Metallic MOS: Difference between revisions

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The Stern-Brocot tree can be recursed indefinitely, so an infinite number of metallic generators exist. And since each parent ratio branches into two child ratios, each new recursive level of the Stern-Brocot tree offers the next power of 2 more intervals.

Despite this exponential profusion of scales, however, as one ~~traverses~~ deeper down the tree, the generators lose musical interest. They become less and less metallic. We can quantify their metallicity in terms of how many iterations of their scale sequence are required before they reach the scale which

Despite this exponential profusion of scales, however, as one travels deeper down the tree, the generators lose musical interest. They become less and less metallic. We can quantify their metallicity in terms of how many iterations of their scale sequence are required before they reach the scale which

# begins the periodic phase of the <math>L{:}s</math> sequence for the metal they’re based on, and

# begins supporting the interval pattern for the metal they’re based on.

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As for numbering convention, we are extending the numbering system used by David J. Finnamore on the Elven Minstrel site. His numbering system, as opposed to Wilson's, allows for further levels to be included without requiring renumbering. We extend the numbering system by prefixing numbers for generators for metallic means beyond golden with their metal number and an underscore; e.g. the eighth silver generator would be '''#2_8.'''

Each metal is also accompanied by a diagram of its patterns of generator equivalence, i.e. each generator has infinite different ways of being expressed as we ~~traverse~~ deeper and deeper into the tree, but those infinite sequences of different ways follow patterns. In each of the boxes of these diagrams, the top half is a simple code that explains the type of equivalence:

Each metal is also accompanied by a diagram of its patterns of generator equivalence, i.e. each generator has infinite different ways of being expressed as we travel deeper and deeper into the tree, but those infinite sequences of different ways follow patterns. In each of the boxes of these diagrams, the top half is a simple code that explains the type of equivalence:

# '''P''' or '''C''': Parent or Child. Does the interval lean toward the parent ratio or the child ratio?

# '''L''' or '''G''': Lesser or Greater: Does the interval run from the child ratio to its lesser parent or to its greater parent?

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It may seem odd that the most popular use of the silver mean uses its isotope rather than the mean directly. However, if we consider the ratio of the generator to the period here, that ratio is the silver mean. In the golden case, there was no difference between these two conceptions; both splitting the period into two segments in the ratio of <math>φ</math> and having the generator to period ratio be <math>φ</math> produce the same result. So while in this discussion from the beginning we put things in terms of splitting intervals (in order to smoothly transition from the golden generator into noble generators), it is probably the case that those who first brought us the Fibonacci generator and Argent Temperament were thinking in terms of the ratio of the generator to the period.

Argent means “silver” in French, which explains the name (it has also been called Arguros, which is “silver” in Greek). Wilson called this generator "2-Zig/2-Zag", after the pattern of the lines in the Stern-Brocot tree that results as we ~~traverse~~ deeper into the tree searching for better approximations of it: we move twice to the right, then twice to the left, and repeat.

Argent means “silver” in French, which explains the name (it has also been called Arguros, which is “silver” in Greek). Wilson called this generator "2-Zig/2-Zag", after the pattern of the lines in the Stern-Brocot tree that results as we travel deeper into the tree searching for better approximations of it: we move twice to the right, then twice to the left, and repeat.

== Imaginary ==

@@ Line 400: / Line 400: @@
 The Stern-Brocot tree can be recursed indefinitely, so an infinite number of metallic generators exist. And since each parent ratio branches into two child ratios, each new recursive level of the Stern-Brocot tree offers the next power of 2 more intervals.
-Despite this exponential profusion of scales, however, as one traverses deeper down the tree, the generators lose musical interest. They become less and less metallic. We can quantify their metallicity in terms of how many iterations of their scale sequence are required before they reach the scale which
+Despite this exponential profusion of scales, however, as one travels deeper down the tree, the generators lose musical interest. They become less and less metallic. We can quantify their metallicity in terms of how many iterations of their scale sequence are required before they reach the scale which
 # begins the periodic phase of the <math>L{:}s</math> sequence for the metal they’re based on, and
 # begins supporting the interval pattern for the metal they’re based on.
@@ Line 946: / Line 946: @@
 As for numbering convention, we are extending the numbering system used by David J. Finnamore on the Elven Minstrel site. His numbering system, as opposed to Wilson's, allows for further levels to be included without requiring renumbering. We extend the numbering system by prefixing numbers for generators for metallic means beyond golden with their metal number and an underscore; e.g. the eighth silver generator would be '''#2_8.'''
-Each metal is also accompanied by a diagram of its patterns of generator equivalence, i.e. each generator has infinite different ways of being expressed as we traverse deeper and deeper into the tree, but those infinite sequences of different ways follow patterns. In each of the boxes of these diagrams, the top half is a simple code that explains the type of equivalence:
+Each metal is also accompanied by a diagram of its patterns of generator equivalence, i.e. each generator has infinite different ways of being expressed as we travel deeper and deeper into the tree, but those infinite sequences of different ways follow patterns. In each of the boxes of these diagrams, the top half is a simple code that explains the type of equivalence:
 # '''P''' or '''C''': Parent or Child. Does the interval lean toward the parent ratio or the child ratio?
 # '''L''' or '''G''': Lesser or Greater: Does the interval run from the child ratio to its lesser parent or to its greater parent?
@@ Line 1,058: / Line 1,058: @@
 It may seem odd that the most popular use of the silver mean uses its isotope rather than the mean directly. However, if we consider the ratio of the generator to the period here, that ratio is the silver mean. In the golden case, there was no difference between these two conceptions; both splitting the period into two segments in the ratio of <math>φ</math> and having the generator to period ratio be <math>φ</math> produce the same result. So while in this discussion from the beginning we put things in terms of splitting intervals (in order to smoothly transition from the golden generator into noble generators), it is probably the case that those who first brought us the Fibonacci generator and Argent Temperament were thinking in terms of the ratio of the generator to the period.
-Argent means “silver” in French, which explains the name (it has also been called Arguros, which is “silver” in Greek). Wilson called this generator "2-Zig/2-Zag", after the pattern of the lines in the Stern-Brocot tree that results as we traverse deeper into the tree searching for better approximations of it: we move twice to the right, then twice to the left, and repeat.
+Argent means “silver” in French, which explains the name (it has also been called Arguros, which is “silver” in Greek). Wilson called this generator "2-Zig/2-Zag", after the pattern of the lines in the Stern-Brocot tree that results as we travel deeper into the tree searching for better approximations of it: we move twice to the right, then twice to the left, and repeat.
 == Imaginary ==