Metallic MOS: Difference between revisions

Line 1,018:

The two segments are actually in a ratio equal to the weight multiplied by the ratio of the denominators of their respective bounds.

We’ll test this out on the example from before. We know that the weighted mediant formula with <math>φ</math> as weight, the interval between 1/3 and 1/2, and weight leaning toward the parent ratio (1/2) gives the value 0.419821. So our two segments are:

We’ll test this out on the example from before. We know that the weighted mediant formula with <math>φ</math> as weight, the interval between <math>\frac 13</math> and <math>\frac 12</math>, and weight leaning toward the parent ratio (<math>\frac 12</math>) gives the value <math>≈ 0.419821</math>. So our two segments are:

<math>

0.419821 - 0.333333 = 0.086487 \\

s_1 = 0.419821 - 0.333333 = 0.086487 \\

0.500000 - 0.419821 = 0.080179 \\

s_2 = 0.500000 - 0.419821 = 0.080179

</math>

Their ratio is

<math> 0.086487 / 0.080179 = 1.078674

<math> r = \frac{s_1}{s_2} = \frac{0.086487}{0.080179} = 1.078674

</math>

Which we can see is the weight multiplied by the bounding ratios’ denominators:

<math> 1.078674 = φ * 2/3

<math> 1.078674 = φ · \frac 23

</math>

In particular, we see that the denominator of the weighted ratio finds itself on the same side of the ratio as the weight.

The full derivation ~~is here:~~

The full derivation follows. With lower bounding ratio <math>\frac{a_1}{a_2}</math> and upper bounding ratio <math>\frac{b_1}{b_2}</math>, we have a mediant of <math>\frac{φa_1 + b_1}{φa_2 + b_2}</math>. So then the segment from the lower bounding ratio to the mediant has length

<math>

\require{cancel}

\begin{align}

s_1 &= \frac{φa_1 + b_1}{φa_2 + b_2} - \frac{a_1}{a_2} \\

&= \frac{a_2(φa_1 + b_1)}{a_2(φa_2 + b_2)} - \frac{a_1(φa_2 + b_2)}{a_2(φa_2 + b_2)} \\

&= \frac{a_2(φa_1 + b_1) - a_1(φa_2 + b_2)}{a_2(φa_2 + b_2)} \\

&= \frac{φa_1a_2 + a_2b_1 - φa_1a_2 - a_1b_2}{φa_2^2 + a_2b_2} \\

&= \frac{\cancel{φa_1a_2} + a_2b_1 - \cancel{φa_1a_2} - a_1b_2}{φa_2^2 + a_2b_2} \\

&= \frac{a_2b_1 - a_1b_2}{φa_2^2 + a_2b_2}

\end{align}

</math>

bounding ratio ~~1 = a1/a2~~

and the segment from the mediant to the upper bounding ratio has length

~~bounding ratio~~ 2 = ~~b1/b~~2</~~sub~~>

<math>

\require{cancel}

\begin{align}

s_2 &= \frac{b_1}{b_2} - \frac{φa_1 + b_1}{φa_2 + b_2} \\

&= \frac{b_1(φa_2 + b_2)}{b_2(φa_2 + b_2)} - \frac{b_2(φa_1 + b_1)}{b_2(φa_2 + b_2)} \\

&= \frac{b_1(φa_2 + b_2) - b_2(φa_1 + b_1)}{b_2(φa_2 + b_2)} \\

&= \frac{φa_2b_1 + b_1b_2 - φa_1b_2 - b_1b_2}{φa_2b_2 + b_2^2} \\

&= \frac{φa_2b_1 + \cancel{b_1b_2} - φa_1b_2 - \cancel{b_1b_2}}{φa_2b_2 + b_2^2} \\

&= \frac{φa_2b_1 - φa_1b_2}{φa_2b_2 + b_2^2}

\end{align}

</math>

~~their mediant = (φa1 + b1) / (φa2 + b2)~~

so then their ratio is

~~segment from bounding ratio 1 to mediant = (φa1 + b1) / (φa2 + b2) - a1/a2~~

~~a2(φa1 + b1) / a2(φa2 + b2) - a1(φa2 + b2)/a2(φa2 + b2)~~

~~( a2(φa1 + b1) - a1(φa2 + b2) ) / a2(φa2 + b2)~~

~~( φa1a2 + a2b1 - φa1a2 - a1b2 ) / (φa22 + a2b2)~~

~~( a2b1 - a1b2 ) / (φa22 + a2b2)~~

~~segment from bounding ratio 2 to mediant = b1/b2 - (φa1 + b1) / (φa2 + b2)~~

~~b1(φa2 + b2)/b2(φa2 + b2) - b2(φa1 + b1) / b2(φa2 + b2)~~

~~( b1(φa2 + b2) - b2(φa1 + b1) ) / b2(φa2 + b2)~~

~~( φa2b1 + b1b2 - φa1b2 - b1b2 ) / (φa2b2 + b22)~~

~~( φa2b1 - φa1b2 ) / (φa2b2 + b22)~~

so then their ratio is

( φa2b1 - φa1b2 )(φa22 + a2b2) / (φa2b2 + b22)( a2b1 - a1b2 )

