Metallic MOS: Difference between revisions

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Sure, depending on the context, the generator complement greater than <math>0.5</math> may be the one we want to describe our scale in terms of. For example, we may be thinking of the generator as the perfect fifth instead of the perfect fourth. Or we may want to use <math>0.618034</math> instead of its complement <math>0.381966</math> (we’ve been using the latter and calling it the golden generator, but some readers may be more familiar with the former, known as “logarithmic phi”, which is 741.64¢ when the period is an octave). But for purposes of cataloging we prefer the smaller, or ''reduced'' of the two complements.

And this is a subtle point, but it’s another reason to prefer leaning intervals parentward. We have a potential problem: we don’t want to find generators <math>> 0.5</math>. Almost every interval we include does not even allow for that possibility, but one interval does threaten this: the interval <math>\frac 01</math> to <math>\frac 11</math>. We include this interval because it occupies space between <math>\frac 01</math> and <math>\frac 12</math> — so has potential to find useful generators — but we have to be careful with it to avoid finding generators <math>> 0.5</math>. The method for this is simple. First, note that the unweighted mediant in the interval <math>\frac 01</math> to <math>\frac 11</math> is <math>\frac 12</math>, or exactly <math>0.5</math>. So if we want to avoid generators <math>> 0.5</math>, all we must do is make sure to weight more toward <math>\frac 01</math>. Since of these two ratios <math>\frac 01</math> and <math>\frac 11</math>, <math>\frac 01</math> ~~is the parent ratio~~, weighting parentward is the solution.

And this is a subtle point, but it’s another reason to prefer leaning intervals parentward. We have a potential problem: we don’t want to find generators <math>> 0.5</math>. Almost every interval we include does not even allow for that possibility, but one interval does threaten this: the interval <math>\frac 01</math> to <math>\frac 11</math>. We include this interval because it occupies space between <math>\frac 01</math> and <math>\frac 12</math> — so has potential to find useful generators — but we have to be careful with it to avoid finding generators <math>> 0.5</math>. The method for this is simple. First, note that the unweighted mediant in the interval <math>\frac 01</math> to <math>\frac 11</math> is <math>\frac 12</math>, or exactly <math>0.5</math>. So if we want to avoid generators <math>> 0.5</math>, all we must do is make sure to weight more toward <math>\frac 01</math>. Since of these two ratios <math>\frac 01</math> and <math>\frac 11</math>, the parent ratio is <math>\frac 01</math>, weighting parentward is the solution.

== Isotopic arithmetic progression ==

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 Sure, depending on the context, the generator complement greater than <span><math>0.5</math></span> may be the one we want to describe our scale in terms of. For example, we may be thinking of the generator as the perfect fifth instead of the perfect fourth. Or we may want to use <span><math>0.618034</math></span> instead of its complement <span><math>0.381966</math></span> (we’ve been using the latter and calling it the golden generator, but some readers may be more familiar with the former, known as “logarithmic phi”, which is 741.64¢ when the period is an octave). But for purposes of cataloging we prefer the smaller, or ''reduced'' of the two complements.
-And this is a subtle point, but it’s another reason to prefer leaning intervals parentward. We have a potential problem: we don’t want to find generators <span><math>> 0.5</math></span>. Almost every interval we include does not even allow for that possibility, but one interval does threaten this: the interval <span><math>\frac 01</math></span> to <span><math>\frac 11</math></span>. We include this interval because it occupies space between <span><math>\frac 01</math></span> and <span><math>\frac 12</math></span> — so has potential to find useful generators — but we have to be careful with it to avoid finding generators <span><math>> 0.5</math></span>. The method for this is simple. First, note that the unweighted mediant in the interval <span><math>\frac 01</math></span> to <span><math>\frac 11</math></span> is <span><math>\frac 12</math></span>, or exactly <span><math>0.5</math></span>. So if we want to avoid generators <span><math>> 0.5</math></span>, all we must do is make sure to weight more toward <span><math>\frac 01</math></span>. Since of these two ratios <span><math>\frac 01</math></span> and <span><math>\frac 11</math></span>, <span><math>\frac 01</math></span> is the parent ratio, weighting parentward is the solution.
+And this is a subtle point, but it’s another reason to prefer leaning intervals parentward. We have a potential problem: we don’t want to find generators <span><math>> 0.5</math></span>. Almost every interval we include does not even allow for that possibility, but one interval does threaten this: the interval <span><math>\frac 01</math></span> to <span><math>\frac 11</math></span>. We include this interval because it occupies space between <span><math>\frac 01</math></span> and <span><math>\frac 12</math></span> — so has potential to find useful generators — but we have to be careful with it to avoid finding generators <span><math>> 0.5</math></span>. The method for this is simple. First, note that the unweighted mediant in the interval <span><math>\frac 01</math></span> to <span><math>\frac 11</math></span> is <span><math>\frac 12</math></span>, or exactly <span><math>0.5</math></span>. So if we want to avoid generators <span><math>> 0.5</math></span>, all we must do is make sure to weight more toward <span><math>\frac 01</math></span>. Since of these two ratios <span><math>\frac 01</math></span> and <span><math>\frac 11</math></span>, the parent ratio is <span><math>\frac 01</math></span>, weighting parentward is the solution.
 == Isotopic arithmetic progression ==