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{{texcases}}
Metallic MOS scales are a family of [[MOS scales]] generated by {{w|metallic means}}. These scales offer interesting musical possibilities due to special mathematical properties of the metallic means.  
Metallic MOS scales are a family of [[MOS scales]] generated by {{w|metallic means}}. These scales offer interesting musical possibilities due to special mathematical properties of the metallic means.  


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<math>
<math>
\newenvironment{rcases}
\newenvironment{rcases}
   {\left.\begin{aligned}}
   {\left.\begin{align*}}
   {\end{aligned}\right\rbrace}
   {\end{align*}\right\rbrace}


\begin{rcases}
\begin{rcases}
L&:s \\
L &: s \\
L+s&:L \\
L + s &: L \\
2L+s&:L+s \\
2L + s &: L + s \\
3L+2s&:2L+s \\
3L + 2s &: 2L + s \\
5L+3s&:3L+2s \\
5L + 3s &: 3L + 2s \\
&\vdots \\
&\vdots \\
\end{rcases}
\end{rcases}
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<math>
<math>
\newenvironment{rcases}
\newenvironment{rcases}
   {\left.\begin{aligned}}
   {\left.\begin{align*}}
   {\end{aligned}\right\rbrace}
   {\end{align*}\right\rbrace}


\begin{rcases}
\begin{rcases}
L&:s \\
L &: s \\
2L+s&:L \\
2L + s &: L \\
5L+2s&:2L+s \\
5L + 2s &: 2L + s \\
12L+5s&:5L+2s \\
12L + 5s &: 5L + 2s \\
29L+12s&:12L+5s \\
29L + 12s &: 12L + 5s \\
\vdots \\
&\vdots \\
\end{rcases}
\end{rcases}
= \delta_s
= \delta_s
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<math>
<math>
\newenvironment{rcases}
\newenvironment{rcases}
   {\left.\begin{aligned}}
   {\left.\begin{align*}}
   {\end{aligned}\right\rbrace}
   {\end{align*}\right\rbrace}


\begin{rcases}
\begin{rcases}
L&:s \\
L &: s \\
L+2s&:L+s \\
L + 2s &: L + s \\
3L+4s&:2L+3s \\
3L + 4s &: 2L + 3s \\
7L+10s&:5L+7s \\
7L + 10s &: 5L + 7s \\
17L+24s&:12L+17s \\
17L + 24s &: 12L + 17s \\
\vdots \\
&\vdots \\
\end{rcases}
\end{rcases}
= \delta_s - 1
= \delta_s - 1
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We know that the golden generator's {{nowrap|''L'':''s'' {{=}} &phi;}}, but we can also say this about them:
We know that the golden generator's {{nowrap|''L'':''s'' {{=}} &phi;}}, but we can also say this about them:


<math>\qquad (L+s){:}L = \phi
<math>\qquad (L + s){:}L = \phi
</math>
</math>


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This fact follows from one of the many ways of defining the golden mean: the value for which  
This fact follows from one of the many ways of defining the golden mean: the value for which  


<math>\qquad a{:}b = (a+b){:}a = \phi
<math>\qquad a{:}b = (a + b){:}a = \phi
</math>
</math>


We can substitute into this equation our large and small scale step sizes in place of ''a'' and ''b'', respectively, to see that
We can substitute into this equation our large and small scale step sizes in place of ''a'' and ''b'', respectively, to see that


<math>\qquad L{:}s = (L+s){:}L = \phi
<math>\qquad L{:}s = (L + s){:}L = \phi
</math>
</math>


But we're only getting started. This situation has recursive potential. We can now substitute <math>L+s</math> in for <math>L</math> as long as we also substitute in ''L'' for ''s'', and we'll still get a ratio that equals &phi;:
But we're only getting started. This situation has recursive potential. We can now substitute <math>L + s</math> in for <math>L</math> as long as we also substitute in ''L'' for ''s'', and we'll still get a ratio that equals &phi;:


<math>
<math>
\begin{align}
\begin{align}
((L+s)+(L)){:}(L+s) &= \\
((L + s)+(L)){:}(L + s) &= \\
(2L+s){:}(L+s) &= \\
(2L + s){:}(L + s) &= \\
\phi
\phi
\end{align}
\end{align}
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\begin{rcases}
\begin{rcases}
L&:s \\
L &: s \\
L+s&:L \\
L + s &: L \\
2L+s&:L+s \\
2L + s &: L + s \\
3L+2s&:2L+s \\
3L + 2s &: 2L + s \\
5L+3s&:3L+2s \\
5L + 3s &: 3L + 2s \\
&\vdots
&\vdots
\end{rcases}
\end{rcases}
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\begin{aligned}
\begin{aligned}
L \\
L \\
L+s \\
L + s \\
2L+s \\
2L + s \\
3L+2s \\
3L + 2s \\
5L+3s \\
5L + 3s \\
\vdots
\vdots
\end{aligned}
\end{aligned}
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s \\
s \\
L \\
L \\
L+s \\
L + s \\
2L+s \\
2L + s \\
3L+2s \\
3L + 2s \\
\vdots
\vdots
</math>
</math>
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If the golden mean is the value for which {{nowrap|''a'':''b'' {{=}} (''a'' + ''b''):''a''}}, then the silver mean is the value for which
If the golden mean is the value for which {{nowrap|''a'':''b'' {{=}} (''a'' + ''b''):''a''}}, then the silver mean is the value for which


