Frequency temperament: Difference between revisions

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

~~== Rank-2 arithmetic temperaments ==~~

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 ~~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 ~~a rank-2~~ arithmetic temperament, it will eventually start repeating the same intervals.

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.

=== List ~~===~~

== List ==

~~==== Rank-2 temperaments ==~~==

* [[Euler]]

@@ Line 3: / Line 3: @@
 '''Arithmetic temperaments''' are the arithmetic counterpart to [[regular temperament]]s.
-== Rank-2 arithmetic temperaments ==
+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 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 a rank-2 arithmetic temperament, it will eventually start repeating the same intervals.
+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.
-=== List ===
+== List ==
-==== Rank-2 temperaments ====
 * [[Euler]]