Frequency temperament: Difference between revisions
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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. | 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. | ||
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 rank-2 temperament. If a just interval is used as a generator for an arithmetic temperament, it will eventually repeat. | 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 rank-2 temperament. If a just interval is used as a generator for an arithmetic temperament, it will eventually repeat the same intervals. | ||
Revision as of 02:10, 2 March 2023
WIP
Arithmetic temperaments are the arithmetic counterpart to regular temperaments.
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.
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 rank-2 temperament. If a just interval is used as a generator for an arithmetic temperament, it will eventually repeat the same intervals.