Metallic MOS: Difference between revisions
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But that’s not all. Due to the mathemagic of phi, we also get a recursive interval relationship pattern: | But that’s not all. Due to the mathemagic of phi, we also get a recursive interval relationship pattern: | ||
<math>\ | <math> | ||
\begin{align} | |||
φ = | |||
\begin{cases} | |||
L&:s \\ | |||
L+s&:L \\ | |||
2L+s&:L+s \\ | |||
3L+2s&:2L+s \\ | |||
5L+3s&:3L+2s \\ | |||
… \\ | |||
\end{cases} | |||
\end{align} | |||
</math> | </math> | ||
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In our very first case — that of the golden generator — “related by” ''could'' be defined as simply “having a ratio of”. We’ll check the segment lengths to confirm this. One of the two segments is, of course, equal to the golden generator, approximately 0.381966. The other is equal to the remainder of the golden generator with the period: | In our very first case — that of the golden generator — “related by” ''could'' be defined as simply “having a ratio of”. We’ll check the segment lengths to confirm this. One of the two segments is, of course, equal to the golden generator, approximately 0.381966. The other is equal to the remainder of the golden generator with the period: | ||
<math>\qquad 1 - 0.381966 ≈ 0.618034 | |||
</math> | |||
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 / 0.381966 ≈ φ | |||
</math> | |||
However enticingly simple this definition may be, it unfortunately does not work in general. In fact, it ''only'' gives the correct value when the interval being split is the entire period. | However enticingly simple this definition may be, it unfortunately does not work in general. In fact, it ''only'' gives the correct value when the interval being split is the entire period. | ||
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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{num1 + num2}{den1 + den2} | |||
</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: | ||