Metallic MOS: Difference between revisions

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=== Beyond golden cases ===

If the golden mean is the value for which a:b = (a+b):a, then the silver mean is the value for which

If the golden mean is the value for which <math>a{:}b = (a+b){:}a</math>, then the silver mean is the value for which

<math> a:b = (2a+b):a = δ_s

<math> a{:}b = (2a+b){:}a = δ_s

</math>

Following the same logic as we followed for the golden case,

<math> L{:}s = (2L+s):L

<math> L{:}s = (2L+s){:}L = δ_s

</math>

So, wherever we have a scale where <math>L{:}s = δ_s</math>, we’ll also see the interval pattern

<math> L{:}s = (2L+s):L = (5L+2s):(2L+s) = (12L+5s):(5L+2s) = (29L+12s):(12L+5s) ~~= …~~ = δ_s

<math>

\newenvironment{rcases}

{\left.\begin{aligned}}

{\end{aligned}\right\rbrace}

\begin{rcases}

L&{:}s \\

(2L+s)&{:}L \\

(5L+2s)&{:}(2L+s) \\

(12L+5s)&{:}(5L+2s) \\

(29L+12s)&{:}(12L+5s) \\

&\vdots

\end{rcases}

= δ_s

</math>

Every other scale the silver generator generates has an <math>L{:}s</math> other than <math>δ_s</math>, namely, its isotope, <math>δ_s - 1</math>. These scales have a different pattern:

<math> L{:}s = (L+2s):(L+s) = δ_s - 1

<math> L{:}s = (L+2s){:}(L+s) = δ_s - 1

</math>

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) ~~= …~~ = δ_s - 1

<math>

\newenvironment{rcases}

{\left.\begin{aligned}}

{\end{aligned}\right\rbrace}

\begin{rcases}

L&{:}s \\

(L+2s)&{:}(L+s) \\

(3L+4s)&{:}(2L+3s) \\

(7L+10s)&{:}(5L+7s) \\

(17L+24s)&{:}(12L+17s) \\

&\vdots

\end{rcases}

= δ_s - 1

</math>

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<math>

\begin{aligned}

L \\

2L+s \\

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12L+5s \\

29L+12s \\

… \\

\vdots \\

\end{aligned}

</math>

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5L+2s \\

12L+5s \\

… \\

\vdots \\

</math>

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 <math>L</math> 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 <math>s</math>.

And if we want to understand the interval pattern for <math>δ_s - 1</math>, we’ll look at the right and left sides separately:

<math>

\begin{aligned}

L \\

L+2s \\

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Line 900:

7L+10s \\

17L+24s \\

… \\

\vdots \\

\end{aligned}

</math>

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5L+7s \\

12L+17s \\

… \\

\vdots \\

</math>

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There’s something a bit different about the interval pattern for <math>δ_s - 1</math> from the other two we’ve looked at so far. The interval patterns for <math>δ_s</math> and <math>φ</math> 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

</math>

here we do not see such overlapping; the pattern of intervals looks more like

<math> a:b = c:d = e:f = … = δ_s - 1

<math> a{:}b = c{:}d = e{:}f = … = δ_s - 1

</math>

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 <math>L</math>, which is the same as the <math>L</math> size of the preceding ratio. However, for the silver mean’s first isotope here, no such link exists, since s is substituted not for <math>L</math>, but <math>(L+s)</math>.

Another way of looking at this is: for <math>δ_s</math> and <math>φ</math>, 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 <math>δ_s - 1</math>, s and ~~L’s~~ interval sequences are completely different.

Another way of looking at this is: for <math>δ_s</math> and <math>φ</math>, it was the case that both <math>s</math> and <math>L</math>’s interval sequences were the same, just offset from each other by a step. Whereas for <math>δ_s - 1</math>, <math>s</math> and <math>L</math>’s interval sequences are completely different.

Finally, for the bronze ratio,

<math> a:b = (3a+b):a

<math> a{:}b = (3a+b){:}a

</math>

As expected, <math>L{:}s = (3L+s):L</math> is only true of every ''third'' scale the bronze generator generates. The remaining interval relationships are left as an exercise for the reader.

As expected, <math>L{:}s = (3L+s){:}L</math> is only true of every ''third'' scale the bronze generator generates. The remaining interval relationships are left as an exercise for the reader.

