Douglas Blumeyer's RTT How-To: Difference between revisions
Cmloegcmluin (talk | contribs) m →tempering out commas: include trailing 0s here too |
Cmloegcmluin (talk | contribs) m →approximating JI: add link to search for yourself |
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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 1c)''. And that’s a pretty cool thing to find! | 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 1c)''. And that’s a pretty cool thing to find! | ||
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 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 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. | ||
Why is this rare? Well, it’s like a game of trying to get these numbers to line up ''(see Figure 1d)'': | Why is this rare? Well, it’s like a game of trying to get these numbers to line up ''(see Figure 1d)'': | ||
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And why is this cool? Well, if 12-ET approximates the harmonic building blocks well individually, then JI intervals built out of them, like 16/15, 5/4, 10/9, etc. should also be reasonably well-approximated overall, and thus recognizable as their JI counterparts in musical context. You could think of it like taking all the primes in a prime factorization and swapping in their approximations. For example, if 16/15 = 2⁴ × 3⁻¹ × 5⁻¹ ≈ 1.067, and in 12-ET 2, 3, and 5 are approximated by 1.059¹² ≈ 1.998, 1.059¹⁹ ≈ 2.992, and 1.059²⁸ ≈ 5.029, respectively, then in 12-ET 16/15 maps to 1.998⁴ × 2.992⁻¹ × 5.029⁻¹ ≈ 1.059, which is indeed pretty close to 1.067. Of course, we should also note that 1.059 is the same as our step of 12-ET, which checks out with our calculation we made in the previous section that the best approximation of 16/15 in 12-ET would be 1 step. | And why is this cool? Well, if 12-ET approximates the harmonic building blocks well individually, then JI intervals built out of them, like 16/15, 5/4, 10/9, etc. should also be reasonably well-approximated overall, and thus recognizable as their JI counterparts in musical context. You could think of it like taking all the primes in a prime factorization and swapping in their approximations. For example, if 16/15 = 2⁴ × 3⁻¹ × 5⁻¹ ≈ 1.067, and in 12-ET 2, 3, and 5 are approximated by 1.059¹² ≈ 1.998, 1.059¹⁹ ≈ 2.992, and 1.059²⁸ ≈ 5.029, respectively, then in 12-ET 16/15 maps to 1.998⁴ × 2.992⁻¹ × 5.029⁻¹ ≈ 1.059, which is indeed pretty close to 1.067. Of course, we should also note that 1.059 is the same as our step of 12-ET, which checks out with our calculation we made in the previous section that the best approximation of 16/15 in 12-ET would be 1 step. | ||
=== tuning & pure octaves === | === tuning & pure octaves === | ||