φ( a2b1 - a1b2 )a2(φa2 + b2) / b2(φa2 + b2)( a2b1 - a1b2 )

~~φa2/b2~~

<math>

\require{cancel}

\begin{align}

r &= \frac{( φa_2b_1 - φa_1b_2 )(φa_2^2 + a_2b_2)}{(φa_2b_2 + b_2^2)( a_2b_1 - a_1b_2 )} \\

&= \frac{φ( a_2b_1 - a_1b_2 )a_2(φa_2 + b_2)}{b_2(φa_2 + b_2)( a_2b_1 - a_1b_2 )} \\

&= \frac{φ\cancel{( a_2b_1 - a_1b_2 )}a_2\cancel{(φa_2 + b_2)}}{b_2\cancel{(φa_2 + b_2)}\cancel{( a_2b_1 - a_1b_2 )}} \\

&= \frac{φa_2}{b_2}

\end{align}

</math>

The fact that the numerators do not figure into the result at all speaks to the impossibility of finding more than one interval on the tree with the same two bounding ratio denominators.

We can also infer why the ratio worked out to exactly <math>φ</math> in the case of the entire period: both of the denominators of 0/1 and 1/1 are 1, so the scalar on <math>φ</math> was 1.

We can also infer why the ratio worked out to exactly <math>φ</math> in the case of the entire period: both of the denominators of <math>\frac 01</math> and <math>\frac 11</math> are 1, so the scalar on <math>φ</math> was 1.

And from this we also ascertain that weighted mediants sometimes fall toward the edges of the interval and sometimes toward the middle. I.e. if we choose the interval 8/21 to 5/13, weighted by <math>φ</math> toward 5/13, the ratio between the two split segments would be 13φ/21 = 1.001640, making that split almost right down the middle; on the other hand, if we chose the interval 0/1 to 1/7, weighted by <math>φ</math> toward 1/7, the ratio between the two split segments would be 7φ/1 = 11.326238, extremely off.

And from this we also ascertain that weighted mediants sometimes fall toward the edges of the interval and sometimes toward the middle. I.e. if we choose the interval <math>\frac{8}{21}</math> to <math>\frac{5}{13}</math>, weighted by <math>φ</math> toward <math>\frac{5}{13}</math>, the ratio between the two split segments would be <math>\frac{13φ}{21} ≈ 1.001640</math>, making that split almost right down the middle; on the other hand, if we chose the interval <math>\frac 01</math> to <math>\frac 17</math>, weighted by <math>φ</math> toward <math>\frac 17</math>, the ratio between the two split segments would be <math>\frac{7φ}{1} ≈ 11.326238</math>, extremely off.

