Talk:Frequency temperament: Difference between revisions

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::::: More than a month later, but I finally realized what is the frequency-space equivalent of monzos and mappings, although they would be related to my original definition of "arithmetic temperament" (both generator and period as AFS) rather than the current one. Keep in mind that, in frequency space, intervals are reduced to numbers between 0 and 1 if the period is 1, like how in pitch space intervals are reduced to numbers between 1 and 2 if the period is an octave.

::::: If we want to preserve uniqueness, the frequency equivalent of a monzo (the sum of the multiples of some basis elements) is not possible unless we restrict the multiplying factor to a certain range, resulting in what is essentially place value systems (like binary and decimal). I think the most useful of these systems as a frequency monzo would be the [https://en.wikipedia.org/wiki/Factorial_number_system factorial number system], where the place values (basis elements) are the factorials and reciprocals of them, which allows for representation any rational number ~~given enough digits~~.

::::: If we want to preserve uniqueness, the frequency equivalent of a monzo (the sum of the multiples of some basis elements) is not possible unless we restrict the multiplying factor to a certain range, resulting in what is essentially place value systems (like binary and decimal). I think the most useful of these systems as a "frequency monzo" would be the [https://en.wikipedia.org/wiki/Factorial_number_system factorial number system], where the place values (basis elements) are the factorials and reciprocals of them, which allows for representation any rational number, just as with normal monzos.

::::: Mappings and patent vals would work essentially the same except using reciprocals of factorials instead of primes as the basis elements, and using AFSes instead of ETs. "Tempering out" a comma also now means to reduce it to 0 instead of 1. Prime limits are replaced with "factorial limits".

::::: [[User:CompactStar|CompactStar]] ([[User talk:CompactStar|talk]]) 09:42, 17 May 2023 (UTC)

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 ::::: More than a month later, but I finally realized what is the frequency-space equivalent of monzos and mappings, although they would be related to my original definition of "arithmetic temperament" (both generator and period as AFS) rather than the current one. Keep in mind that, in frequency space, intervals are reduced to numbers between 0 and 1 if the period is 1, like how in pitch space intervals are reduced to numbers between 1 and 2 if the period is an octave.
-::::: If we want to preserve uniqueness, the frequency equivalent of a monzo (the sum of the multiples of some basis elements) is not possible unless we restrict the multiplying factor to a certain range, resulting in what is essentially place value systems (like binary and decimal). I think the most useful of these systems as a frequency monzo would be the [https://en.wikipedia.org/wiki/Factorial_number_system factorial number system], where the place values (basis elements) are the factorials and reciprocals of them, which allows for representation any rational number given enough digits.
+::::: If we want to preserve uniqueness, the frequency equivalent of a monzo (the sum of the multiples of some basis elements) is not possible unless we restrict the multiplying factor to a certain range, resulting in what is essentially place value systems (like binary and decimal). I think the most useful of these systems as a "frequency monzo" would be the [https://en.wikipedia.org/wiki/Factorial_number_system factorial number system], where the place values (basis elements) are the factorials and reciprocals of them, which allows for representation any rational number, just as with normal monzos.
 ::::: Mappings and patent vals would work essentially the same except using reciprocals of factorials instead of primes as the basis elements, and using AFSes instead of ETs. "Tempering out" a comma also now means to reduce it to 0 instead of 1. Prime limits are replaced with "factorial limits".
 ::::: [[User:CompactStar|CompactStar]] ([[User talk:CompactStar|talk]]) 09:42, 17 May 2023 (UTC)