Talk:Frequency temperament - Revision history

CompactStar: CompactStar moved page User talk:CompactStar/Arithmetic interval chain to Talk:Frequency temperament

2023-05-19T09:06:48Z

CompactStar moved page User talk:CompactStar/Arithmetic interval chain to Talk:Frequency temperament

← Older revision	Revision as of 09:06, 19 May 2023
(No difference)

CompactStar at 09:44, 17 May 2023

2023-05-17T09:44:40Z

← Older revision		Revision as of 09:44, 17 May 2023
Line 15:		Line 15:

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

CompactStar at 09:44, 17 May 2023

2023-05-17T09:44:27Z

← Older revision		Revision as of 09:44, 17 May 2023
Line 15:		Line 15:

	::::: 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.		::::: 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 given enough digits.

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

CompactStar at 09:44, 17 May 2023

2023-05-17T09:44:09Z

← Older revision		Revision as of 09:44, 17 May 2023
Line 17:		Line 17:
	::::: 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 given enough digits.

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

CompactStar at 09:43, 17 May 2023

2023-05-17T09:43:53Z

← Older revision		Revision as of 09:43, 17 May 2023
Line 15:		Line 15:

	::::: 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.		::::: 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:		::::: 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.

	~~::::: 1/2 = 1/2! (a "frequency monzo" of {{monzo\|1}})~~
	~~::::: 1/3 = 2/3! (a "frequency monzo" of {{monzo\|0 2}})~~
	~~::::: 1/4 = 1/3! + 2/4! (a "frequency monzo" of {{monzo\|0 1 2}})~~
	~~::::: 1/5 = 1/3! + 4/5! (a "frequency monzo" of {{monzo\|0 1 0 4}})~~

	::::: 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.		::::: 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.

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

CompactStar at 09:43, 17 May 2023

2023-05-17T09:43:20Z

← Older revision		Revision as of 09:43, 17 May 2023
Line 15:		Line 15:

	::::: 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.		::::: 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 ~~such system~~ 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:		::::: 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:

	::::: 1/2 = 1/2! (a "frequency monzo" of {{monzo\|1}})		::::: 1/2 = 1/2! (a "frequency monzo" of {{monzo\|1}})

CompactStar at 09:42, 17 May 2023

2023-05-17T09:42:32Z

← Older revision		Revision as of 09:42, 17 May 2023
Line 24:		Line 24:
	::::: 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.		::::: 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.

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

CompactStar at 09:42, 17 May 2023

2023-05-17T09:42:26Z

← Older revision		Revision as of 09:42, 17 May 2023
Line 17:		Line 17:
	::::: 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 such system 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:		::::: 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 such system 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:

	1/2 = 1/2! (a "frequency monzo" of {{monzo\|1}})		::::: 1/2 = 1/2! (a "frequency monzo" of {{monzo\|1}})
	1/3 = 2/3! (a "frequency monzo" of {{monzo\|0 2}})		::::: 1/3 = 2/3! (a "frequency monzo" of {{monzo\|0 2}})
	1/4 = 1/3! + 2/4! (a "frequency monzo" of {{monzo\|0 1 2}})		::::: 1/4 = 1/3! + 2/4! (a "frequency monzo" of {{monzo\|0 1 2}})
	1/5 = 1/3! + 4/5! (a "frequency monzo" of {{monzo\|0 1 0 4}})		::::: 1/5 = 1/3! + 4/5! (a "frequency monzo" of {{monzo\|0 1 0 4}})

	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.		::::: 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.

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

CompactStar at 09:42, 17 May 2023

2023-05-17T09:42:08Z

← Older revision		Revision as of 09:42, 17 May 2023
Line 13:		Line 13:

	:::: 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)		:::: 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 such system 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:

			1/2 = 1/2! (a "frequency monzo" of {{monzo\|1}})
			1/3 = 2/3! (a "frequency monzo" of {{monzo\|0 2}})
			1/4 = 1/3! + 2/4! (a "frequency monzo" of {{monzo\|0 1 2}})
			1/5 = 1/3! + 4/5! (a "frequency monzo" of {{monzo\|0 1 0 4}})

			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.

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

FloraC: FloraC moved page Talk:Arithmetic interval chain to User talk:CompactStar/Arithmetic interval chain without leaving a redirect: Not ready for the main namespace. See talk

2023-04-23T05:59:14Z

FloraC moved page Talk:Arithmetic interval chain to User talk:CompactStar/Arithmetic interval chain without leaving a redirect: Not ready for the main namespace. See talk

← Older revision	Revision as of 05:59, 23 April 2023
(No difference)

← Older revision		Revision as of 09:42, 17 May 2023
Line 17:		Line 17:
	::::: 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 such system 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:		::::: 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 such system 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:

	1/2 = 1/2! (a "frequency monzo" of {{monzo\|1}})		::::: 1/2 = 1/2! (a "frequency monzo" of {{monzo\|1}})
	1/3 = 2/3! (a "frequency monzo" of {{monzo\|0 2}})		::::: 1/3 = 2/3! (a "frequency monzo" of {{monzo\|0 2}})
	1/4 = 1/3! + 2/4! (a "frequency monzo" of {{monzo\|0 1 2}})		::::: 1/4 = 1/3! + 2/4! (a "frequency monzo" of {{monzo\|0 1 2}})
	1/5 = 1/3! + 4/5! (a "frequency monzo" of {{monzo\|0 1 0 4}})		::::: 1/5 = 1/3! + 4/5! (a "frequency monzo" of {{monzo\|0 1 0 4}})

	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.		::::: 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.

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