Patent val: Difference between revisions

(2 intermediate revisions by 2 users not shown)

Line 5:

| ja = 特徴的なヴァル

}}

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

== Generalized patent val ==

Line 18:

Line 20:

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 × log25 = 39.705, which rounds up to 40. Essentially this is showing that there does exist some generator size, 21/17.1, 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 × log25 = 39.705, which rounds up to 40. Essentially this is showing that there does exist some generator size, 21/17.1, 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]] ~~(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 ==

Line 134:

Line 136:

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 ==

@@ Line 5: / Line 5: @@
 | ja = 特徴的なヴァル
 }}
-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 an explanation of what a val is, see [[val]].'''
 == Generalized patent val ==
@@ Line 18: / Line 20: @@
 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]] (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 ==
@@ Line 134: / Line 136: @@
 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 ==