# MOS Scale Family Tree

Todo: expand

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 distinct from other trees, such as the scale tree (which organizes scales based on the Farey tree) and the Stern-Brocot tree, in that the mos family tree organizes scales regardless of temperament, equave, step size, or period. In other words, this is a family tree of step patterns.

The family tree of moment-of-symmetry scales, recreated by a xen wiki user.

## History

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 tree, the term "family tree" is used instead.

### Differences and conventions

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:

• 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.
• 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:

• 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 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, not 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:

• The upper child will have x large steps and x+y small steps, for a mos of xL (x+y)s.
• The lower 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 8

## 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.