Patent val - Revision history

Godtone: this is important to link to here. im not sure how to factor it in in the most obvious way possible or how to refactor the existing text to account for this so this is my suggestion

2026-05-08T13:47:54Z

this is important to link to here. im not sure how to factor it in in the most obvious way possible or how to refactor the existing text to account for this so this is my suggestion

← Older revision		Revision as of 13:47, 8 May 2026
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	For example, the patent val for 17edo is {{val\| 17 27 39 }}, indicating that the closest mapping for 2/1 is 17 steps, the closest mapping for 3/1 is 27 steps, and the closest mapping for 5/1 is 39 steps. This means, if octaves are pure, that 3/2 is 706 cents, which is what you get if you round off 3/2 to the closest location in 17-equal, and that 5/4 is 353 cents, which is what you get is you round off 5/4 to the closest location in 17-equal.		For example, the patent val for 17edo is {{val\| 17 27 39 }}, indicating that the closest mapping for 2/1 is 17 steps, the closest mapping for 3/1 is 27 steps, and the closest mapping for 5/1 is 39 steps. This means, if octaves are pure, that 3/2 is 706 cents, which is what you get if you round off 3/2 to the closest location in 17-equal, and that 5/4 is 353 cents, which is what you get is you round off 5/4 to the closest location in 17-equal.

			'''For an explanation of what a val is, see [[val]].'''

	== Generalized patent val ==		== Generalized patent val ==

Squib: /* Generalized patent val */ clarification

2026-03-08T16:55:24Z

Generalized patent val: clarification

← Older revision		Revision as of 16:55, 8 March 2026
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	There are other vals worth considering besides the patent val. Consider the case of 5-limit 17et. {{val\| 17 27 39}} is the patent val, meaning each prime individually is as closely approximated as possible (again, assuming pure octaves). However, if that constraint is lifted, and we're allowed to choose the next-closest approximations for prime 5, the overall damage to the consonances we care about can be reduced; in other words, even though 39 steps of 17edo can take you just a tiny bit closer to prime 5 than 40 steps can, this is a naïve choice which does not take into account whether the errors tend to cancel or reinforce in simple ratios that combine different primes. Considering the problem more deeply in this manner may lead to choosing {{val\|17 27 40}} instead. And there are other harmonic reasons to choose {{val\| 17 27 40 }} over {{val\| 17 27 39 }} as well; it tempers different commas.		There are other vals worth considering besides the patent val. Consider the case of 5-limit 17et. {{val\| 17 27 39}} is the patent val, meaning each prime individually is as closely approximated as possible (again, assuming pure octaves). However, if that constraint is lifted, and we're allowed to choose the next-closest approximations for prime 5, the overall damage to the consonances we care about can be reduced; in other words, even though 39 steps of 17edo can take you just a tiny bit closer to prime 5 than 40 steps can, this is a naïve choice which does not take into account whether the errors tend to cancel or reinforce in simple ratios that combine different primes. Considering the problem more deeply in this manner may lead to choosing {{val\|17 27 40}} instead. And there are other harmonic reasons to choose {{val\| 17 27 40 }} over {{val\| 17 27 39 }} as well; it tempers different commas.

	We can show that {{val\| 17 27 40 }} is a generalized patent val because it would be the patent val for 17.1et: 17.1 × log<sub>2</sub>5 = 39.705, which rounds up to 40. Essentially this is showing that there does exist some generator size, 2<sup>1/17.1</sup>, for which it is truly the case that 17, 27, and 40 are the respective best approximations of primes 2, 3, and 5. That is, we are not "forcing" an interpretation of a prime which is not closest to the truth. A counterexample would be {{val\| 17 27 41 }}: it is possible to find a generator that maps 2 to 17 steps and 5 to 41 steps, but it would ~~require~~ 3 to be 28 steps (this type of information can be read easily off the nearby visualization).		We can show that {{val\| 17 27 40 }} is a generalized patent val because it would be the patent val for 17.1et: 17.1 × log<sub>2</sub>5 = 39.705, which rounds up to 40. Essentially this is showing that there does exist some generator size, 2<sup>1/17.1</sup>, for which it is truly the case that 17, 27, and 40 are the respective best approximations of primes 2, 3, and 5. That is, we are not "forcing" an interpretation of a prime which is not closest to the truth. A counterexample would be {{val\| 17 27 41 }}: it is possible to find a generator that maps 2 to 17 steps and 5 to 41 steps, but it would map 3 to be 28 steps, not 27 (this type of information can be read easily off the nearby visualization). Therefore {{val\| 17 '''28''' 41 }} is a GPV, but {{val\| 17 27 41 }} is not.

