Frequency temperament: Difference between revisions

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An '''~~arithmetic~~ temperament''' is a type of [[temperament]] ~~which generates a, when period-reduced~~, ~~arithmetic progression of frequency. This is~~ in contrast to [[regular temperaments]] which generate ~~a geometric progression instead~~. ~~Arithmetic~~ 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 ==~~

Like how regular temperaments are based on monzos, vals, and mappings, arithmetic temperaments are based on their arithmetic counterparts. The arithmetic 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.

~~Such arithmetic~~ monzos naturally ~~extend~~ into the ~~arithmetic~~ equivalent of mappings, using the place values of the selected positional numeral system as the basis elements instead of primes. Thus, ~~arithmetic~~ temperaments can "[[temper out]]" commas in a similar way to regular temperaments, but because period reduction is now performed through addition/subtraction rather than multiplication/division, tempering out a [[comma]] means mapping it to 0 (the additive identity) instead of 1 (the multiplicative identity).

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 serve as a frequency-based equivalent 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.

This notion of the frequency equivalent of monzos naturally extends into the frequency equivalent of mappings, using the place values of the selected positional numeral system as the basis elements instead of primes. Thus, frequency temperaments can "[[temper out]]" commas in a similar way to regular temperaments, but because period reduction is now performed through addition/subtraction rather than multiplication/division, tempering out a [[comma]] means mapping it to 0 (the additive identity) instead of 1 (the multiplicative identity).

== Examples ==

* [[Neutrino]]

* [[Sqrttwo]]

[[Category:Frequency temperaments]]

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-An '''arithmetic  temperament''' is a type of [[temperament]] which generates a, when period-reduced, arithmetic progression of frequency. This is in contrast to [[regular temperaments]] which generate a geometric progression instead. Arithmetic 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 ==
-Like how regular temperaments are based on monzos, vals, and mappings, arithmetic temperaments are based on their arithmetic counterparts. The arithmetic 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.
-Such arithmetic monzos naturally extend into the arithmetic equivalent of mappings, using the place values of the selected positional numeral system as the basis elements instead of primes. Thus, arithmetic temperaments can "[[temper out]]" commas in a similar way to regular temperaments, but because period reduction is now performed through addition/subtraction rather than multiplication/division, tempering out a [[comma]] means mapping it to 0 (the additive identity) instead of 1 (the multiplicative identity).
+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 serve as a frequency-based equivalent 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.
+This notion of the frequency equivalent of monzos naturally extends into the frequency equivalent of mappings, using the place values of the selected positional numeral system as the basis elements instead of primes. Thus, frequency temperaments can "[[temper out]]" commas in a similar way to regular temperaments, but because period reduction is now performed through addition/subtraction rather than multiplication/division, tempering out a [[comma]] means mapping it to 0 (the additive identity) instead of 1 (the multiplicative identity).
+== Examples ==
+* [[Neutrino]]
+* [[Sqrttwo]]
+[[Category:Frequency temperaments]]