== Generator introduction counts ==

@@ Line 1,018: / Line 1,018: @@
 The two segments are actually in a ratio equal to the weight multiplied by the ratio of the denominators of their respective bounds.
-We’ll test this out on the example from before. We know that the weighted mediant formula with <span><math>φ</math></span> as weight, the interval between 1/3 and 1/2, and weight leaning toward the parent ratio (1/2) gives the value 0.419821. So our two segments are:
+We’ll test this out on the example from before. We know that the weighted mediant formula with <span><math>φ</math></span> as weight, the interval between <span><math>\frac 13</math></span> and <span><math>\frac 12</math></span>, and weight leaning toward the parent ratio (<span><math>\frac 12</math></span>) gives the value <span><math>≈ 0.419821</math></span>. So our two segments are:
 <math>
-.419821 - 0.333333 = 0.086487 \\
+s_1 = 0.419821 - 0.333333 = 0.086487 \\
-.500000 - 0.419821 = 0.080179 \\
+s_2 = 0.500000 - 0.419821 = 0.080179
 </math>
 Their ratio is
-<math> 0.086487 / 0.080179 = 1.078674
+<math> r = \frac{s_1}{s_2} = \frac{0.086487}{0.080179} = 1.078674
 </math>
 Which we can see is the weight multiplied by the bounding ratios’ denominators:
-<math> 1.078674 = φ * 2/3
+<math> 1.078674 = φ · \frac 23
 </math>
 In particular, we see that the denominator of the weighted ratio finds itself on the same side of the ratio as the weight.
-The full derivation is here:
+The full derivation follows. With lower bounding ratio <span><math>\frac{a_1}{a_2}</math></span> and upper bounding ratio <span><math>\frac{b_1}{b_2}</math></span>, we have a mediant of <span><math>\frac{φa_1 + b_1}{φa_2 + b_2}</math></span>. So then the segment from the lower bounding ratio to the mediant has length
 <math>
+\require{cancel}
+\begin{align}
+s_1 &= \frac{φa_1 + b_1}{φa_2 + b_2} - \frac{a_1}{a_2} \\
+&= \frac{a_2(φa_1 + b_1)}{a_2(φa_2 + b_2)} - \frac{a_1(φa_2 + b_2)}{a_2(φa_2 + b_2)} \\
+&= \frac{a_2(φa_1 + b_1) - a_1(φa_2 + b_2)}{a_2(φa_2 + b_2)} \\
+&= \frac{φa_1a_2 + a_2b_1 - φa_1a_2 - a_1b_2}{φa_2^2 + a_2b_2} \\
+&= \frac{\cancel{φa_1a_2} + a_2b_1 - \cancel{φa_1a_2} - a_1b_2}{φa_2^2 + a_2b_2} \\
+&= \frac{a_2b_1 - a_1b_2}{φa_2^2 + a_2b_2}
+\end{align}
+</math>
-bounding ratio 1 = a<sub>1</sub>/a<sub>2</sub>
+and the segment from the mediant to the upper bounding ratio has length
-bounding ratio 2 = b<sub>1</sub>/b<sub>2</sub>
+<math>
+\require{cancel}
+\begin{align}
+s_2 &= \frac{b_1}{b_2} - \frac{φa_1 + b_1}{φa_2 + b_2} \\
+&= \frac{b_1(φa_2 + b_2)}{b_2(φa_2 + b_2)} - \frac{b_2(φa_1 + b_1)}{b_2(φa_2 + b_2)} \\
+&= \frac{b_1(φa_2 + b_2) - b_2(φa_1 + b_1)}{b_2(φa_2 + b_2)} \\
+&= \frac{φa_2b_1 + b_1b_2 - φa_1b_2 - b_1b_2}{φa_2b_2 + b_2^2} \\
+&= \frac{φa_2b_1 + \cancel{b_1b_2} - φa_1b_2 - \cancel{b_1b_2}}{φa_2b_2 + b_2^2} \\
+&= \frac{φa_2b_1 - φa_1b_2}{φa_2b_2 + b_2^2}
+\end{align}
+</math>
-their mediant = (φa<sub>1</sub> + b<sub>1</sub>) / (φa<sub>2</sub> + b<sub>2</sub>)
+so then their ratio is
-segment from bounding ratio 1 to mediant = (φa<sub>1</sub> + b<sub>1</sub>) / (φa<sub>2</sub> + b<sub>2</sub>) - a<sub>1</sub>/a<sub>2</sub>
-a<sub>2</sub>(φa<sub>1</sub> + b<sub>1</sub>) / a<sub>2</sub>(φa<sub>2</sub> + b<sub>2</sub>) - a<sub>1</sub>(φa<sub>2</sub> + b<sub>2</sub>)/a<sub>2</sub>(φa<sub>2</sub> + b<sub>2</sub>)
-( a<sub>2</sub>(φa<sub>1</sub> + b<sub>1</sub>) - a<sub>1</sub>(φa<sub>2</sub> + b<sub>2</sub>) ) / a<sub>2</sub>(φa<sub>2</sub> + b<sub>2</sub>)
-( φa<sub>1</sub>a<sub>2</sub> + a<sub>2</sub>b<sub>1</sub> - φa<sub>1</sub>a<sub>2</sub> - a<sub>1</sub>b<sub>2</sub> ) / (φa<sub>2</sub><sup>2</sup> + a<sub>2</sub>b<sub>2</sub>)
-( a<sub>2</sub>b<sub>1</sub> - a<sub>1</sub>b<sub>2</sub> ) / (φa<sub>2</sub><sup>2</sup> + a<sub>2</sub>b<sub>2</sub>)
