Frequency temperament: Difference between revisions
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'''Arithmetic temperaments''' are the arithmetic counterpart to [[regular temperament]]s. | '''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. | ||
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 | 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 | ||
Revision as of 05:10, 2 March 2023
Arithmetic temperaments are the arithmetic counterpart to regular temperaments. Whereas regular temperaments are created by taking integer powers of a generator, an arithmetic temperament is created by taking integer multiples of a generator.
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.