Frequency temperament: Difference between revisions

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'''~~Arithmetic temperaments~~''' are ~~the arithmetic counterpart~~ to [[~~regular temperament~~]]s~~. Whereas~~ regular temperaments are ~~created by reducing integer powers of a~~ [[~~generator~~]], an arithmetic temperament is created by reducing integer multiples of a generator. The n-th interval in an arithmetic temperament prior to octave-reduction is given by n*g + 1, where g is the generator.

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 [[AFS]]s as regular temperaments are to [[ET]]s.

~~For example~~, ~~this~~ is the ~~interval chain~~ of ~~an arithmetic temperament with~~ a ~~generator~~ of 0.~~29 and period~~ [[2/1]]:

Frequency temperaments are based on the frequency counterparts of monzos, vals, and mappings. The frequency 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 (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 [https://en.wikipedia.org/wiki/Factorial_number_system "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.

~~<pre>~~

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).

~~1+0~~.~~29 = 1.29~~

~~1+2*0.29 = 1.58~~

1+3*0~~.29 =~~ 1.87

~~1+4*0.29 = 2.16 -> 1.08~~

~~1+5*0.29 = 2.45 -> 1.225~~

~~1+6*0.29 = 2.74 -> 1.37~~

~~...~~

~~</pre>~~

~~Arithmetic temperaments also temper out~~ [[~~comma~~]]s, but these commas represent differences between intervals rather than ratios between them. For example, equating [[9/7]] with [[13/10]] in an arithmetic temperament tempers out the "arithmetic comma" (13/10)-(9/7) = 1/70.

== Examples ==

~~== List of arithmetic temperaments ==~~

* [[Neutrino]]

* [[Sqrttwo]]

[[Category:~~Temperaments]]~~

[[Category:Frequency temperaments]]

~~[[Category:Arithmetic~~ temperaments]]

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-'''Arithmetic temperaments''' are the arithmetic counterpart to [[regular temperament]]s. Whereas regular temperaments are created by reducing integer powers of a [[generator]], an arithmetic temperament is created by reducing integer multiples of a generator. The n-th interval in an arithmetic temperament prior to octave-reduction is given by n*g + 1, where g is the generator.
+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 [[AFS]]s as regular temperaments are to [[ET]]s.
-For example, this is the interval chain of an arithmetic temperament with a generator of 0.29 and period [[2/1]]:
+Frequency temperaments are based on the frequency counterparts of monzos, vals, and mappings. The frequency 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 (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 [https://en.wikipedia.org/wiki/Factorial_number_system "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.
-<pre>
+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).
-+0.29 = 1.29
-+2*0.29 = 1.58
-+3*0.29 = 1.87
-+4*0.29 = 2.16 -> 1.08
-+5*0.29 = 2.45 -> 1.225
-+6*0.29 = 2.74 -> 1.37
-...
-</pre>
-Arithmetic temperaments also temper out [[comma]]s, but these commas represent differences between intervals rather than ratios between them. For example, equating [[9/7]] with [[13/10]] in an arithmetic temperament tempers out the "arithmetic comma" (13/10)-(9/7) = 1/70.
+== Examples ==
-== List of arithmetic temperaments ==
+* [[Neutrino]]
 * [[Sqrttwo]]
-[[Category:Temperaments]]
+[[Category:Frequency temperaments]]
-[[Category:Arithmetic temperaments]]