How likely is it that the sequence yielding the temperament properties described above would occur by chance?
The structure of the spiral consists of four contiguous ones, three contiguous zeros and six contiguous twos:
1 1 1 1 0 0 0 2 2 2 2 2 2
Combinatorics shows that the number of possible sequences of length 13 is 1'594'323. So assuming a random decoration was drawn with thirteen symbols of three kinds, the probability of the spiral at the top of the cover page of Das Wohltemperierte Clavier occurring by chance is 1 in 1'594'323 or less than one chance in a million. If the drawing were random, however, there is no reason why it should contain exactly thirteen symbols. If we assume, for example, that a random decoration is equally likely to have between ten and fifteen symbols, where each symbol is one of three kinds, then the total number of patterns is:
59'049 + 177'147 + 531'441 + 1'594'323 + 4'782'969 + 14'348'907 = 21'493'836
So a better estimate is that the probability of the needed sequence of thirteen items occurring by chance is less than one in twenty million.