Metallic MOS: Difference between revisions
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=== As a bizarro average === | === As a bizarro average === | ||
We can think of a mediant like a bizarro average of two ratios: however we may choose to weight one, it will always lie somewhere between the two ratios. That is why we can call these two ratios its bounds. This fact is easy enough to intuit: as the weight tends toward zero, the effects of <span><math>a_1</math></span> and <span><math>a_2</math></span> drops off to nothing, and as it tends toward infinity, their effects begin to utterly overwhelm <span><math>b_1</math></span> and <span><math>b_2</math></span>. | We can think of a mediant like a bizarro average of two ratios: however we may choose to weight one, it will always lie somewhere between the two ratios. That is why we can call these two ratios its bounds. This fact is easy enough to intuit: as the weight tends toward zero, the effects of <span><math>a_1</math></span> and <span><math>a_2</math></span> drops off to nothing, and as it tends toward infinity, their effects begin to utterly overwhelm <span><math>b_1</math></span> and <span><math>b_2</math></span>. | ||
<math> | <math> | ||
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Another benefit of finding by the child ratio is that every ratio has exactly two parent ratios, while its count of child ratios is variable (consider how many child ratios <span><math>\frac 01</math></span> has). | Another benefit of finding by the child ratio is that every ratio has exactly two parent ratios, while its count of child ratios is variable (consider how many child ratios <span><math>\frac 01</math></span> has). | ||
Each interval spans two levels of the tree, because a parent ratio will always be one level less than its child ratio. When classifying intervals by level, then, we should classify them by the child ratio. For example, we should consider the interval <span><math>\frac 17</math></span> to <span><math>\frac 16</math></span> a seventh-level interval, because it would not be available until we included the seventh level. | Each interval spans two levels of the tree, because a parent ratio will always be one level less than its child ratio. When classifying intervals by level, then, we should classify them by the child ratio. For example, we should consider the interval <span><math>\frac 17</math></span> to <span><math>\frac 16</math></span> a seventh-level interval, because it would not be available until we included the seventh level. | ||
By the way, here is an easy way to identify which level of the tree a ratio is on: we can scan along the level to the left until we find the unit fraction which appears in the initial position, closest to <span><math>\frac 01</math></span>; if our level starts with <span><math>\frac 1n</math></span>, then the ratio is in the <span><math>n</math></span>th level. | By the way, here is an easy way to identify which level of the tree a ratio is on: we can scan along the level to the left until we find the unit fraction which appears in the initial position, closest to <span><math>\frac 01</math></span>; if our level starts with <span><math>\frac 1n</math></span>, then the ratio is in the <span><math>n</math></span>th level. | ||
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We’ve stated that <span><math>L{:}s = φ</math></span> for every golden scale, while <span><math>L{:}s</math></span> for noble scales eventually do, just not at first. Noble <span><math>L{:}s</math></span> sequences lock onto <span><math>φ</math></span> at the point where depleting the continued fraction more no longer changes it (removing a <span><math>1</math></span> from the beginning of an infinite string of <span><math>1</math></span>’s is a no-op). | We’ve stated that <span><math>L{:}s = φ</math></span> for every golden scale, while <span><math>L{:}s</math></span> for noble scales eventually do, just not at first. Noble <span><math>L{:}s</math></span> sequences lock onto <span><math>φ</math></span> at the point where depleting the continued fraction more no longer changes it (removing a <span><math>1</math></span> from the beginning of an infinite string of <span><math>1</math></span>’s is a no-op). | ||
Thus it makes sense that logarithmic phi’s <span><math>L{:}s</math></span> sequence remains fixed from the beginning, because with a continued fraction of <span><math>[0; 1]</math></span> we get the <span><math>L{:}s</math></span> sequence | Thus it makes sense that logarithmic phi’s <span><math>L{:}s</math></span> sequence remains fixed from the beginning, because with a continued fraction of <span><math>[0; 1]</math></span> we get the <span><math>L{:}s</math></span> sequence | ||
<math> | <math> | ||
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=== μ notation === | === μ notation === | ||
