Metallic MOS: Difference between revisions

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We know that the golden generator’s <math>L{:}s = φ</math>, but we can also say this about them:

(L+s):L = φ

<math>\qquad (L+s){:}L = φ

</math>

In other words, any interval in the scale which spans exactly one large and one small step is <math>φ</math> times the size of one large step.

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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 = φ

<math>\qquad a{:}b = (a+b){:}a = φ

</math>

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 = φ

<math>\qquad L{:}s = (L+s){:}L = φ

</math>

But we’re only getting started. This situation has recursive potential. We can now substitute (L+s) in for L as long as we also substitute in L for s, and we’ll still get a ratio that = φ:

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 <math>L</math> for <math>s</math>, and we’ll still get a ratio that <math>= φ</math>:

<math>\~~qquad~~ ((L+s)+(L)):(L+s) = (2L+s):(L+s) = φ

<math>

\begin{align}

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

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

φ

\end{align}

</math>

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[[File:Golden horogram.png|alt=horogram for g ≈ 0.381966, 7 iterations|right|392x392px|horogram for g ≈ 0.381966, 7 iterations]]

Horograms depict the scale sequences of MOS generators. To understand how the horogram illustrates the interval pattern, too, first consider just the left side of the interval pattern, for L:

Horograms depict the scale sequences of MOS generators. To understand how the horogram illustrates the interval pattern, too, first consider just the left side of the interval pattern, for <math>L</math>:

<math>

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</math>

Now find any L in the horogram and observe how it gets split up as we iterate through the scale sequence. In the next iteration, L will be replaced with an L and an s. After two iterations, the original L interval is now represented by two ~~L’s~~ and an s. And so forth.

Now find any <math>L</math> in the horogram and observe how it gets split up as we iterate through the scale sequence. In the next iteration, <math>L</math> will be replaced with an <math>L</math> and an <math>s</math>. After two iterations, the original <math>L</math> interval is now represented by two <math>L</math>’s and an <math>s</math>. And so forth.

The same will hold for the right side of the interval pattern, for s:

The same will hold for the right side of the interval pattern, for <math>s</math>:

<math>

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Find any s in the horogram and observe how it gets split up as we iterate through the scale sequence. In the next iteration, s will be replaced with L. After two iterations, the original s interval is now represented by an L and an s. And so forth.

Find any <math>s</math> in the horogram and observe how it gets split up as we iterate through the scale sequence. In the next iteration, <math>s</math> will be replaced with <math>L</math>. After two iterations, the original <math>s</math> interval is now represented by an <math>L</math> and an <math>s</math>. And so forth.

Every MOS scale contains every scale earlier in its scale sequence. In other words, any interval that existed in an earlier scale will remain in all later scales. These earlier ~~L’s~~ and ~~s’s~~ that remain ⁠— only now spanning many ~~L’s~~ and ~~s’s~~ each — are precisely the larger intervals in the scale that also exhibit the <math>φ</math> ratio to each other.

Every MOS scale contains every scale earlier in its scale sequence. In other words, any interval that existed in an earlier scale will remain in all later scales. These earlier <math>L</math>’s and <math>s</math>’s that remain ⁠— only now spanning many <math>L</math>’s and <math>s</math>’s each — are precisely the larger intervals in the scale that also exhibit the <math>φ</math> ratio to each other.

=== Beyond golden cases ===

@@ Line 738: / Line 738: @@
 We know that the golden generator’s <span><math>L{:}s = φ</math></span>, but we can also say this about them:
-(L+s):L = φ
+<math>\qquad (L+s){:}L = φ
+</math>
 In other words, any interval in the scale which spans exactly one large and one small step is <span><math>φ</math></span> times the size of one large step.
@@ Line 744: / Line 745: @@
 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 = φ
+<math>\qquad a{:}b = (a+b){:}a = φ
 </math>
 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 = φ
+<math>\qquad L{:}s = (L+s){:}L = φ
 </math>
-But we’re only getting started. This situation has recursive potential. We can now substitute (L+s) in for L as long as we also substitute in L for s, and we’ll still get a ratio that = φ:
+But we’re only getting started. This situation has recursive potential. We can now substitute <span><math>L+s</math></span> in for <span><math>L</math></span> as long as we also substitute in <span><math>L</math></span> for <span><math>s</math></span>, and we’ll still get a ratio that <span><math>= φ</math></span>:
-<math>\qquad ((L+s)+(L)):(L+s) = (2L+s):(L+s) = φ
+<math>
+\begin{align}
+((L+s)+(L)){:}(L+s) &= \\
+(2L+s){:}(L+s) &= \\
+φ
+\end{align}
 </math>
@@ Line 776: / Line 782: @@
 [[File:Golden horogram.png|alt=horogram for g ≈ 0.381966, 7 iterations|right|392x392px|horogram for g ≈ 0.381966, 7 iterations]]
-Horograms depict the scale sequences of MOS generators. To understand how the horogram illustrates the interval pattern, too, first consider just the left side of the interval pattern, for L:
+Horograms depict the scale sequences of MOS generators. To understand how the horogram illustrates the interval pattern, too, first consider just the left side of the interval pattern, for <span><math>L</math></span>:
 <math>
@@ Line 787: / Line 793: @@
 </math>
-Now find any L in the horogram and observe how it gets split up as we iterate through the scale sequence. In the next iteration, L will be replaced with an L and an s. After two iterations, the original L interval is now represented by two L’s and an s. And so forth.
+Now find any <span><math>L</math></span> in the horogram and observe how it gets split up as we iterate through the scale sequence. In the next iteration, <span><math>L</math></span> will be replaced with an <span><math>L</math></span> and an <span><math>s</math></span>. After two iterations, the original <span><math>L</math></span> interval is now represented by two <span><math>L</math></span>’s and an <span><math>s</math></span>. And so forth.
-The same will hold for the right side of the interval pattern, for s:
+The same will hold for the right side of the interval pattern, for <span><math>s</math></span>:
 <math>
@@ Line 801: / Line 807: @@
-Find any s in the horogram and observe how it gets split up as we iterate through the scale sequence. In the next iteration, s will be replaced with L. After two iterations, the original s interval is now represented by an L and an s. And so forth.
+Find any <span><math>s</math></span> in the horogram and observe how it gets split up as we iterate through the scale sequence. In the next iteration, <span><math>s</math></span> will be replaced with <span><math>L</math></span>. After two iterations, the original <span><math>s</math></span> interval is now represented by an <span><math>L</math></span> and an <span><math>s</math></span>. And so forth.
-Every MOS scale contains every scale earlier in its scale sequence. In other words, any interval that existed in an earlier scale will remain in all later scales. These earlier L’s and s’s that remain ⁠— only now spanning many L’s and s’s each — are precisely the larger intervals in the scale that also exhibit the <span><math>φ</math></span> ratio to each other.
+Every MOS scale contains every scale earlier in its scale sequence. In other words, any interval that existed in an earlier scale will remain in all later scales. These earlier <span><math>L</math></span>’s and <span><math>s</math></span>’s that remain ⁠— only now spanning many <span><math>L</math></span>’s and <span><math>s</math></span>’s each — are precisely the larger intervals in the scale that also exhibit the <span><math>φ</math></span> ratio to each other.
 === Beyond golden cases ===