	Another name for generalized patent val is [[uniform map]].		Another name for generalized patent val is [[uniform map]].

	== Further explanation ==		== Further explanation ==

FloraC: Not seeing the term nearest edomapping catch on

2025-11-28T12:07:08Z

Not seeing the term *nearest edomapping* catch on

← Older revision		Revision as of 12:07, 28 November 2025
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	}}		}}
	The '''patent val''' ~~(a.k.a. '''nearest edomapping''')~~ for an [[edo]] is a list of numbers you obtain by finding the closest rounded approximation to each [[prime harmonic]] in the tuning, assuming [[2/1\|octaves]] are pure (or in other words, assuming the edo number is an integer). The basic application of a patent val is that you round prime harmonics to edosteps, and then deduce the number of steps of an arbitrary just interval based on its [[prime factorization]].		The '''patent val''' for an [[edo]] is a list of numbers you obtain by finding the closest rounded approximation to each [[prime harmonic]] in the tuning, assuming [[2/1\|octaves]] are pure (or in other words, assuming the edo number is an integer). The basic application of a patent val is that you round prime harmonics to edosteps, and then deduce the number of steps of an arbitrary just interval based on its [[prime factorization]].

	For example, the patent val for 17edo is {{val\| 17 27 39 }}, indicating that the closest mapping for 2/1 is 17 steps, the closest mapping for 3/1 is 27 steps, and the closest mapping for 5/1 is 39 steps. This means, if octaves are pure, that 3/2 is 706 cents, which is what you get if you round off 3/2 to the closest location in 17-equal, and that 5/4 is 353 cents, which is what you get is you round off 5/4 to the closest location in 17-equal.		For example, the patent val for 17edo is {{val\| 17 27 39 }}, indicating that the closest mapping for 2/1 is 17 steps, the closest mapping for 3/1 is 27 steps, and the closest mapping for 5/1 is 39 steps. This means, if octaves are pure, that 3/2 is 706 cents, which is what you get if you round off 3/2 to the closest location in 17-equal, and that 5/4 is 353 cents, which is what you get is you round off 5/4 to the closest location in 17-equal.
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	We can show that {{val\| 17 27 40 }} is a generalized patent val because it would be the patent val for 17.1et: 17.1 × log<sub>2</sub>5 = 39.705, which rounds up to 40. Essentially this is showing that there does exist some generator size, 2<sup>1/17.1</sup>, for which it is truly the case that 17, 27, and 40 are the respective best approximations of primes 2, 3, and 5. That is, we are not "forcing" an interpretation of a prime which is not closest to the truth. A counterexample would be {{val\| 17 27 41 }}: it is possible to find a generator that maps 2 to 17 steps and 5 to 41 steps, but it would require 3 to be 28 steps (this type of information can be read easily off the nearby visualization).		We can show that {{val\| 17 27 40 }} is a generalized patent val because it would be the patent val for 17.1et: 17.1 × log<sub>2</sub>5 = 39.705, which rounds up to 40. Essentially this is showing that there does exist some generator size, 2<sup>1/17.1</sup>, for which it is truly the case that 17, 27, and 40 are the respective best approximations of primes 2, 3, and 5. That is, we are not "forcing" an interpretation of a prime which is not closest to the truth. A counterexample would be {{val\| 17 27 41 }}: it is possible to find a generator that maps 2 to 17 steps and 5 to 41 steps, but it would require 3 to be 28 steps (this type of information can be read easily off the nearby visualization).

