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.

== Theory ==

Much like how regular temperaments are based on monzos and vals, arithmetic temperaments are based on their arithmetic conuterparts. 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 [[Factorial_number_system|"factorial number system"]] where the place values are factorials and reciprocals of them, ~~since~~ it ~~exactly~~ can represent any rational number in a finite string ~~like monzos~~.

Much like how regular temperaments are based on monzos and vals, arithmetic temperaments are based on their arithmetic conuterparts. 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 [[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.

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 to equate it to 0 (the additive identity) instead of 1 (the multiplicative identity).

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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.
 == Theory ==
-Much like how regular temperaments are based on monzos and vals, arithmetic temperaments are based on their arithmetic conuterparts. 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 [[Factorial_number_system|"factorial number system"]] where the place values are factorials and reciprocals of them, since it exactly can represent any rational number in a finite string like monzos.
+Much like how regular temperaments are based on monzos and vals, arithmetic temperaments are based on their arithmetic conuterparts. 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 [[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.
 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 to equate it to 0 (the additive identity) instead of 1 (the multiplicative identity).