Douglas Blumeyer's RTT How-To: Difference between revisions

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So when I say 12:19:28 ≈ log(2:3:5) what I’m saying is that there is indeed some shared generator g for which logg2 ≈ 12, logg3 ≈ 19, and logg5 ≈ 28 are all good approximations all at the same time, or, equivalently, a shared generator g for which g¹² ≈ 2, g¹⁹ ≈ 3, and g²⁸ ≈ 5 are all good approximations at the same time ''(see Figure 2c)''. And that’s a pretty cool thing to find! To be clear, with g = 1.059, we get g¹² ≈ 1.9982, g¹⁹ ≈ 2.9923, and g²⁸ ≈ 5.0291.

Another glowing example is the map {{val|53 84 123}}, for which a good generator will give you ~~g¹²~~ ≈ 2.0002, ~~g¹⁹~~ ≈ 3.0005, ~~g²⁸~~ ≈ 4.9974. This speaks to historical attention given to [[53edo|53-ET]]. So while 53:84:123 is an even better approximation of log(2:3:5) (and [https://en.xen.wiki/images/a/a2/Generalized_Patent_Vals.png you won’t find a better one] until 118:187:274), of course its integers aren’t as low, so that lessens its appeal.

Another glowing example is the map {{val|53 84 123}}, for which a good generator will give you g⁵³ ≈ 2.0002, g⁸⁴ ≈ 3.0005, g¹²³ ≈ 4.9974. This speaks to historical attention given to [[53edo|53-ET]]. So while 53:84:123 is an even better approximation of log(2:3:5) (and [https://en.xen.wiki/images/a/a2/Generalized_Patent_Vals.png you won’t find a better one] until 118:187:274), of course its integers aren’t as low, so that lessens its appeal.

Why is this rare? Well, it’s like a game of trying to get these numbers to line up ''(see Figure 2d)'':

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 So when I say 12:19:28 ≈ log(2:3:5) what I’m saying is that there is indeed some shared generator g for which log<sub>g</sub>2 ≈ 12, log<sub>g</sub>3 ≈ 19, and log<sub>g</sub>5 ≈ 28 are all good approximations all at the same time, or, equivalently, a shared generator g for which g¹² ≈ 2, g¹⁹ ≈ 3, and g²⁸ ≈ 5 are all good approximations at the same time ''(see Figure 2c)''. And that’s a pretty cool thing to find! To be clear, with g = 1.059, we get g¹² ≈ 1.9982, g¹⁹ ≈ 2.9923, and g²⁸ ≈ 5.0291.
-Another glowing example is the map {{val|53 84 123}}, for which a good generator will give you g¹² ≈ 2.0002, g¹⁹ ≈ 3.0005, g²⁸ ≈ 4.9974. This speaks to historical attention given to [[53edo|53-ET]]. So while 53:84:123 is an even better approximation of log(2:3:5) (and [https://en.xen.wiki/images/a/a2/Generalized_Patent_Vals.png you won’t find a better one] until 118:187:274), of course its integers aren’t as low, so that lessens its appeal.
+Another glowing example is the map {{val|53 84 123}}, for which a good generator will give you g⁵³ ≈ 2.0002, g⁸⁴ ≈ 3.0005, g¹²³ ≈ 4.9974. This speaks to historical attention given to [[53edo|53-ET]]. So while 53:84:123 is an even better approximation of log(2:3:5) (and [https://en.xen.wiki/images/a/a2/Generalized_Patent_Vals.png you won’t find a better one] until 118:187:274), of course its integers aren’t as low, so that lessens its appeal.
 Why is this rare? Well, it’s like a game of trying to get these numbers to line up ''(see Figure 2d)'':