	Another name for generalized patent val is [[uniform map]] ~~(and an [[integer uniform map]], or [[simple map]], is another name for patent val)~~.		Another name for generalized patent val is [[uniform map]].

	== Further explanation ==		== Further explanation ==
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	You're dividing 81 by 80, so (assuming we're starting at zero, though it works no matter where you start) you add the steps for 81 (+196) and subtract the steps for 80 (-196). 196-196 = 0. This means that it takes zero steps to reach 81/80 – in other words, 81/80 "vanishes".		You're dividing 81 by 80, so (assuming we're starting at zero, though it works no matter where you start) you add the steps for 81 (+196) and subtract the steps for 80 (-196). 196-196 = 0. This means that it takes zero steps to reach 81/80 – in other words, 81/80 "vanishes".

			== Terminology ==
			[[Kite Giedraitis]] proposed the term ''nearest edomapping'' for this concept. [[Dave Keenan]] and [[Douglas Blumeyer]] use ''[[integer uniform map]]'' and ''[[simple map]]'', notably in their [[Dave Keenan & Douglas Blumeyer's guide to RTT\|guides]].

	== See also ==		== See also ==

Dummy index: /* Generalized patent val */ clarity

2025-08-19T13:28:44Z

Generalized patent val: clarity

← Older revision		Revision as of 13:28, 19 August 2025
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	[[File:Generalized Patent Vals.png\|thumb\|a visualization of all possible GPVs through the 13-limit up to 99et (any vertical slice is a GPV)]]		[[File:Generalized Patent Vals.png\|thumb\|a visualization of all possible GPVs through the 13-limit up to 99et (any vertical slice is a GPV)]]

	There are other vals worth considering besides the patent val. Consider the case of 5-limit 17et. {{val\| 17 27 39}} is the patent val, meaning each prime individually is as closely approximated as possible (again, assuming pure octaves). However, if that constraint is lifted, and we're allowed to choose the next-closest approximations for prime 5, the overall damage to the consonances we care about can be reduced; in other words, even though 39 steps can take you just a tiny bit closer to prime 5 than 40 steps can, this is a naïve choice which does not take into account whether the errors tend to cancel or reinforce in simple ratios that combine different primes. Considering the problem more deeply in this manner may lead to choosing {{val\|17 27 40}} instead. And there are other harmonic reasons to choose {{val\| 17 27 40 }} over {{val\| 17 27 39 }} as well; it tempers different commas.		There are other vals worth considering besides the patent val. Consider the case of 5-limit 17et. {{val\| 17 27 39}} is the patent val, meaning each prime individually is as closely approximated as possible (again, assuming pure octaves). However, if that constraint is lifted, and we're allowed to choose the next-closest approximations for prime 5, the overall damage to the consonances we care about can be reduced; in other words, even though 39 steps of 17edo can take you just a tiny bit closer to prime 5 than 40 steps can, this is a naïve choice which does not take into account whether the errors tend to cancel or reinforce in simple ratios that combine different primes. Considering the problem more deeply in this manner may lead to choosing {{val\|17 27 40}} instead. And there are other harmonic reasons to choose {{val\| 17 27 40 }} over {{val\| 17 27 39 }} as well; it tempers different commas.

	We can show that {{val\| 17 27 40 }} is a generalized patent val because it would be the patent val for 17.1et: 17.1 × log<sub>2</sub>5 = 39.705, which rounds up to 40. Essentially this is showing that there does exist some generator size, 2<sup>1/17.1</sup>, for which it is truly the case that 17, 27, and 40 are the respective best approximations of primes 2, 3, and 5. That is, we are not "forcing" an interpretation of a prime which is not closest to the truth. A counterexample would be {{val\| 17 27 41 }}: it is possible to find a generator that maps 2 to 17 steps and 5 to 41 steps, but it would require 3 to be 28 steps (this type of information can be read easily off the nearby visualization).		We can show that {{val\| 17 27 40 }} is a generalized patent val because it would be the patent val for 17.1et: 17.1 × log<sub>2</sub>5 = 39.705, which rounds up to 40. Essentially this is showing that there does exist some generator size, 2<sup>1/17.1</sup>, for which it is truly the case that 17, 27, and 40 are the respective best approximations of primes 2, 3, and 5. That is, we are not "forcing" an interpretation of a prime which is not closest to the truth. A counterexample would be {{val\| 17 27 41 }}: it is possible to find a generator that maps 2 to 17 steps and 5 to 41 steps, but it would require 3 to be 28 steps (this type of information can be read easily off the nearby visualization).