A number of names and symbols have historically been used to denote metallic means. But many of them are ambiguous, or outright conflict with each other, and unfortunately none of them are optimal for met-MOS purposes. We’ve gotten by alright so far using traditional names and symbols in this discussion, but for the master charts we’re going to need to break from tradition in order to most clearly convey the patterns therein. So, we here propose a new notation using the Greek letter <span><math>μ</math></span>, read “mu” (<span><math>μ</math></span> because “m” figures so prominently in this domain: “m” for metallic, mean, or moment). | A number of names and symbols have historically been used to denote metallic means. But many of them are ambiguous, or outright conflict with each other, and unfortunately none of them are optimal for met-MOS purposes. We’ve gotten by alright so far using traditional names and symbols in this discussion, but for the master charts we’re going to need to break from tradition in order to most clearly convey the patterns therein. So, we here propose a new notation using the Greek letter <span><math>μ</math></span>, read “mu” (<span><math>μ</math></span> because “m” figures so prominently in this domain: “m” for metallic, mean, or moment). | ||
This <span><math>μ</math></span> notation is a direct mapping of the continued fraction for the metallic mean or isotope, as can clearly be seen in the following chart. | This <span><math>μ</math></span> notation is a direct mapping of the continued fraction for the metallic mean or isotope, as can clearly be seen in the following chart. | ||
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</math> | </math> | ||
Find any <span><math>s</math></span> in the horogram and observe how it gets split up as we iterate through the scale sequence. In the next iteration, <span><math>s</math></span> will be replaced with <span><math>L</math></span>. After two iterations, the original <span><math>s</math></span> interval is now represented by an <span><math>L</math></span> and an <span><math>s</math></span>. And so forth. | Find any <span><math>s</math></span> in the horogram and observe how it gets split up as we iterate through the scale sequence. In the next iteration, <span><math>s</math></span> will be replaced with <span><math>L</math></span>. After two iterations, the original <span><math>s</math></span> interval is now represented by an <span><math>L</math></span> and an <span><math>s</math></span>. And so forth. | ||
Every MOS scale contains every scale earlier in its scale sequence. In other words, any interval that existed in an earlier scale will remain in all later scales. These earlier <span><math>L</math></span>’s and <span><math>s</math></span>’s that remain — only now spanning many <span><math>L</math></span>’s and <span><math>s</math></span>’s each — are precisely the larger intervals in the scale that also exhibit the <span><math>φ</math></span> ratio to each other. | Every MOS scale contains every scale earlier in its scale sequence. In other words, any interval that existed in an earlier scale will remain in all later scales. These earlier <span><math>L</math></span>’s and <span><math>s</math></span>’s that remain — only now spanning many <span><math>L</math></span>’s and <span><math>s</math></span>’s each — are precisely the larger intervals in the scale that also exhibit the <span><math>φ</math></span> ratio to each other. | ||
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[ insert a table version of above chart here; too big to paste in Google docs ] | [ insert a table version of above chart here; too big to paste in Google docs ] | ||
[[File:Generator Equivalence Patterns - Golden Mean.png|none|thumb|150x150px| | |||
generator equivalence patterns - golden | |||
]] | |||
== Silver == | == Silver == | ||
[ insert a table version of above chart here; too big to paste in Google docs ] | [ insert a table version of above chart here; too big to paste in Google docs ] | ||
[[File:Generator Equivalence Patterns - Silver Mean.png|none|thumb|150x150px| | |||
generator equivalence patterns - silver | |||
]] | |||
== Bronze == | == Bronze == | ||
[ insert a table version of above chart here; too big to paste in Google docs ] | [ insert a table version of above chart here; too big to paste in Google docs ] | ||
[[File:Generator Equivalence Patterns - Bronze Mean.png|none|thumb|153x153px| | |||
generator equivalence patterns - bronze | |||
]] | |||
== Beyond bronze == | == Beyond bronze == | ||
Including scale trees beyond bronze is outside the scope of this present work. However, an additional generator equivalence pattern diagram for the fourth metallic mean is illustrative of that meta-pattern as it continues to expand. | Including scale trees beyond bronze is outside the scope of this present work. However, an additional generator equivalence pattern diagram for the fourth metallic mean is illustrative of that meta-pattern as it continues to expand. | ||
[[File:Generator Equivalence Patterns - 4th Metallic Mean.png|none|thumb|150x150px| | |||
generator equivalence patterns - 4th metal | |||
]] | |||
== Master scale table == | == Master scale table == | ||