Metallic MOS: Difference between revisions

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But if we follow the policy of finding by the child ratio first and then the parent ratio, we only have to avoid the currently shallowest level of the tree. This is much easier, because:

# it never changes — it’s always at the root of the tree;

# it’s much smaller, including only a single ratio, 0/1;

# it’s much smaller, including only a single ratio, <math>\frac 01</math>;

# that single ratio doesn’t even have a parent ratio anyway, so it’s easy to avoid.

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 0/1 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 <math>\frac 01</math> 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 1/7 to 1/6 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 <math>\frac 17</math> to <math>\frac 16</math> 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 0/1; if our level starts with 1/n, then the ratio is in the ~~nth~~ 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 <math>\frac 01</math>; if our level starts with <math>\frac 1n</math>, then the ratio is in the <math>n</math>th level.

=== Interval lean ===

@@ Line 354: / Line 354: @@
 But if we follow the policy of finding by the child ratio first and then the parent ratio, we only have to avoid the currently shallowest level of the tree. This is much easier, because:
 # it never changes — it’s always at the root of the tree;
-# it’s much smaller, including only a single ratio, 0/1;
+# it’s much smaller, including only a single ratio, <span><math>\frac 01</math></span>;
 # that single ratio doesn’t even have a parent ratio anyway, so it’s easy to avoid.
-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 0/1 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 1/7 to 1/6 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 0/1; if our level starts with 1/n, then the ratio is in the nth 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.
 === Interval lean ===