FloraC: -integer uniform map and simple map from the lemma for now since they don't redirect here

2025-03-31T08:47:38Z

-integer uniform map and simple map from the lemma for now since they don't redirect here

← Older revision		Revision as of 08:47, 31 March 2025
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	\| ja = 特徴的なヴァル		\| ja = 特徴的なヴァル
	}}		}}
	The '''patent val''' (a.k.a. '''nearest edomapping~~''', '''integer uniform map''', and '''simple map~~''') for an [[edo]] is a list of numbers you obtain by finding the closest rounded approximation to each [[prime harmonic]] in the tuning, assuming [[2/1\|octaves]] are pure (or in other words, assuming the edo number is an integer). The basic application of a patent val is that you round prime harmonics to edosteps, and then deduce the number of steps of an arbitrary just interval based on its [[prime factorization]].		The '''patent val''' (a.k.a. '''nearest edomapping''') for an [[edo]] is a list of numbers you obtain by finding the closest rounded approximation to each [[prime harmonic]] in the tuning, assuming [[2/1\|octaves]] are pure (or in other words, assuming the edo number is an integer). The basic application of a patent val is that you round prime harmonics to edosteps, and then deduce the number of steps of an arbitrary just interval based on its [[prime factorization]].

	For example, the patent val for 17edo is {{val\| 17 27 39 }}, indicating that the closest mapping for 2/1 is 17 steps, the closest mapping for 3/1 is 27 steps, and the closest mapping for 5/1 is 39 steps. This means, if octaves are pure, that 3/2 is 706 cents, which is what you get if you round off 3/2 to the closest location in 17-equal, and that 5/4 is 353 cents, which is what you get is you round off 5/4 to the closest location in 17-equal.		For example, the patent val for 17edo is {{val\| 17 27 39 }}, indicating that the closest mapping for 2/1 is 17 steps, the closest mapping for 3/1 is 27 steps, and the closest mapping for 5/1 is 39 steps. This means, if octaves are pure, that 3/2 is 706 cents, which is what you get if you round off 3/2 to the closest location in 17-equal, and that 5/4 is 353 cents, which is what you get is you round off 5/4 to the closest location in 17-equal.

FloraC: It makes no sense to introduce how to use something before explaining what it is

2025-03-31T08:46:18Z

It makes no sense to introduce how to use something before explaining what it is

← Older revision		Revision as of 08:46, 31 March 2025
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	}}		}}
	The ~~basic principle of~~ ''~~using~~'' a ~~patent val is that you round prime harmonics to edosteps~~, ~~and then deduce the "mapping" of an arbitrary interval based on its prime factorization.~~		The '''patent val''' (a.k.a. '''nearest edomapping''', '''integer uniform map''', and '''simple map''') for an [[edo]] is a list of numbers you obtain by finding the closest rounded approximation to each [[prime harmonic]] in the tuning, assuming [[2/1\|octaves]] are pure (or in other words, assuming the edo number is an integer). The basic application of a patent val is that you round prime harmonics to edosteps, and then deduce the number of steps of an arbitrary just interval based on its [[prime factorization]].

	~~The~~ '''~~patent [[val]]~~''' ~~(aka~~ '''~~nearest edomapping~~''') for ~~some~~ edo is ~~thus the~~ list of numbers ~~you get that~~ you obtain by finding the closest rounded approximation to each [[prime]] in the tuning, assuming octaves are pure (or in other words, assuming the edo number is an integer).

