Scale tree
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Todo: expand This page is a work-in-progress; feel free to edit as needed. Idea: give musical examples instead of the 0/1 to 1/0 case, which can be found on Wikipedia. |
The scale tree, usually referred to as the Stern-Brocot tree, is an infinite binary tree that lists every possible reduced positive rational number. The scale tree is commonly used in the context of MOS scales and regular temperament theory.
Construction
The easiest way of producing the scale tree is by finding the mediants, or freshman sums, of adjacent ratios, starting with 0/1, 1/0, and the mediant of 1/1 in between. The next level of the tree contains these ratios, as well as the next mediants of 1/2 and 2/1. This process can be repeated to produce a tree of any depth.
| Level 1 | Level 2 | Level 3 | Level 4 |
|---|---|---|---|
| 0/1 | 0/1 | 0/1 | 0/1 |
| 1/4 | |||
| 1/3 | 1/3 | ||
| 2/5 | |||
| 1/2 | 1/2 | 1/2 | |
| 3/5 | |||
| 2/3 | 2/3 | ||
| 3/4 | |||
| 1/1 | 1/1 | 1/1 | 1/1 |
| 4/3 | |||
| 3/2 | 3/2 | ||
| 5/3 | |||
| 2/1 | 2/1 | 2/1 | |
| 5/2 | |||
| 3/1 | 3/1 | ||
| 4/1 | |||
| 1/0 | 1/0 | 1/0 | 1/0 |
If 0/1 and 1/0 are separated into their own level before 1/1 and duplicate ratios in successive levels are removed, the tree structure becomes more apparent.
| Level 0 | Level 1 | Level 2 | Level 3 | Level 4 |
|---|---|---|---|---|
| 0/1 | ||||
| 1/4 | ||||
| 1/3 | ||||
| 2/5 | ||||
| 1/2 | ||||
| 3/5 | ||||
| 2/3 | ||||
| 3/4 | ||||
| 1/1 | ||||
| 4/3 | ||||
| 3/2 | ||||
| 5/3 | ||||
| 2/1 | ||||
| 5/2 | ||||
| 3/1 | ||||
| 4/1 | ||||
| 1/0 |
The two starting ratios of 0/1 and 1/1 may be replaced with any other ratios to produce a new tree that is a subset of the original tree, where the mediant of those ratios represents the root.
Diagrams
The scale tree, or Stern-Brocot tree, provides a visual map of the world of EDOs, based on fifth size.
The regular EDOs, up to 72edo:
See also
- MOS family tree, a similar tree for organizing MOS scales by production rules.