Scale tree

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Stern–Brocot tree

The scale tree is an infinite binary tree that lists every possible interval in an equal tuning, assuming a given equave (on this page, the octave). It is the xenharmonic application, commonly used in the context of MOS scales and regular temperament theory, of the Stern-Brocot tree, an infinite tree containing every (positive, reduced) rational number.

Structure

The scale tree is a diagram of every EDO interval, but it can be taken as a chart of EDOs based on their best approximations of some interval in particular, for example tunings of the perfect fifth.

Every branching point on the scale tree (that is, every unique edo interval) is the head of a kite. Visually on the diagram, the kite of an interval is the line connecting the head to the left child of the head (the left shoulder), the left shoulder's rightmost branch (the left side), the line to the right shoulder, and the right side. This forms the distinctive "kite" shape from which the concept get its name. The dotted line descending vertically from the head is the spine of the kite. The entire scale tree is made up of kites, each infinitely tall. The kite's sides are the set of edo intervals that produce the closest approximations to the circle of the head's interval for their size, being sharp or flat by a single edostep for their edo where the head interval would close the circle. For example, the 7\12 kite contains 11\19 and 10\17, but not 13\22 and 15\26 as those end up off by 2 edosteps. The spine consists of multi-ring edos, e.g. 10edo and 15edo.

The ripple lines of a kite are lines which are roughly parallel to the sides of a kite. For example, the sharpness lines are ripple lines of the 4\7 kite. Likewise penta-sharpness lines are ripple lines of the 3\5 kite. The ripple lines connect edos with a similar edostepspan

Diagrams

The complete scale tree runs from 0\1 to 1\1, a full octave, as seen here: File:Scale Tree Complete for edos 2-41.pdf

However, often only the region of the scale tree in the neighborhood of 3/2 is shown. Furthermore, the tree is often pruned to show only the best approximations of 3/2.

This scale tree provides a visual map of the world of EDOs, based on fifth size. Two kites are colored here: the pentatonic (3\5) and heptatonic (4\7) kites.

The regular EDOs, up to 72edo:

Construction

The easiest way of producing the scale tree for the intervals within the octave is by finding the mediants of adjacent ratios, starting with 0\1 (the unison, 0 steps of 1edo) and 1\1 (the octave, 1 step of 1edo). One can also use 0\1 (zero cents) and 1\0 (infinite cents) to get every possible ascending edo interval. Then, treat the logarithmic ratios as normal fractions and proceed take the mediant, which is 1\2 (the semioctave), putting it in between. The next level of the tree contains these ratios, as well as the next mediants 1\3 and 2\3 (the major third and minor sixth of 12edo). This process can be repeated to produce a tree of any depth. Note that these are actually logarithmic ratios, and should be written with a backslash (¥) instead of a forward slash.

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

If duplicate ratios in successive levels are removed, the tree structure becomes more apparent.

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

The two starting intervals of 0\1 and 1\1 may be replaced with any other edo intervals to produce a new tree that is a subset of the original tree, where the mediant of those ratios represents the root.

See also

• MOS family tree, a similar tree for organizing MOS scales by production rules.
• Sharpness

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. Idea: add a section explaining what “kites” are.