	For example, the patent val for 17edo is {{val\| 17 27 39 }}, indicating that the closest mapping for 2/1 is 17 steps, the closest mapping for 3/1 is 27 steps, and the closest mapping for 5/1 is 39 steps. This means, if octaves are pure, that 3/2 is 706 cents, which is what you get if you round off 3/2 to the closest location in 17-equal, and that 5/4 is 353 cents, which is what you get is you round off 5/4 to the closest location in 17-equal.		For example, the patent val for 17edo is {{val\| 17 27 39 }}, indicating that the closest mapping for 2/1 is 17 steps, the closest mapping for 3/1 is 27 steps, and the closest mapping for 5/1 is 39 steps. This means, if octaves are pure, that 3/2 is 706 cents, which is what you get if you round off 3/2 to the closest location in 17-equal, and that 5/4 is 353 cents, which is what you get is you round off 5/4 to the closest location in 17-equal.

Godtone: clarify

2025-01-30T18:51:32Z

clarify

← Older revision		Revision as of 18:51, 30 January 2025
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	}}		}}
	The basic principle of ''using'' a patent val is that you round prime harmonics to edosteps, and then deduce the "mapping" of an arbitrary interval based on its prime factorization. The '''patent val''' (aka '''nearest edomapping''') for some edo is thus the ~~[[val]]~~ that you obtain by finding the closest rounded approximation to each [[prime]] in the tuning, assuming octaves are pure (or in other words, assuming the edo number is an integer).		The basic principle of ''using'' a patent val is that you round prime harmonics to edosteps, and then deduce the "mapping" of an arbitrary interval based on its prime factorization.

			The '''patent [[val]]''' (aka '''nearest edomapping''') for some edo is thus the list of numbers you get that you obtain by finding the closest rounded approximation to each [[prime]] in the tuning, assuming octaves are pure (or in other words, assuming the edo number is an integer).

	For example, the patent val for 17edo is {{val\| 17 27 39 }}, indicating that the closest mapping for 2/1 is 17 steps, the closest mapping for 3/1 is 27 steps, and the closest mapping for 5/1 is 39 steps. This means, if octaves are pure, that 3/2 is 706 cents, which is what you get if you round off 3/2 to the closest location in 17-equal, and that 5/4 is 353 cents, which is what you get is you round off 5/4 to the closest location in 17-equal.		For example, the patent val for 17edo is {{val\| 17 27 39 }}, indicating that the closest mapping for 2/1 is 17 steps, the closest mapping for 3/1 is 27 steps, and the closest mapping for 5/1 is 39 steps. This means, if octaves are pure, that 3/2 is 706 cents, which is what you get if you round off 3/2 to the closest location in 17-equal, and that 5/4 is 353 cents, which is what you get is you round off 5/4 to the closest location in 17-equal.

Godtone: add simple explanation to give the reader an idea

2025-01-30T18:50:06Z

add simple explanation to give the reader an idea

← Older revision		Revision as of 18:50, 30 January 2025
Line 5:		Line 5:
	\| ja = 特徴的なヴァル		\| ja = 特徴的なヴァル
	}}		}}
	The '''patent val''' (aka '''nearest edomapping''') for some edo is the [[val]] that you obtain by finding the closest rounded approximation to each [[prime]] in the tuning, assuming octaves are pure (or in other words, assuming the edo number is an integer).		The basic principle of ''using'' a patent val is that you round prime harmonics to edosteps, and then deduce the "mapping" of an arbitrary interval based on its prime factorization. The '''patent val''' (aka '''nearest edomapping''') for some edo is thus the [[val]] that you obtain by finding the closest rounded approximation to each [[prime]] in the tuning, assuming octaves are pure (or in other words, assuming the edo number is an integer).

	For example, the patent val for 17edo is {{val\| 17 27 39 }}, indicating that the closest mapping for 2/1 is 17 steps, the closest mapping for 3/1 is 27 steps, and the closest mapping for 5/1 is 39 steps. This means, if octaves are pure, that 3/2 is 706 cents, which is what you get if you round off 3/2 to the closest location in 17-equal, and that 5/4 is 353 cents, which is what you get is you round off 5/4 to the closest location in 17-equal.		For example, the patent val for 17edo is {{val\| 17 27 39 }}, indicating that the closest mapping for 2/1 is 17 steps, the closest mapping for 3/1 is 27 steps, and the closest mapping for 5/1 is 39 steps. This means, if octaves are pure, that 3/2 is 706 cents, which is what you get if you round off 3/2 to the closest location in 17-equal, and that 5/4 is 353 cents, which is what you get is you round off 5/4 to the closest location in 17-equal.

