Frequency temperament: Difference between revisions

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'''Arithmetic temperaments''' are the arithmetic counterpart to [[regular temperament]]s.

A regular 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.

A regular 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

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 '''Arithmetic temperaments''' are the arithmetic counterpart to [[regular temperament]]s.
-A regular 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.
+A regular 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