# Clock arithmetic

Clock arithmetic is a term sometimes used for modular arithmetic when the modulus is 12. It refers to the behaviour of addition of hours on a clock face: for example, if we begin at 7 o'clock and add 8 hours, then rather than ending at 15 o'clock (as in usual addition), we are at 3 o'clock. Likewise, if we start at noon and count off 7 hours three times ([itex]3 \times 7[itex]), we end up at 9 o'clock (rather than 21). Essentially, when we reach 12, we start over; 12 is called the modulus, hence the name "modular arithmetic." We can easily pretend that the clock face contains any number of hours, and calculate according to the new modulus.

However, unlike clock faces, numbers in modular arithmetic may start with any number. Musical set theory finds it convenient to use mod 12: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11; to represent 12-tone equal temperament.

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