Frequency temperament: Difference between revisions

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A '''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. 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/~~di vision~~, tempering out a comma means to equate it to 0 (the ~~additve~~ identity) instead of 1 (the multiplicative identity).

A '''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 ==

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).

The arithmetic equivalent of [[monzos]]s is, in a sense, [https://en.wikipedia.org/wiki/Positional_notation positional notation] systems–monzos represent numbers as a product of the powers of the base elements (primes), whereas positional notation systems represent numbers as a sum of the multiples of the base elements.

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-A '''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. 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/di vision, tempering out a comma means to equate it to 0 (the additve identity) instead of 1 (the multiplicative identity).
+A '''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 ==
+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).
+The arithmetic equivalent of [[monzos]]s is, in a sense, [https://en.wikipedia.org/wiki/Positional_notation positional notation] systems–monzos represent numbers as a product of the powers of the base elements (primes), whereas positional notation systems represent numbers as a sum of the multiples of the base elements.