Patent val: Difference between revisions

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This val can be extended to the case where the number of steps in an octave is a real number rather than an integer; this is called a '''generalized patent val''', or '''GPV'''. For instance the 7-limit generalized patent val for 16.9 is {{val| 17 27 39 47 }}, since 16.9 × log27 = 47.444, which rounds down to 47.

There are other vals ~~or edomappings~~ besides the patent ~~or nearest one~~. ~~You may prefer to use~~ the {{val| 17 27 40 }} val as the 5-~~limit 17-equal val instead~~, which ~~rather~~ than {{val| 17 27 39 }} ~~treats 424 cents as 5/4~~. ~~This~~ val ~~has lower Tenney-Euclidean error than the~~ 17~~-EDO patent val. However, while~~ {{val| 17 27 39 }} ~~may not necessarily be the "best" val for 17-equal for all purposes,~~ it ~~is the obvious, or "patent" val,~~ that ~~you get by naively rounding primes off within the EDO and taking no further considerations into account. However,~~ {{val| 17 27 40 }} is the ~~generalized~~ patent val for 17.1~~, since~~ 17.1 × log25 = 39.705, which rounds up to 40.

[[File:Generalized Patent Vals.png|thumb|a visualization of all possible GPVs through the 13-limit up to 99-ET (any vertical slice is a GPV)]]

There are other vals worth considering besides the patent val. Consider the case of 5-limit 17-ET. {{val| 17 27 39}} is the patent val, meaning each prime individually is as closely approximated as possible. However, if that constraint is lifted, and we're allowed to choose the next-closest approximations for prime 5, the overall Tenney-Euclidean error can actually be reduced; in other words, even though 39 steps can take you just a tiny bit closer to prime 5 than 40 steps can, the tiny amount by which it is closer is less than the improvements to the tuning of primes 2 and 3 you can get by using {{val|17 27 40}}. 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 17.1-ET: 17.1 × log25 = 39.705, which rounds up to 40. Essentially this is showing that there does exist some generator size, 2^(1/17.1), 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, that 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).

== Further explanation ==

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 This val can be extended to the case where the number of steps in an octave is a real number rather than an integer; this is called a '''generalized patent val''', or '''GPV'''. For instance the 7-limit generalized patent val for 16.9 is {{val| 17 27 39 47 }}, since 16.9 × log<sub>2</sub>7 = 47.444, which rounds down to 47.
-There are other vals or edomappings besides the patent or nearest one. You may prefer to use the {{val| 17 27 40 }} val as the 5-limit 17-equal val instead, which rather than {{val| 17 27 39 }} treats 424 cents as 5/4. This val has lower Tenney-Euclidean error than the 17-EDO patent val. However, while {{val| 17 27 39 }} may not necessarily be the "best" val for 17-equal for all purposes, it is the obvious, or "patent" val, that you get by naively rounding primes off within the EDO and taking no further considerations into account. However, {{val| 17 27 40 }} is the generalized patent val for 17.1, since 17.1 × log<sub>2</sub>5 = 39.705, which rounds up to 40.
+[[File:Generalized Patent Vals.png|thumb|a visualization of all possible GPVs through the 13-limit up to 99-ET (any vertical slice is a GPV)]]
+There are other vals worth considering besides the patent val. Consider the case of 5-limit 17-ET. {{val| 17 27 39}} is the patent val, meaning each prime individually is as closely approximated as possible. However, if that constraint is lifted, and we're allowed to choose the next-closest approximations for prime 5, the overall Tenney-Euclidean error can actually be reduced; in other words, even though 39 steps can take you just a tiny bit closer to prime 5 than 40 steps can, the tiny amount by which it is closer is less than the improvements to the tuning of primes 2 and 3 you can get by using {{val|17 27 40}}. 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 17.1-ET: 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^(1/17.1), 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, that 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).
 == Further explanation ==