Frequency temperament

A frequency temperament is a type of temperament based on frequency, in contrast to regular temperaments which are based on pitch. They generate frequency MOS scales. Frequency temperaments are to AFSs as regular temperaments are to ETs.

Frequency temperaments are based on the frequency counterparts of monzos, vals, and mappings. The frequency 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 serve as a frequency-based equivalent 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.

This notion of the frequency equivalent of monzos naturally extends into the frequency equivalent of mappings, using the place values of the selected positional numeral system as the basis elements instead of primes. Thus, frequency 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).