MOS scale family tree: Difference between revisions

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The MOS scale family tree (or mos family tree) is an infinite binary tree that organizes [[MOS scale|moment-of-symmetry scales]] based on the parent-to-child relationship between scales. This tree is not to be confused with other scale trees, such as those based on the Stern-Brocot or tree or Farey tree. Rather, this tree organizes MOS scales quite differently, depicting a family tree of step patterns.

== History ==

[[File:Family Tree of MOS-MV2 Scales.svg|thumb|~~663x663px~~|The family tree of moment-of-symmetry scales, recreated by a xen wiki user. Note that the construction of this tree uses an upper and lower child, as opposed to a left and right child.]][[Erv Wilson]] was the first to describe such a tree using Fibonacci rabbit patterns. One version of his tree is referred to the scale/rhythm tree, and it's this tree that shows the parent-child relationship between all (single-period) moment-of-symmetry scales.

[[File:Family Tree of MOS-MV2 Scales.svg|thumb|811x811px|The family tree of moment-of-symmetry scales, recreated by a xen wiki user. Note that the construction of this tree uses an upper and lower child, as opposed to a left and right child.]][[Erv Wilson]] was the first to describe such a tree using Fibonacci rabbit patterns. One version of his tree is referred to the scale/rhythm tree, and it's this tree that shows the parent-child relationship between all (single-period) moment-of-symmetry scales.

Since the term "scale tree" is already used to describe scales arranged using the Farey or Stern-Brocot trees, the term "family tree" is used instead.

=== ~~Differences~~ and ~~conventions~~ ===

=== Conventions and other differences ===

~~Although~~ the tree ~~described here~~ is ~~ultimately based on~~ Wilson~~'s description and therefore shows the same information~~, ~~the tree described in this article deviates from Wilson's description. The main~~ differences ~~are~~ listed below:

For the purposes of this article, the mos family tree will be depicted sideways and with an "upper" and "lower" child mos, rather than a left and right child, as typical with binary trees. This is exactly how Wilson initially described his tree, but with a few additional differences, listed below:

* Scale step patterns may not be shown; preferably, the mos in its xL ys form will be shown instead of a step pattern.

* Scale step patterns may not be shown. If ~~they~~ are shown, ~~scale step patterns~~ will always be shown in both its brightest and darkest modes. One way to think of this is the brightest mode will have as many L's to the left as possible while still preserving the mos property, and the darkest mode will have as many L's to the right as possible while still preserving the mos property.

* If step patterns are shown, they will always be shown in both its brightest and darkest modes.

* The construction rules described here result in a tree that is upside-down relative to Wilson's description.

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At the root of the tree is the step pattern Ls, representing the mos 1L 1s. The child scales of any node can be constructed as such:

* One child starts with a copy of the step pattern of its parent and has every "L" replaced with "Ls" and every "s" is replaced with "s".

* One child (the upper child) starts with a copy of the step pattern of its parent and has every "L" replaced with "Ls" and every "s" is replaced with "s".

* The other child starts with a reversed copy of the parent's step pattern and has every "L" replaced with "Ls" and every "s" replaced with "L".

* The other child (the lower child) starts with a reversed copy of the parent's step pattern and has every "L" replaced with "Ls" and every "s" replaced with "L".

This pattern is repeated indefinitely to each new node added to the tree, or for however many generations are desired.

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|

|-

|[[13L 8]]

|[[13L 8s]]

|}

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Since a tree structure is built such that each node connects back to a unique parent (except for the root), there are no looping paths, so every path between any two nodes is unique. In [[regular temperament theory]], scales are described as being generated from stacking an interval repeatedly, with moment-of-symmetry scales resulting from this process. Since the sizes of the generating intervals are necessarily described, this means temperaments describe a specific path down the family tree.

=== Relation to ~~edos~~ ===

=== Relation to EDOs ===

Since every interval available to an [[EDO~~|edo~~]] can be used as a generating interval, repeatedly stacking such an interval will necessarily produce mosses. Each mos produced this way will describe a unique path on the mos family tree, starting at 1L 1s and terminating right before a pair of sister scales whose note count is equal to the number of equal divisions. Combining all of these paths into a tree will form a subset of the infinite mos family tree, where each path represents a different sequence of mosses that all have the same generating intervals.

Since every interval available to an [[EDO]] can be used as a generating interval, repeatedly stacking such an interval will necessarily produce mosses. Each mos produced this way will describe a unique path on the mos family tree, starting at 1L 1s and terminating right before a pair of sister scales whose note count is equal to the number of equal divisions. Combining all of these paths into a tree will form a subset of the infinite mos family tree, where each path represents a different sequence of mosses that all have the same generating intervals.

