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 == | == 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. 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. | |||
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). | 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). | ||
Revision as of 08:53, 19 May 2023
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 AFSs as regular temperaments are to ETs.
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, 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. 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.
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).