Frequency temperament: Difference between revisions
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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). | ||
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 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. | ||
Revision as of 08:52, 19 May 2023
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 AFSs as regular temperaments are to ETs.
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 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.