Frequency temperament: Difference between revisions
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'''Arithmetic temperaments''' are the arithmetic counterpart to [[regular temperament]]s. Whereas regular temperaments are created by reducing integer powers of a [[generator]], an arithmetic temperament is created by reducing integer multiples of a generator. The n-th interval in an arithmetic temperament is given by ng mod (p - 1) + 1, where g is the generator and p is the period. | '''Arithmetic temperaments''' are the arithmetic counterpart to [[regular temperament]]s. Whereas regular temperaments are created by reducing integer powers of a [[generator]], an arithmetic temperament is created by reducing integer multiples of a generator. The n-th interval in an arithmetic temperament is given by ng mod (p - 1) + 1, where g is the generator and p is the period. | ||
For example, these will be the intervals of an arithmetic temperament with a generator of 1.29 (440 cents) and period [[2/1]]: | |||
<pre> | |||
1.29 ≈ 440¢ | |||
2*1.29 - 1 = 1.58 ≈ 791¢ | |||
3*1.29 - 2 = 1.87 ≈ 1084¢ | |||
4*1.29 - 4 = 1.16 ≈ 257¢ | |||
5*1.29 - 5 = 1.45 ≈ 643¢ | |||
6*1.29 - 6 = 1.74 ≈ 960¢ | |||
... | |||
</pre> | |||
== List of arithmetic temperaments == | == List of arithmetic temperaments == | ||
* [[Sqrttwo]] | * [[Sqrttwo]] | ||
Revision as of 05:18, 2 March 2023
Arithmetic temperaments are the arithmetic counterpart to regular temperaments. Whereas regular temperaments are created by reducing integer powers of a generator, an arithmetic temperament is created by reducing integer multiples of a generator. The n-th interval in an arithmetic temperament is given by ng mod (p - 1) + 1, where g is the generator and p is the period.
For example, these will be the intervals of an arithmetic temperament with a generator of 1.29 (440 cents) and period 2/1:
1.29 ≈ 440¢ 2*1.29 - 1 = 1.58 ≈ 791¢ 3*1.29 - 2 = 1.87 ≈ 1084¢ 4*1.29 - 4 = 1.16 ≈ 257¢ 5*1.29 - 5 = 1.45 ≈ 643¢ 6*1.29 - 6 = 1.74 ≈ 960¢ ...