Frequency temperament

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, vals, and mappings, 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 (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" 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).