Frequency temperament: Difference between revisions

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'''Arithmetic temperaments''' are the arithmetic counterpart to [[rank-~~2 temperament~~]]s. A 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.

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

== 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. Whereas ra

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.

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-'''Arithmetic temperaments''' are the arithmetic counterpart to [[rank-2 temperament]]s. A 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.
+'''Arithmetic temperaments''' are the arithmetic counterpart to [[regular temperament]]s.
+== 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. Whereas ra
 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.