Implementation [of sound] ListPlay[data] in Mathematica generates sound output by treating the elements of data as successive samples in the waveform of the sound, typically with a default sample rate of 8000 Hz.

Chords Two pure tones played together exhibit beats at the difference of their frequencies—a consequence of the fact that Sin[ ω 1 t] + Sin[ ω 2 t]  2 Sin[1/2( ω 1 + ω 2 ) t] Cos[( ω 1 - ω 2 ) t] With ω ≃ 500 Hz , one can explicitly hear the time variation of the beats if their frequency is below about 15 Hz, and the result is quite pleasant. … The mechanics of the ear imply that if two tones of reasonable amplitude are played together, progressively smaller additional signals will effectively be generated at frequencies Abs[n 1 ω 1 ± n 2 ω 2 ] .

However, if one uses the function to generate a score—say playing a note at the position of each peak—then no such simplicity can be recognized.

This yields a chord such as Play[Evaluate[Apply[Plus, Flatten[Map[Sin[1000 # t] &, N[2 1/12 ]^Position[list, 1]]]]], {t, 0, 0.2}] A sequence of such chords can sometimes provide a useful representation of cellular automaton evolution.

[Sounds based on] musical scores Instead of taking a sequence to correspond directly to the waveform of a sound, one can consider it to give a musical score in which each element represents a note of a certain frequency, played for some specific short time.

There are certainly some tasks—such as playing chess or doing algebra—that at one time were considered indicative of human-like thinking, but which are now routinely done by computer.

Properties of numbers and certain elementary aspects of number theory have also always played a central role in amateur and recreational mathematics.

And here again computer technology played a crucial role.

One issue—beyond the obvious fact that sounds cannot be included directly in a printed book—is that while one can study the details of a picture at whatever pace one wants, a sound is in a sense gone as soon as it has finished playing.

(A simpler game—certainly played since antiquity—is Penny Matching or Evens and Odds, with m = {{1, -1}, {-1, 1}} .)