Metallic MOS: Difference between revisions
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Another way to think about this is that a small-step-sized chunk has been split off of each of the former large steps. The remainder can be either larger or smaller than the small step. If it is larger, then it stays the large step. If it is smaller, then it becomes the new small step, and everything that used to be a small step is now a large step. | Another way to think about this is that a small-step-sized chunk has been split off of each of the former large steps. The remainder can be either larger or smaller than the small step. If it is larger, then it stays the large step. If it is smaller, then it becomes the new small step, and everything that used to be a small step is now a large step. | ||
[[File:MOS iteration rules for L and s.png|452x452px]] | |||
We are reasoning about MOS concepts in the abstract here. These truths about large and small steps are true whether they are 100¢ or 4516.8¢, and all we really care about are their ratios. So if we treat our small steps’ size as 1 then we can treat our large steps’ size as equal to the L:s ratio. | We are reasoning about MOS concepts in the abstract here. These truths about large and small steps are true whether they are 100¢ or 4516.8¢, and all we really care about are their ratios. So if we treat our small steps’ size as 1 then we can treat our large steps’ size as equal to the L:s ratio. | ||
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We can visualize the interval pattern using what is called a horogram: | We can visualize the interval pattern using what is called a horogram: | ||
[[File:Golden horogram.png|alt=horogram for g ≈ 0.381966, 7 iterations|right|392x392px|horogram for g ≈ 0.381966, 7 iterations]] | |||
[ horogram for g ≈ 0. | |||
Horograms depict the scale sequences of MOS generators. To understand how the horogram illustrates the interval pattern, too, first consider just the left side of the interval pattern, for L: | Horograms depict the scale sequences of MOS generators. To understand how the horogram illustrates the interval pattern, too, first consider just the left side of the interval pattern, for L: | ||
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L:s = (L+2s):(L+s) = (3L+4s):(2L+3s) = (7L+10s):(5L+7s) = (17L+24s):(12L+17s) = ... = δ<sub>s</sub> - 1 | L:s = (L+2s):(L+s) = (3L+4s):(2L+3s) = (7L+10s):(5L+7s) = (17L+24s):(12L+17s) = ... = δ<sub>s</sub> - 1 | ||
[[File:Silver horogram.png|alt=horogram for g = 0.292893, 7 iterations|right|396x396px|horogram for g = 0.292893, 7 iterations]] | |||
[ horogram for g = 0. | |||
We can use the horogram for the silver generator to see how its interval pattern cycle is length 2, i.e. that it alternates between two different interval patterns. If we want to understand the interval pattern for δ<sub>s</sub>, we’ll look at the right and left sides separately, as we did with the golden: | We can use the horogram for the silver generator to see how its interval pattern cycle is length 2, i.e. that it alternates between two different interval patterns. If we want to understand the interval pattern for δ<sub>s</sub>, we’ll look at the right and left sides separately, as we did with the golden: | ||
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== Argent Temperament == | == Argent Temperament == | ||
[[File:Argent horogram.png|alt=horogram for g = 0.414214, 7 iterations|right|388x388px]] | |||
Scales based on the bronze mean and metallic means beyond it have not been extensively explored. However, the silver mean has gotten some attention. | Scales based on the bronze mean and metallic means beyond it have not been extensively explored. However, the silver mean has gotten some attention. | ||
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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 traverse deeper into the tree searching for better approximations of it: we move twice to the right, then twice to the left, and repeat. | ||
== Imaginary == | == Imaginary == | ||
If Argent temperament splits the period into segments in the ratio of the silver ratio’s isotope, what if we split the period into segments in the ratio of the silver mean itself? That gives us a generator of ≈ 0.292893, which has been used by Billy Stiltner, who calls it Imaginary. | If Argent temperament splits the period into segments in the ratio of the silver ratio’s isotope, what if we split the period into segments in the ratio of the silver mean itself? That gives us a generator of ≈ 0.292893, which has been used by Billy Stiltner, who calls it Imaginary. | ||
Twice 0.292893 is equal to 0.585786, which is the complement of Argent Temperament’s generator, 0.414214. Despite this similarity, these two generate tremendously different scales | Twice 0.292893 is equal to 0.585786, which is the complement of Argent Temperament’s generator, 0.414214. Despite this similarity, these two generate tremendously different scales (compare with the silver horogram shown earlier). | ||
== Other Wilson scales == | == Other Wilson scales == | ||