Frequency temperament: Difference between revisions

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A '''frequency temperament''' is a type of [[temperament]] based on [[frequency]], in contrast to [[regular temperaments]] which are based on [[pitch]]. They generate [[frequency MOS]] scales. Frequency temperaments are to [[AFS]]s as regular temperaments are to [[ET]]s.

A '''frequency temperament''' is a type of [[temperament]] based on frequency, in contrast to [[regular temperaments]] which are based on [[pitch]]. They generate [[frequency MOS]] scales. Frequency temperaments are to [[AFS]]s as regular temperaments are to [[ET]]s.

== Theory ==

Frequency temperaments are based on the frequency counterparts of monzos, vals, and mappings. The frequency equivalent of [[monzos]] is, in a way, [https://en.wikipedia.org/wiki/Positional_notation positional numeral systems] like the decimal or binary system–monzos represent numbers as a product of the powers of the base elements (primes), whereas positional numeral systems represent numbers as a sum of the multiples of the base elements (place values). The only major difference is that, in monzos, the power a prime can be raised to is unlimited, whereas in positional numeral systems, the multiplying factors (digits) are restricted to a certain range. Theoretically, any positional numeral system could be used as a substitute for monzos, but the best option would likely be the [https://en.wikipedia.org/wiki/Factorial_number_system "factorial number system"] where the place values are factorials and reciprocals of them, because, like monzos, it can represent any rational number exactly in a finite string.

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-A '''frequency  temperament''' is a type of [[temperament]] based on [[frequency]], in contrast to [[regular temperaments]] which are based on [[pitch]]. They generate [[frequency MOS]] scales.  Frequency temperaments are to [[AFS]]s as regular temperaments are to [[ET]]s.
+A '''frequency  temperament''' is a type of [[temperament]] based on frequency, in contrast to [[regular temperaments]] which are based on [[pitch]]. They generate [[frequency MOS]] scales.  Frequency temperaments are to [[AFS]]s as regular temperaments are to [[ET]]s.
 == Theory ==
 Frequency temperaments are based on the frequency counterparts of monzos, vals, and mappings. The frequency equivalent of [[monzos]] is, in a way, [https://en.wikipedia.org/wiki/Positional_notation positional numeral systems] like the decimal or binary system–monzos represent numbers as a product of the powers of the base elements (primes), whereas positional numeral systems represent numbers as a sum of the multiples of the base elements (place values). The only major difference is that, in monzos, the power a prime can be raised to is unlimited, whereas in positional numeral systems, the multiplying factors (digits) are restricted to a certain range. Theoretically, any positional numeral system could be used as a substitute for monzos, but the best option would likely be the [https://en.wikipedia.org/wiki/Factorial_number_system "factorial number system"] where the place values are factorials and reciprocals of them, because, like monzos, it can represent any rational number exactly in a finite string.