Frequency temperament: Difference between revisions
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'''Arithmetic temperaments''' are the arithmetic counterpart to [[rank- | '''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. | 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. | ||
Revision as of 01:53, 2 March 2023
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. 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.