Metallic MOS: Difference between revisions
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Every other scale the silver generator generates has an L:s other than | Every other scale the silver generator generates has an L:s other than <span><math>δ_s</math></span>, namely, its isotope, <span><math>δ_s - 1</math></span>. These scales have a different pattern: | ||
<math>L:s = (L+2s):(L+s) = δ_s - 1 | |||
</math> | |||
Due to this different pattern, we’ll see the different interval pattern | Due to this different pattern, we’ll see the different interval pattern | ||
<math> L:s = (L+2s):(L+s) = (3L+4s):(2L+3s) = (7L+10s):(5L+7s) = (17L+24s):(12L+17s) = … = | <math> L:s = (L+2s):(L+s) = (3L+4s):(2L+3s) = (7L+10s):(5L+7s) = (17L+24s):(12L+17s) = … = δ_s - 1 | ||
</math> | </math> | ||
[[File:Silver horogram.png|alt=horogram for g = 0.292893, 7 iterations|right|396x396px|horogram for g = 0.292893, 7 iterations]] | [[File:Silver horogram.png|alt=horogram for g = 0.292893, 7 iterations|right|396x396px|horogram for g = 0.292893, 7 iterations]] | ||
We can use the horogram for the silver generator to see how its interval pattern cycle is length 2, i.e. that it alternates between two different interval patterns. If we want to understand the interval pattern for | We can use the horogram for the silver generator to see how its interval pattern cycle is length 2, i.e. that it alternates between two different interval patterns. If we want to understand the interval pattern for <span><math>δ_s</math></span>, we’ll look at the right and left sides separately, as we did with the golden: | ||
<math> | <math> | ||
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We’ll repeat the technique we used for the golden case: find any L in the horogram and observe how it gets split up as we iterate through the scale sequence. However, the complexity that silver introduces is that we don’t look to the next iteration to see the next entry in the interval pattern; we have to skip an iteration. So if we just look at all the odd rings, ring 1, 3, 5, 7, etc. then we’ll see the pattern. The same is true of s. | We’ll repeat the technique we used for the golden case: find any L in the horogram and observe how it gets split up as we iterate through the scale sequence. However, the complexity that silver introduces is that we don’t look to the next iteration to see the next entry in the interval pattern; we have to skip an iteration. So if we just look at all the odd rings, ring 1, 3, 5, 7, etc. then we’ll see the pattern. The same is true of s. | ||
And if we want to understand the interval pattern for | And if we want to understand the interval pattern for <span><math>δ_s - 1</math></span>, we’ll look at the right and left sides separately: | ||
<math> | <math> | ||
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And now we’ll look not at the odd, but at the even iterations, rings 2, 4, 6, 8, etc. to see the pattern visualized. | And now we’ll look not at the odd, but at the even iterations, rings 2, 4, 6, 8, etc. to see the pattern visualized. | ||
There’s something a bit different about the interval pattern for | There’s something a bit different about the interval pattern for <span><math>δ_s - 1</math></span> from the other two we’ve looked at so far. The interval patterns for <span><math>δ_s</math></span> and <span><math>φ</math></span> exhibited overlap, i.e. we saw something like | ||
<math> a:b = b:c = c:d = … = δ_s | <math> a:b = b:c = c:d = … = δ_s | ||
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The reason the other cases exhibited such overlapping is that the small step size of the next ratio in the equivalence pattern became an L, which is the same as the L size of the preceding ratio. However, for the silver mean’s first isotope here, no such link exists, since s is substituted not for L, but (L+s). | The reason the other cases exhibited such overlapping is that the small step size of the next ratio in the equivalence pattern became an L, which is the same as the L size of the preceding ratio. However, for the silver mean’s first isotope here, no such link exists, since s is substituted not for L, but (L+s). | ||
Another way of looking at this is: for | Another way of looking at this is: for <span><math>δ_s</math></span> and <span><math>φ</math></span>, it was the case that both s and L’s interval sequences were the same, just offset from each other by a step. Whereas for <span><math>δ_s - 1</math></span>, s and L’s interval sequences are completely different. | ||
Finally, for the bronze ratio, | Finally, for the bronze ratio, | ||