<math> a{:}b = (2a+b){:}a = \delta_s
<math> a{:}b = (2a + b){:}a = \delta_s
</math>
</math>


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


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


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<math>
<math>
\newenvironment{rcases}
\newenvironment{rcases}
   {\left.\begin{aligned}}
   {\left.\begin{align*}}
   {\end{aligned}\right\rbrace}
   {\end{align*}\right\rbrace}


\begin{rcases}
\begin{rcases}
L&{:}s \\
L &: s \\
(2L+s)&{:}L \\
(2L + s) &: L \\
(5L+2s)&{:}(2L+s) \\
(5L + 2s) &: (2L + s) \\
(12L+5s)&{:}(5L+2s) \\
(12L + 5s) &: (5L + 2s) \\
(29L+12s)&{:}(12L+5s) \\
(29L + 12s) &: (12L + 5s) \\
&\vdots  
&\vdots
\end{rcases}
\end{rcases}
= \delta_s
= \delta_s
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Every other scale the silver generator generates has an ''L'':''s'' other than &delta;<sub>''s''</sub>, namely, its isotope, {{nowrap|&delta;<sub>''s''</sub> &minus; 1}}. These scales have a different pattern:
Every other scale the silver generator generates has an ''L'':''s'' other than &delta;<sub>''s''</sub>, namely, its isotope, {{nowrap|&delta;<sub>''s''</sub> &minus; 1}}. These scales have a different pattern:


<math> L{:}s = (L+2s){:}(L+s) = \delta_s - 1
<math> L{:}s = (L + 2s){:}(L + s) = \delta_s - 1
</math>
</math>


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<math>
<math>
\newenvironment{rcases}
\newenvironment{rcases}
   {\left.\begin{aligned}}
   {\left.\begin{align*}}
   {\end{aligned}\right\rbrace}
   {\end{align*}\right\rbrace}


\begin{rcases}
\begin{rcases}
L&{:}s \\
L&{:}s \\
(L+2s)&{:}(L+s) \\
(L + 2s)&{:}(L + s) \\
(3L+4s)&{:}(2L+3s) \\
(3L + 4s)&{:}(2L + 3s) \\
(7L+10s)&{:}(5L+7s) \\
(7L + 10s)&{:}(5L + 7s) \\
(17L+24s)&{:}(12L+17s) \\
(17L + 24s)&{:}(12L + 17s) \\
&\vdots  
&\vdots  
\end{rcases}
\end{rcases}
Line 834: Line 835:
\begin{aligned}
\begin{aligned}
L \\
L \\
2L+s \\
2L + s \\
5L+2s \\
5L + 2s \\
12L+5s \\
12L + 5s \\
29L+12s \\
29L + 12s \\
\vdots \\
\vdots \\
\end{aligned}
\end{aligned}
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s \\
s \\
L \\
L \\
2L+s \\
2L + s \\
5L+2s \\
5L + 2s \\
12L+5s \\
12L + 5s \\
\vdots \\
\vdots \\
</math>
</math>
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\begin{aligned}
\begin{aligned}
L \\
L \\
L+2s \\
L + 2s \\
3L+4s \\
3L + 4s \\
7L+10s \\
7L + 10s \\
17L+24s \\
17L + 24s \\
\vdots \\
\vdots \\
\end{aligned}
\end{aligned}
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<math>
<math>
s \\
s \\
L+s \\
L + s \\
2L+3s \\
2L + 3s \\
5L+7s \\
5L + 7s \\
12L+17s \\
12L + 17s \\
\vdots \\
\vdots \\
</math>
</math>
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Finally, for the bronze ratio,  
Finally, for the bronze ratio,  


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


Line 1,220: Line 1,221:
So by the 7th level, when looking at the bronze scale tree, we should see a total of  
So by the 7th level, when looking at the bronze scale tree, we should see a total of  


<math> 3+3+6+12+24+48+96 = 192
<math> 3 + 3 + 6 + 12 + 24 + 48 + 96 = 192
</math>
</math>


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