Patent val: Difference between revisions

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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).

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

== Further explanation ==

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