== External links ==

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== See also ==

* [[Catalog of MOS]]

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 The MOS scale family tree (or mos family tree<!-- Are there any other names this tree goes by? If so, add them here. -->) is an infinite binary tree that organizes [[MOS scale|moment-of-symmetry scales]] based on the parent-to-child relationship between scales. This tree is not to be confused with other scale trees, such as those based on the Stern-Brocot or tree or Farey tree. Rather, this tree organizes MOS scales quite differently, depicting a family tree of step patterns.
 == History ==
-[[File:Family Tree of MOS-MV2 Scales.svg|thumb|663x663px|The family tree of moment-of-symmetry scales, recreated by a xen wiki user. Note that the construction of this tree uses an upper and lower child, as opposed to a left and right child.]][[Erv Wilson]] was the first to describe such a tree using Fibonacci rabbit patterns. One version of his tree is referred to the scale/rhythm tree, and it's this tree that shows the parent-child relationship between all (single-period) moment-of-symmetry scales.
+[[File:Family Tree of MOS-MV2 Scales.svg|thumb|811x811px|The family tree of moment-of-symmetry scales, recreated by a xen wiki user. Note that the construction of this tree uses an upper and lower child, as opposed to a left and right child.]][[Erv Wilson]] was the first to describe such a tree using Fibonacci rabbit patterns. One version of his tree is referred to the scale/rhythm tree, and it's this tree that shows the parent-child relationship between all (single-period) moment-of-symmetry scales.
 Since the term "scale tree" is already used to describe scales arranged using the Farey or Stern-Brocot trees, the term "family tree" is used instead.
-=== Differences and conventions ===
+=== Conventions and other differences ===
-Although the tree described here is ultimately based on Wilson's description and therefore shows the same information, the tree described in this article deviates from Wilson's description. The main differences are listed below:
+For the purposes of this article, the mos family tree will be depicted sideways and with an "upper" and "lower" child mos, rather than a left and right child, as typical with binary trees. This is exactly how Wilson initially described his tree, but with a few additional differences, listed below:
+* Scale step patterns may not be shown; preferably, the mos in its xL ys form will be shown instead of a step pattern.
-* Scale step patterns may not be shown. If they are shown, scale step patterns will always be shown in both its brightest and darkest modes. One way to think of this is the brightest mode will have as many L's to the left as possible while still preserving the mos property, and the darkest mode will have as many L's to the right as possible while still preserving the mos property.
+* If step patterns are shown, they will always be shown in both its brightest and darkest modes.
 * The construction rules described here result in a tree that is upside-down relative to Wilson's description.
@@ Line 16: / Line 16: @@
 At the root of the tree is the step pattern Ls, representing the mos 1L 1s. The child scales of any node can be constructed as such:
-* One child starts with a copy of the step pattern of its parent and has every "L" replaced with "Ls" and every "s" is replaced with "s".
+* One child (the upper child) starts with a copy of the step pattern of its parent and has every "L" replaced with "Ls" and every "s" is replaced with "s".
-* The other child starts with a reversed copy of the parent's step pattern and has every "L" replaced with "Ls" and every "s" replaced with "L".
+* The other child (the lower child) starts with a reversed copy of the parent's step pattern and has every "L" replaced with "Ls" and every "s" replaced with "L".
 This pattern is repeated indefinitely to each new node added to the tree, or for however many generations are desired.
@@ Line 217: / Line 217: @@
 |
 |-
-|[[13L 8]]
+|[[13L 8s]]
 |}
@@ Line 229: / Line 229: @@
 Since a tree structure is built such that each node connects back to a unique parent (except for the root), there are no looping paths, so every path between any two nodes is unique. In [[regular temperament theory]], scales are described as being generated from stacking an interval repeatedly, with moment-of-symmetry scales resulting from this process. Since the sizes of the generating intervals are necessarily described, this means temperaments describe a specific path down the family tree.
-=== Relation to edos ===
+=== Relation to EDOs ===
-Since every interval available to an [[EDO|edo]] can be used as a generating interval, repeatedly stacking such an interval will necessarily produce mosses. Each mos produced this way will describe a unique path on the mos family tree, starting at 1L 1s and terminating right before a pair of sister scales whose note count is equal to the number of equal divisions. Combining all of these paths into a tree will form a subset of the infinite mos family tree, where each path represents a different sequence of mosses that all have the same generating intervals.
+Since every interval available to an [[EDO]] can be used as a generating interval, repeatedly stacking such an interval will necessarily produce mosses. Each mos produced this way will describe a unique path on the mos family tree, starting at 1L 1s and terminating right before a pair of sister scales whose note count is equal to the number of equal divisions. Combining all of these paths into a tree will form a subset of the infinite mos family tree, where each path represents a different sequence of mosses that all have the same generating intervals.
 == External links ==
@@ Line 238: / Line 238: @@
 == See also ==
 * [[Catalog of MOS]]