 and 3400 Hz, so they'll fit in a phone line's bandwidth. See https://en.wikipedia.org/wiki/Dual-tone_multi-frequency_signaling for more information, including the eight frequencies. Can you do better on selecting eight frequencies with those constraints? What about other numbers of frequencies? Is there a good algorithm for selecting them?