Metallic MOS: Difference between revisions

Line 278:

Recall that in order to find a metallic generator, we find a value which splits an interval into two segments related by a metallic mean. When this technique was introduced earlier, we left the definition of “related by” vague. Well, now is the time to make it explicit.

In our very first case — that of the golden generator — “related by” ''could'' be defined as simply “having a ratio of”. We’ll check the segment lengths to confirm this. One of the two segments is, of course, equal to the golden generator, approximately ~~<math>~~0.381966~~</math>~~. The other is equal to the remainder of the golden generator with the period:

In our very first case — that of the golden generator — “related by” ''could'' be defined as simply “having a ratio of”. We’ll check the segment lengths to confirm this. One of the two segments is, of course, equal to the golden generator, approximately 0.381966. The other is equal to the remainder of the golden generator with the period:

<math>\qquad 1 - 0.381966 = 0.618034

<math>\qquad 1 - 0.381966 ≈ 0.618034

</math>

Line 292:

The correct general definition of a metallic generator is actually a [[wikipedia:Mediant_(mathematics)|mediant]] of the two ratios which bound the interval.

But even then it’s not quite that simple, because it’s not a ''simple'' mediant ~~between <math>\frac{a_1}{a_2}</math> and <math>\frac{b_1}{b_2}</math>~~, which would look like this:

But even then it’s not quite that simple, because it’s not a ''simple'' mediant, which would look like this:

<math>\qquad \frac{a_1 + b_1}{a_2 + b_2}

</math>

Rather, it’s a [https://www.mathpages.com/home/kmath055/kmath055.htm ''weighted'' mediant], which looks like this ~~(where the weight is <math>\frac{w_1}{w_2}</math>)~~:

Rather, it’s a [https://www.mathpages.com/home/kmath055/kmath055.htm ''weighted'' mediant], which looks like this:

<math>\qquad \frac{a_1w_1 + b_1w_2}{a_2w_1 + b_2w_2}

</math>

In particular, it is the mediant which is weighted by the desired metallic mean ~~<math>~~μ~~</math>~~:

In particular, it is the mediant which is weighted by the desired metallic mean μ:

<math>\qquad \frac{a_1μ + b_1}{a_2μ + b_2}

@@ Line 278: / Line 278: @@
 Recall that in order to find a metallic generator, we find a value which splits an interval into two segments related by a metallic mean. When this technique was introduced earlier, we left the definition of “related by” vague. Well, now is the time to make it explicit.
-In our very first case — that of the golden generator — “related by” ''could'' be defined as simply “having a ratio of”. We’ll check the segment lengths to confirm this. One of the two segments is, of course, equal to the golden generator, approximately <span><math>0.381966</math></span>. The other is equal to the remainder of the golden generator with the period:
+In our very first case — that of the golden generator — “related by” ''could'' be defined as simply “having a ratio of”. We’ll check the segment lengths to confirm this. One of the two segments is, of course, equal to the golden generator, approximately 0.381966. The other is equal to the remainder of the golden generator with the period:
-<math>\qquad 1 - 0.381966 = 0.618034
+<math>\qquad 1 - 0.381966 ≈ 0.618034
 </math>
@@ Line 292: / Line 292: @@
 The correct general definition of a metallic generator is actually a [[wikipedia:Mediant_(mathematics)|mediant]] of the two ratios which bound the interval.
-But even then it’s not quite that simple, because it’s not a ''simple'' mediant between <span><math>\frac{a_1}{a_2}</math></span> and <span><math>\frac{b_1}{b_2}</math></span>, which would look like this:
+But even then it’s not quite that simple, because it’s not a ''simple'' mediant, which would look like this:
 <math>\qquad \frac{a_1 + b_1}{a_2 + b_2}
 </math>
-Rather, it’s a [https://www.mathpages.com/home/kmath055/kmath055.htm ''weighted'' mediant], which looks like this (where the weight is <span><math>\frac{w_1}{w_2}</math></span>):
+Rather, it’s a [https://www.mathpages.com/home/kmath055/kmath055.htm ''weighted'' mediant], which looks like this:
 <math>\qquad \frac{a_1w_1 + b_1w_2}{a_2w_1 + b_2w_2}
 </math>
-In particular, it is the mediant which is weighted by the desired metallic mean <span><math>μ</math></span>:
+In particular, it is the mediant which is weighted by the desired metallic mean μ:
 <math>\qquad \frac{a_1μ + b_1}{a_2μ + b_2}