Talk:Frequency temperament: Difference between revisions

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:: I suppose one important difference between this and AFS is its equave repeating feature. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 07:59, 21 April 2023 (UTC)

::: In that case, should this concept be redefined as a 2D lattice with at least one dimension being an AFS? I believe CompactStar originally had in mind that both the generator and the period would be AFS, and after discussing with him on Discord, he changed his definition to use regular octave equivalence, which is an APS, but having an AFS instead would simply result in a different system. In that case, the concept would yet again need to be renamed, and the original name "arithmetic temperament" still wouldn't work, especially when working with just intervals in the AFS, unless someone shows that it can behave like a regular temperament, or possibly that it classifies more generally as a [[temperament]] (in the sense that includes well temperaments). --[[User:Fredg999|Fredg999]] ([[User talk:Fredg999|talk]]) 04:03, 22 April 2023 (UTC)

:::: Ah, interesting. Yes, a "Frequency MOS" suggests to me that the entirety of the MOS concept is converted from pitch to frequency. If the intention here is only to have an octave-repeating structure, i.e. still pitch-based, but within each octave is frequency-based, that's indeed different, and a bit messy, and wouldn't be best called "Frequency MOS". (Wait, does the amount of frequency iterated by change by a factor of 2 in each octave? In other words, does each octave have the same count of pitches? My typical interpretation of "octave-equivalent" would say "yes" to that question, but keeping it the same amount of frequency iterated by in each octave also makes sense in a different way and is a potentially more interesting structure.) I agree there's still insufficient justification for "temperament" in the name. I also repeat my suggestion that this be moved away from a main page until the concept is ironed out better and expressed more clearly with an acceptable name. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 20:43, 22 April 2023 (UTC)

::::: 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, 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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 :: I suppose one important difference between this and AFS is its equave repeating feature. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 07:59, 21 April 2023 (UTC)
 ::: In that case, should this concept be redefined as a 2D lattice with at least one dimension being an AFS? I believe CompactStar originally had in mind that both the generator and the period would be AFS, and after discussing with him on Discord, he changed his definition to use regular octave equivalence, which is an APS, but having an AFS instead would simply result in a different system. In that case, the concept would yet again need to be renamed, and the original name "arithmetic temperament" still wouldn't work, especially when working with just intervals in the AFS, unless someone shows that it can behave like a regular temperament, or possibly that it classifies more generally as a [[temperament]] (in the sense that includes well temperaments). --[[User:Fredg999|Fredg999]] ([[User talk:Fredg999|talk]]) 04:03, 22 April 2023 (UTC)
+:::: Ah, interesting. Yes, a "Frequency MOS" suggests to me that the entirety of the MOS concept is converted from pitch to frequency. If the intention here is only to have an octave-repeating structure, i.e. still pitch-based, but within each octave is frequency-based, that's indeed different, and a bit messy, and wouldn't be best called "Frequency MOS". (Wait, does the amount of frequency iterated by change by a factor of 2 in each octave? In other words, does each octave have the same count of pitches? My typical interpretation of "octave-equivalent" would say "yes" to that question, but keeping it the same amount of frequency iterated by in each octave also makes sense in a different way and is a potentially more interesting structure.) I agree there's still insufficient justification for "temperament" in the name. I also repeat my suggestion that this be moved away from a main page until the concept is ironed out better and expressed more clearly with an acceptable name. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 20:43, 22 April 2023 (UTC)
+::::: 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, 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)