Metallic MOS: Difference between revisions

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=== Why they cycle ===

That is true of scale iterations where L - s > s. For the other type of scale iteration, where L - s < s, the result is simply reciprocated:

That is true of scale iterations where <math>L - s > s</math>. For the other type of scale iteration, where <math>L - s < s</math>, the result is simply reciprocated:

<math>\~~qquad~~ L’:s’ = s:(L - s) = 1:(L - 1) = 1 ~~/ (~~L - 1)

<math>

\begin{align}

L’{:}s’ &= s{:}(L - s) \\

&= 1{:}(L - 1) \\

&= \frac{1}{L - 1}

\end{align}

</math>

This alone would not suffice to explain how the <math>L{:}s</math> sequences lock into a cycle of isotopes. But here’s where the magic of the metallic means comes into play. <math>φ</math> has the property that

<math>\qquad φ - 1 = 1 / φ

<math>\qquad φ - 1 = \frac{1}{φ}

</math>

So, in the case of <math>φ</math>:

<math>\~~qquad~~ L’:s’ = 1 ~~/ (~~L - 1) = 1 ~~/ (~~φ - 1) = 1 ~~/ (~~1 / φ) = φ

<math>

\begin{align}

L’{:}s’ &= \frac{1}{L - 1} \\

&= \frac{1}{φ - 1} \\

&= \frac{1}{\frac{1}{φ}} \\

&= φ

\end{align}

</math>

Line 481:

Line 492:

A similar case is true for the silver mean, except we have to subtract 1 from it twice before the resulting value’s reciprocal is equal to the silver mean.

<math>\qquad δ_s - 1 - 1 = 1 / δ_s

<math>\qquad δ_s - 1 - 1 = \frac{1}{δ_s}

</math>

And for the bronze ratio, we must subtract thrice:

<math>\qquad ~~δ_s~~ - 1 - 1 - 1 = 1 ~~/ δ_s~~

<math>\qquad δ_b - 1 - 1 - 1 = \frac{1}{δ_b}

</math>

@@ Line 462: / Line 462: @@
 === Why they cycle ===
-That is true of scale iterations where L - s > s. For the other type of scale iteration, where L - s < s, the result is simply reciprocated:
+That is true of scale iterations where <span><math>L - s > s</math></span>. For the other type of scale iteration, where <span><math>L - s < s</math></span>, the result is simply reciprocated:
-<math>\qquad L’:s’ = s:(L - s) = 1:(L - 1) = 1 / (L - 1)
+<math>
+\begin{align}
+L’{:}s’ &= s{:}(L - s) \\
+&= 1{:}(L - 1) \\
+&= \frac{1}{L - 1}
+\end{align}
 </math>
 This alone would not suffice to explain how the <span><math>L{:}s</math></span> sequences lock into a cycle of isotopes. But here’s where the magic of the metallic means comes into play. <span><math>φ</math></span> has the property that
-<math>\qquad  φ - 1 = 1 / φ
+<math>\qquad  φ - 1 = \frac{1}{φ}
 </math>
 So, in the case of <span><math>φ</math></span>:
-<math>\qquad  L’:s’ = 1 / (L - 1) = 1 / (φ - 1) = 1 / (1 / φ) = φ
+<math>
+\begin{align}
+L’{:}s’ &= \frac{1}{L - 1} \\
+&= \frac{1}{φ - 1} \\
+&= \frac{1}{\frac{1}{φ}} \\
+&= φ
+\end{align}
 </math>
@@ Line 481: / Line 492: @@
 A similar case is true for the silver mean, except we have to subtract 1 from it twice before the resulting value’s reciprocal is equal to the silver mean.
-<math>\qquad  δ_s - 1 - 1 = 1 / δ_s
+<math>\qquad  δ_s - 1 - 1 = \frac{1}{δ_s}
 </math>
 And for the bronze ratio, we must subtract thrice:
-<math>\qquad  δ_s - 1 - 1 - 1 = 1 / δ_s
+<math>\qquad  δ_b - 1 - 1 - 1 = \frac{1}{δ_b}
 </math>