Frequency temperament: Difference between revisions

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-'''Arithmetic temperaments''' are the arithmetic counterpart to [[regular temperament]]s. Whereas regular temperaments are created by taking integer powers of a [[generator]], an arithmetic temperament is created by taking integer multiples of a generator.
+'''Arithmetic temperaments''' are the arithmetic counterpart to [[regular temperament]]s. Whereas regular temperaments are created by reducing integer powers of a [[generator]], an arithmetic temperament is created by reducing integer multiples of a generator. The nth step of an arithmetic temperament is given by x<sub>n</sub> = ng mod 1 + 1, where g is the generator andpn is the period.
+== Example ==
+Cconsider an arithmetic temperament with a generator of 1.29 (440 cents) and period [[2/1]].
-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
-temperament. If a just interval is used as a generator for an arithmetic temperament, it will eventually start repeating the same intervals.
 == List of arithmetic temperaments ==