Frequency temperament

WIP

Arithmetic temperaments are the arithmetic counterpart to regular temperaments.

Rank-2 arithmetic temperaments

A standard rank-2 temperament has a generator interval and a period interval, and new intervals are produced by taking powers of the generator, and then reducing them logarithmically to the range from 1/1 to the period. But in arithmetic temperaments, new intervals are produced by taking multiples of the generator and reducing them arithmetically.

For example, consider an arithmetic temperament with a generator of 1.29 (440 cents) and period 2/1. If we want to add a third interval, multiply 1.29 by 2 to obtain 2.58. Since 2.58 is outside of an octave, subtract 1 to get 1.58. To add a fourth interval, multiply 1.29 by 3 and subtract 2 to get 1.87. We can continue generating new intervals this way like in a normal rank-2 temperament. If a just interval is used as a generator for an arithmetic temperament, it will eventually repeat.