Metallic MOS: Difference between revisions

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But is only the first of an infinite sequence of such metallic means which can be used to generate scales offering interesting musical possibilities. And while some attention has been given to silver scales, what we seek to do here is centralize all met-MOS knowledge and generalize principles across all of the metallic means.

MOS concepts are logarithmic, not acoustic. In other words, we are not dealing with frequency ratios here. Frequency ratios related to metallic means, such as ~~“acoustic phi”~~ (approximately 833.09¢), have interesting properties too — creating recursive combination tones, for example — but these musical applications of metallic means will not be discussed here.

MOS concepts are logarithmic, not acoustic. In other words, we are not dealing with frequency ratios here. Frequency ratios related to metallic means, such as “[[acoustic phi]]” (approximately 833.09¢), have interesting properties too — creating recursive combination tones, for example — but these musical applications of metallic means will not be discussed here.

The met-MOS concepts discussed here may be used to define generators as fractions of an octave, as is most common. But these concepts are more abstract than that, and may be used to define generators as fractions of ''any'' period. As such, they only depend on the ratio between the generator and the period, and so for convenience we can lock one of these two values to 1 and only vary the value of the other. We’ll be conforming here with the convention of choosing the period as the interval to lock to 1.

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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 <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.

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>, the parent ratio is <math>\frac 01</math>, weighting parentward is the solution.

@@ Line 12: / Line 12: @@
 But  is only the first of an infinite sequence of such metallic means which can be used to generate scales offering interesting musical possibilities. And while some attention has been given to silver scales, what we seek to do here is centralize all met-MOS knowledge and generalize principles across all of the metallic means.
-MOS concepts are logarithmic, not acoustic. In other words, we are not dealing with frequency ratios here. Frequency ratios related to metallic means, such as “acoustic phi” (approximately 833.09¢), have interesting properties too — creating recursive combination tones, for example — but these musical applications of metallic means will not be discussed here.
+MOS concepts are logarithmic, not acoustic. In other words, we are not dealing with frequency ratios here. Frequency ratios related to metallic means, such as “[[acoustic phi]]” (approximately 833.09¢), have interesting properties too — creating recursive combination tones, for example — but these musical applications of metallic means will not be discussed here.
 The met-MOS concepts discussed here may be used to define generators as fractions of an octave, as is most common. But these concepts are more abstract than that, and may be used to define generators as fractions of ''any'' period. As such, they only depend on the ratio between the generator and the period, and so for convenience we can lock one of these two values to 1 and only vary the value of the other. We’ll be conforming here with the convention of choosing the period as the interval to lock to 1.
@@ Line 440: / Line 440: @@
 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 <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.
+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>, the parent ratio is <span><math>\frac 01</math></span>, weighting parentward is the solution.