Cmloegcmluin: /* See also */ remove broken link

2023-05-07T18:57:49Z

FloraC: Style and link

2023-01-31T05:44:19Z

Style and link

Show changes

← Older revision		Revision as of 13:28, 19 August 2025
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	[[File:Generalized Patent Vals.png\|thumb\|a visualization of all possible GPVs through the 13-limit up to 99et (any vertical slice is a GPV)]]		[[File:Generalized Patent Vals.png\|thumb\|a visualization of all possible GPVs through the 13-limit up to 99et (any vertical slice is a GPV)]]

	There are other vals worth considering besides the patent val. Consider the case of 5-limit 17et. {{val\| 17 27 39}} is the patent val, meaning each prime individually is as closely approximated as possible (again, assuming pure octaves). However, if that constraint is lifted, and we're allowed to choose the next-closest approximations for prime 5, the overall damage to the consonances we care about can be reduced; in other words, even though 39 steps can take you just a tiny bit closer to prime 5 than 40 steps can, this is a naïve choice which does not take into account whether the errors tend to cancel or reinforce in simple ratios that combine different primes. Considering the problem more deeply in this manner may lead to choosing {{val\|17 27 40}} instead. And there are other harmonic reasons to choose {{val\| 17 27 40 }} over {{val\| 17 27 39 }} as well; it tempers different commas.		There are other vals worth considering besides the patent val. Consider the case of 5-limit 17et. {{val\| 17 27 39}} is the patent val, meaning each prime individually is as closely approximated as possible (again, assuming pure octaves). However, if that constraint is lifted, and we're allowed to choose the next-closest approximations for prime 5, the overall damage to the consonances we care about can be reduced; in other words, even though 39 steps of 17edo can take you just a tiny bit closer to prime 5 than 40 steps can, this is a naïve choice which does not take into account whether the errors tend to cancel or reinforce in simple ratios that combine different primes. Considering the problem more deeply in this manner may lead to choosing {{val\|17 27 40}} instead. And there are other harmonic reasons to choose {{val\| 17 27 40 }} over {{val\| 17 27 39 }} as well; it tempers different commas.

	We can show that {{val\| 17 27 40 }} is a generalized patent val because it would be the patent val for 17.1et: 17.1 × log<sub>2</sub>5 = 39.705, which rounds up to 40. Essentially this is showing that there does exist some generator size, 2<sup>1/17.1</sup>, for which it is truly the case that 17, 27, and 40 are the respective best approximations of primes 2, 3, and 5. That is, we are not "forcing" an interpretation of a prime which is not closest to the truth. A counterexample would be {{val\| 17 27 41 }}: it is possible to find a generator that maps 2 to 17 steps and 5 to 41 steps, but it would require 3 to be 28 steps (this type of information can be read easily off the nearby visualization).		We can show that {{val\| 17 27 40 }} is a generalized patent val because it would be the patent val for 17.1et: 17.1 × log<sub>2</sub>5 = 39.705, which rounds up to 40. Essentially this is showing that there does exist some generator size, 2<sup>1/17.1</sup>, for which it is truly the case that 17, 27, and 40 are the respective best approximations of primes 2, 3, and 5. That is, we are not "forcing" an interpretation of a prime which is not closest to the truth. A counterexample would be {{val\| 17 27 41 }}: it is possible to find a generator that maps 2 to 17 steps and 5 to 41 steps, but it would require 3 to be 28 steps (this type of information can be read easily off the nearby visualization).

← Older revision		Revision as of 18:57, 7 May 2023
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	== See also ==		== See also ==
	* [[Patent val list of small EDOs]]
	* [[Optimal patent val]]		* [[Optimal patent val]]