Frequency temperament

WIP

Arithmetic temperaments are the arithmetic counterpart to regular temperaments.

Rank-1 arithmetic temperaments

Rank-1 arithmetic temperaments correspond to ADO systems much like how rank-1 temperaments correspond to EDO systems.

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 generator 9/7 and period 2/1. If we want to add a third interval, then multiply 9/7 by 2 to obtain 18/7. Since 18/7 is greater than an octave, subtract 1 to get 11/7. To get a fourth interval, multiply 9/7 by 3 to get 27/7, then subtract 2 to get 13/7. In contrast to a logarithmic rank-2 temperament, we can only produce 7 intervals this way (creating 7ado) before the intervals will start repeating.