Metallic MOS: Difference between revisions

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By “equivalent”, we mean that they will generate essentially the same scale sequence. And by “essentially the same” we mean that the scales will be mirror images of each other, which for an MOS scale, happens to also mean that they are simply transpositions of each other (different modes of the same scale).

As for why we pick the lower half of the period rather than the upper half, this is somewhat arbitrary, but it seems objectively simpler to keep our lower bound at 0.

As for why we pick the lower half of the period rather than the upper half, this is somewhat arbitrary, but it seems objectively simpler to keep our lower bound at <math>0</math>.

Sure, depending on the context, the generator complement greater than 0.5 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 0.618034 instead of its complement 0.381966 (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 ~~cataloguing~~ we prefer the smaller, or ''reduced'' of the two complements.

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 > 0.5. Almost every interval we include does not even allow for that possibility, but one interval does threaten this: the interval 0/1 to 1/1. We include this interval because it occupies space between 0/1 and 1/2 — so has potential to find useful generators — but we have to be careful with it to avoid finding generators > 0.5. The method for this is simple. First, note that the unweighted mediant in the interval 0/1 to 1/1 is 1/2, or exactly 0.5. So if we want to avoid generators > 0.5, all we must do is make sure to weight more toward 0/1. Since of these two ratios 0/1 and 1/1, 0/1 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>, <math>\frac 01</math> is the parent ratio, weighting parentward is the solution.

== Isotopic arithmetic progression ==

@@ Line 437: / Line 437: @@
 By “equivalent”, we mean that they will generate essentially the same scale sequence. And by “essentially the same” we mean that the scales will be mirror images of each other, which for an MOS scale, happens to also mean that they are simply transpositions of each other (different modes of the same scale).
-As for why we pick the lower half of the period rather than the upper half, this is somewhat arbitrary, but it seems objectively simpler to keep our lower bound at 0.
+As for why we pick the lower half of the period rather than the upper half, this is somewhat arbitrary, but it seems objectively simpler to keep our lower bound at <span><math>0</math></span>.
-Sure, depending on the context, the generator complement greater than 0.5 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 0.618034 instead of its complement 0.381966 (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 cataloguing we prefer the smaller, or ''reduced'' of the two complements.
+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 > 0.5. Almost every interval we include does not even allow for that possibility, but one interval does threaten this: the interval 0/1 to 1/1. We include this interval because it occupies space between 0/1 and 1/2 — so has potential to find useful generators — but we have to be careful with it to avoid finding generators > 0.5. The method for this is simple. First, note that the unweighted mediant in the interval 0/1 to 1/1 is 1/2, or exactly 0.5. So if we want to avoid generators > 0.5, all we must do is make sure to weight more toward 0/1. Since of these two ratios 0/1 and 1/1, 0/1 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>, <span><math>\frac 01</math></span> is the parent ratio, weighting parentward is the solution.
 == Isotopic arithmetic progression ==