-segment from bounding ratio 2 to mediant = b<sub>1</sub>/b<sub>2</sub> - (φa<sub>1</sub> + b<sub>1</sub>) / (φa<sub>2</sub> + b<sub>2</sub>)
-b<sub>1</sub>(φa<sub>2</sub> + b<sub>2</sub>)/b<sub>2</sub>(φa<sub>2</sub> + b<sub>2</sub>) - b<sub>2</sub>(φa<sub>1</sub> + b<sub>1</sub>) / b<sub>2</sub>(φa<sub>2</sub> + b<sub>2</sub>)
-( b<sub>1</sub>(φa<sub>2</sub> + b<sub>2</sub>) - b<sub>2</sub>(φa<sub>1</sub> + b<sub>1</sub>) ) / b<sub>2</sub>(φa<sub>2</sub> + b<sub>2</sub>)
-( φa<sub>2</sub>b<sub>1</sub> + b<sub>1</sub>b<sub>2</sub> - φa<sub>1</sub>b<sub>2</sub> - b<sub>1</sub>b<sub>2</sub> ) / (φa<sub>2</sub>b<sub>2</sub> + b<sub>2</sub><sup>2</sup>)
-( φa<sub>2</sub>b<sub>1</sub> - φa<sub>1</sub>b<sub>2</sub> ) / (φa<sub>2</sub>b<sub>2</sub> + b<sub>2</sub><sup>2</sup>)
-so then their ratio is
-( φa<sub>2</sub>b<sub>1</sub> - φa<sub>1</sub>b<sub>2</sub> )(φa<sub>2</sub><sup>2</sup> + a<sub>2</sub>b<sub>2</sub>) / (φa<sub>2</sub>b<sub>2</sub> + b<sub>2</sub><sup>2</sup>)( a<sub>2</sub>b<sub>1</sub> - a<sub>1</sub>b<sub>2</sub> )
-φ( a<sub>2</sub>b<sub>1</sub> - a<sub>1</sub>b<sub>2</sub> )a<sub>2</sub>(φa<sub>2</sub> + b<sub>2</sub>) / b<sub>2</sub>(φa<sub>2</sub> + b<sub>2</sub>)( a<sub>2</sub>b<sub>1</sub> - a<sub>1</sub>b<sub>2</sub> )
-φa<sub>2</sub>/b<sub>2</sub>
+<math>
+\require{cancel}
+\begin{align}
+r &= \frac{( φa_2b_1 - φa_1b_2 )(φa_2^2 + a_2b_2)}{(φa_2b_2 + b_2^2)( a_2b_1 - a_1b_2 )} \\
+&= \frac{φ( a_2b_1 - a_1b_2 )a_2(φa_2 + b_2)}{b_2(φa_2 + b_2)( a_2b_1 - a_1b_2 )} \\
+&= \frac{φ\cancel{( a_2b_1 - a_1b_2 )}a_2\cancel{(φa_2 + b_2)}}{b_2\cancel{(φa_2 + b_2)}\cancel{( a_2b_1 - a_1b_2 )}} \\
+&= \frac{φa_2}{b_2}
+\end{align}
 </math>
 The fact that the numerators do not figure into the result at all speaks to the impossibility of finding more than one interval on the tree with the same two bounding ratio denominators.
-We can also infer why the ratio worked out to exactly <span><math>φ</math></span> in the case of the entire period: both of the denominators of 0/1 and 1/1 are 1, so the scalar on <span><math>φ</math></span> was 1.
+We can also infer why the ratio worked out to exactly <span><math>φ</math></span> in the case of the entire period: both of the denominators of <span><math>\frac 01</math></span> and <span><math>\frac 11</math></span> are 1, so the scalar on <span><math>φ</math></span> was 1.
-And from this we also ascertain that weighted mediants sometimes fall toward the edges of the interval and sometimes toward the middle. I.e. if we choose the interval 8/21 to 5/13, weighted by <span><math>φ</math></span> toward 5/13, the ratio between the two split segments would be 13φ/21 = 1.001640, making that split almost right down the middle; on the other hand, if we chose the interval 0/1 to 1/7, weighted by <span><math>φ</math></span> toward 1/7, the ratio between the two split segments would be 7φ/1 = 11.326238, extremely off.
+And from this we also ascertain that weighted mediants sometimes fall toward the edges of the interval and sometimes toward the middle. I.e. if we choose the interval <span><math>\frac{8}{21}</math></span> to <span><math>\frac{5}{13}</math></span>, weighted by <span><math>φ</math></span> toward <span><math>\frac{5}{13}</math></span>, the ratio between the two split segments would be <span><math>\frac{13φ}{21} ≈ 1.001640</math></span>, making that split almost right down the middle; on the other hand, if we chose the interval <span><math>\frac 01</math></span> to <span><math>\frac 17</math></span>, weighted by <span><math>φ</math></span> toward <span><math>\frac 17</math></span>, the ratio between the two split segments would be <span><math>\frac{7φ}{1} ≈ 11.326238</math></span>, extremely off.
 == Generator introduction counts ==