Patent val: Difference between revisions

Line 16:

[[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).

@@ Line 16: / Line 16: @@
 [[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).