12 is a special number, it’s a highly composite number but it’s also a small number, this means that a large percentage of the numbers less than 12 divide it. T

12 is a special number, it’s a highly composite number but it’s also a small number, this means that a large percentage of the numbers less than 12 divide it. This gives the group Z12 some interesting properties, there’s a wide range of element orders. Tritones come in pairs, major thirds come in 4 sets of 3 thirds, minor thirds come in 3 sets of diminished seventh chords, etc, major seconds form the two whole tone scales. Fourths (5) and fifths (7) don’t divide 12, so they generate the group, producing the cycle of fourths and fifths. If we look at the groups next door, Z11 and Z13, they are of prime order, so every element generates the group => there is a cycle of “fifths” for each interval. Beautifully symmetric, perhaps, but less structure to enjoy.