Metallic MOS: Difference between revisions
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=== As the formula for generators === | === As the formula for generators === | ||
Recall that in order to find a metallic generator, we find a value which splits an interval into two segments related by a metallic mean. When this technique was introduced earlier, we left the definition of “related by” vague. Well, now is the time to make it explicit. | Recall that in order to find a metallic generator, we find a value which splits an interval into two segments related by a metallic mean. When this technique was introduced earlier, we left the definition of “related by” vague. Well, now is the time to make it explicit. | ||
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And as we can see, the ratio between those two lengths is phi: | And as we can see, the ratio between those two lengths is phi: | ||
<math>\qquad 0.618034 | <math>\qquad \frac{0.618034}{0.381966} ≈ φ | ||
</math> | </math> | ||
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But even then it’s not quite that simple, because it’s not a ''simple'' mediant, which would look like this: | But even then it’s not quite that simple, because it’s not a ''simple'' mediant, which would look like this: | ||
<math>\qquad \frac{ | <math>\qquad \frac{a_1 + b_1}{a_2 + b_2} | ||
</math> | </math> | ||
Rather, it’s a [https://www.mathpages.com/home/kmath055/kmath055.htm ''weighted'' mediant], which looks like this: | Rather, it’s a [https://www.mathpages.com/home/kmath055/kmath055.htm ''weighted'' mediant], which looks like this: | ||
[ math ] | [ math ] \frac{a_1w_1 + b_1w_2}{a_2w_1 + b_2w_2} | ||
In particular, it is the mediant which is weighted by the desired metallic mean: | In particular, it is the mediant which is weighted by the desired metallic mean μ: | ||
[ math ] | [ math ] \frac{a_1μ + b_1}{a_2μ + b_2} | ||
=== As a bizarro average === | === As a bizarro average === | ||