# MOS Scale Family Tree

Todo: expand
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The MOS scale family tree (or mos family tree) is an infinite binary tree that organizes 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

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.

### Conventions and other differences

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.
• 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.

## Construction

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 (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 (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.

An alternate version of the tree can be made by only considering the number of steps there are, ignoring the pattern of steps formed. Again, the tree starts with 1L 1s at the root. The child scales will have the following note counts, given a parent of xL ys:

• One child will have x large steps and x+y small steps, for a mos of xL (x+y)s.
• The other child will have x+y large steps and x small steps, for a mos of (x+y)L xs.

## Family tree as a table

The following table organizes mosses as described. Scales with at least 4 steps have links that go to their corresponding page.

Parent Scale 1st-order child mosses 2nd-order child mosses 3rd-order child mosses 4th-order child mosses 5th-order child mosses
1L 1s 1L 2s 1L 3s 1L 4s 1L 5s 1L 6s
6L 1s
5L 1s 5L 6s
6L 5s
4L 1s 4L 5s 4L 9s
9L 4s
5L 4s 5L 9s
9L 5s
3L 1s 3L 4s 3L 7s 3L 10s
10L 3s
7L 3s 7L 10s
10L 7s
4L 3s 4L 7s 4L 11s
11L 4s
7L 4s 7L 11s
11L 7s
2L 1s 2L 3s 2L 5s 2L 7s 2L 9s
9L 2s
7L 2s 7L 9s
9L 7s
5L 2s 5L 7s 5L 12s
12L 5s
7L 5s 7L 12s
12L 7s
3L 2s 3L 5s 3L 8s 3L 11s
11L 3s
8L 3s 8L 11s
11L 8s
5L 3s 5L 8s 5L 13s
13L 5s
8L 5s 8L 13s
13L 8s

## Observations

The following section is a list of observations that are either mathematical or can be made with respect to other structures described by various xenharmonic theories.

### Growth

The uppermost branch in the tree experiences linear growth, growing by one step each generation, and the lowermost branch experiences exponential growth, in line with the Fibonacci sequence.

### Relation to temperaments

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

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.