= Gallery of generators =

@@ Line 814: / Line 814: @@
 === Beyond golden cases ===
-If the golden mean is the value for which a:b = (a+b):a, then the silver mean is the value for which
+If the golden mean is the value for which <span><math>a{:}b = (a+b){:}a</math></span>, then the silver mean is the value for which
-<math> a:b = (2a+b):a = δ_s
+<math> a{:}b = (2a+b){:}a = δ_s
 </math>
 Following the same logic as we followed for the golden case,
-<math> L{:}s = (2L+s):L
+<math> L{:}s = (2L+s){:}L = δ_s
 </math>
 So, wherever we have a scale where <span><math>L{:}s = δ_s</math></span>, we’ll also see the interval pattern
-<math> L{:}s = (2L+s):L = (5L+2s):(2L+s) = (12L+5s):(5L+2s) = (29L+12s):(12L+5s) = … = δ_s
+<math>
+\newenvironment{rcases}
+  {\left.\begin{aligned}}
+  {\end{aligned}\right\rbrace}
+\begin{rcases}
+L&{:}s \\
+(2L+s)&{:}L \\
+(5L+2s)&{:}(2L+s) \\
+(12L+5s)&{:}(5L+2s) \\
+(29L+12s)&{:}(12L+5s) \\
+&\vdots
+\end{rcases}
+= δ_s
 </math>
 Every other scale the silver generator generates has an <span><math>L{:}s</math></span> 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> L{:}s = (L+2s){:}(L+s) = δ_s - 1
 </math>
 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) = … = δ_s - 1
+<math>
+\newenvironment{rcases}
+  {\left.\begin{aligned}}
+  {\end{aligned}\right\rbrace}
+\begin{rcases}
+L&{:}s \\
+(L+2s)&{:}(L+s) \\
+(3L+4s)&{:}(2L+3s) \\
+(7L+10s)&{:}(5L+7s) \\
+(17L+24s)&{:}(12L+17s) \\
+&\vdots
+\end{rcases}
+= δ_s - 1
 </math>
@@ Line 843: / Line 870: @@
 <math>
+\begin{aligned}
 L \\
 L+s \\
@@ Line 848: / Line 876: @@
 L+5s \\
 L+12s \\
-… \\
+\vdots \\
+\end{aligned}
 </math>
@@ Line 857: / Line 886: @@
 L+2s \\
 L+5s \\
-… \\
+\vdots \\
 </math>
-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 <span><math>L</math></span> 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 <span><math>s</math></span>.
 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>
+\begin{aligned}
 L \\
 L+2s \\
@@ Line 870: / Line 900: @@
 L+10s \\
 L+24s \\
-… \\
+\vdots \\
+\end{aligned}
 </math>
@@ Line 879: / Line 910: @@
 L+7s \\
 L+17s \\
-… \\
+\vdots \\
 </math>
@@ Line 886: / Line 917: @@
 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
 </math>
 here we do not see such overlapping; the pattern of intervals looks more like
-<math> a:b = c:d = e:f = … = δ_s - 1
+<math> a{:}b = c{:}d = e{:}f = … = δ_s - 1
 </math>
-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 <span><math>L</math></span>, which is the same as the <span><math>L</math></span> size of the preceding ratio. However, for the silver mean’s first isotope here, no such link exists, since s is substituted not for <span><math>L</math></span>, but <span><math>(L+s)</math></span>.
-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.
+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 <span><math>s</math></span> and <span><math>L</math></span>’s interval sequences were the same, just offset from each other by a step. Whereas for <span><math>δ_s - 1</math></span>, <span><math>s</math></span> and <span><math>L</math></span>’s interval sequences are completely different.
 Finally, for the bronze ratio,
-<math> a:b = (3a+b):a
+<math> a{:}b = (3a+b){:}a
 </math>
-As expected, <span><math>L{:}s = (3L+s):L</math></span> is only true of every ''third'' scale the bronze generator generates. The remaining interval relationships are left as an exercise for the reader.
+As expected, <span><math>L{:}s = (3L+s){:}L</math></span> is only true of every ''third'' scale the bronze generator generates. The remaining interval relationships are left as an exercise for the reader.
 = Gallery of generators =