There must be a mathematical reason, I've found one that makes sense and is easily proven. I can't find logical reasons to suggest that it doesn't work. I've made a DVD and book about it. www.pianodiy.com.au. I would like to make up some more mathematical models on the subject too.
closed as off topic by Sam, Asaf Karagila, Qiaochu Yuan Feb 4 '12 at 22:36
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No, it's just tradition. There's some interesting mathematics going into why there are 12 keys in an octave, but there's no intrinsic reason for the upper and lower limits of a standard piano keyboard.
There are piano makers that make models with more than 88 keys, extending the keyboard to the left, and more because the extra strings add resonance to the sound than because playing the extra keys is very useful.
In the other direction, electronic keyboards (those that do not aspire to be "digital pianos") often have only 61 keys.