Scale tree - Revision history

TallKite: added a pdf of the entire tree

2026-02-05T02:41:44Z

added a pdf of the entire tree

← Older revision		Revision as of 02:41, 5 February 2026
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	== Diagrams ==		== Diagrams ==
	The complete scale tree runs from 0\1 to 1\1, a full octave. 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.		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.		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.

TallKite: added terminology for parts of the kite

2026-02-05T02:38:51Z

added terminology for parts of the kite

← Older revision		Revision as of 02:38, 5 February 2026
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	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.		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.

	~~Any~~ branching point on the scale tree (that is, ~~any~~ unique edo interval) is the ~~root~~ of a ''kite.'' Visually on the diagram, the kite of an interval is the rightmost branch ~~of its~~ left ~~child~~ and the ~~leftmost branch of its~~ right ~~child, which~~ forms the distinctive "kite" shape from which the concept get its name. The ~~fact that [[Kite Giedraitis]] references~~ the ~~concept extensively~~ is ~~a coincidence~~. ~~Theory-wise~~, ~~the~~ kite is the set of edo intervals that produce the closest approximations to the circle of the ~~root~~ interval for their size, being sharp or flat by a single edostep for their edo where the ~~root~~ 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.		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 [[Ring number\|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 ==		== Diagrams ==
	The 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 complete scale tree runs from 0\1 to 1\1, a full octave. 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.

	[[File:The_Scale_Tree.png\|800x1023px\|The Scale Tree.png]]		[[File:The_Scale_Tree.png\|800x1023px\|The Scale Tree.png]]

VectorGraphics at 01:25, 19 August 2025

2025-08-19T01:25:01Z

Show changes

VectorGraphics: change wording to reflect usage in xenharmonic theory. "scale tree" gives twice the results in the XA discord as "Stern-Brocot tree"

2025-08-18T09:20:35Z

change wording to reflect usage in xenharmonic theory. "scale tree" gives twice the results in the XA discord as "Stern-Brocot tree"

← Older revision		Revision as of 09:20, 18 August 2025
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	{{Wikipedia\|Stern–Brocot tree}}		{{Wikipedia\|Stern–Brocot tree}}
	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 scale\|MOS scales]] and [[regular temperament theory]].		The '''scale tree''', sometimes 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 scale\|MOS scales]] and [[regular temperament theory]].

	== Construction ==		== Construction ==

ArrowHead294 at 19:07, 4 February 2025

2025-02-04T19:07:32Z

Show changes

BudjarnLambeth: Move todo to bottom

2024-12-21T07:14:20Z

Move todo to bottom

← Older revision		Revision as of 07:14, 21 December 2024
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	{{Todo\| expand\| comment=This page is a work-in-progress; feel free to edit as needed.<br>Idea: give musical examples instead of the 0/1 to 1/0 case, which can be found on Wikipedia. <br>Idea: add a section explaining what “kites” are. \| inline=1}}
	{{Wikipedia\|Stern-Brocot tree}}		{{Wikipedia\|Stern-Brocot tree}}
	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 scale\|MOS scales]] and [[regular temperament theory]].		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 scale\|MOS scales]] and [[regular temperament theory]].
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	* [[MOS Scale Family Tree\|MOS family tree]], a similar tree for organizing MOS scales by production rules.		* [[MOS Scale Family Tree\|MOS family tree]], a similar tree for organizing MOS scales by production rules.
	* [[Sharpness]]		* [[Sharpness]]


			{{Todo\| expand\| comment=This page is a work-in-progress; feel free to edit as needed.<br>Idea: give musical examples instead of the 0/1 to 1/0 case, which can be found on Wikipedia. <br>Idea: add a section explaining what “kites” are. \| inline=1}}

	[[Category:Scale]]		[[Category:Scale]]

BudjarnLambeth: /* See also */ Add sharpness

2024-12-20T04:29:54Z

BudjarnLambeth: Add to todo box

2024-12-20T04:21:26Z

Add to todo box

← Older revision		Revision as of 04:21, 20 December 2024
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	{{Todo\| expand\| comment=This page is a work-in-progress; feel free to edit as needed.<br>Idea: give musical examples instead of the 0/1 to 1/0 case, which can be found on Wikipedia. \| inline=1}}		{{Todo\| expand\| comment=This page is a work-in-progress; feel free to edit as needed.<br>Idea: give musical examples instead of the 0/1 to 1/0 case, which can be found on Wikipedia. <br>Idea: add a section explaining what “kites” are. \| inline=1}}
	{{Wikipedia\|Stern-Brocot tree}}		{{Wikipedia\|Stern-Brocot tree}}
	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 scale\|MOS scales]] and [[regular temperament theory]].		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 scale\|MOS scales]] and [[regular temperament theory]].

Mousemambo: +diagram images

2023-10-02T16:50:47Z

+diagram images

@@ Line 208: / Line 208: @@
 |}
 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.
 == See also ==

Fredg999 category edits: Remove stub (todo:expand will be sufficient)

2023-07-12T03:40:20Z

Remove stub (todo:expand will be sufficient)

← Older revision		Revision as of 03:40, 12 July 2023
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	== See also ==		== See also ==

	* [[MOS Scale Family Tree\|MOS family tree]], a similar tree for organizing MOS scales by production rules.		* [[MOS Scale Family Tree\|MOS family tree]], a similar tree for organizing MOS scales by production rules.

	[[Category:Scale]]		[[Category:Scale]]
	~~[[Category:Stub]]~~

← Older revision		Revision as of 04:21, 20 December 2024
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	{{Todo\| expand\| comment=This page is a work-in-progress; feel free to edit as needed.<br>Idea: give musical examples instead of the 0/1 to 1/0 case, which can be found on Wikipedia. \| inline=1}}		{{Todo\| expand\| comment=This page is a work-in-progress; feel free to edit as needed.<br>Idea: give musical examples instead of the 0/1 to 1/0 case, which can be found on Wikipedia. <br>Idea: add a section explaining what “kites” are. \| inline=1}}
	{{Wikipedia\|Stern-Brocot tree}}		{{Wikipedia\|Stern-Brocot tree}}
	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 scale\|MOS scales]] and [[regular temperament theory]].		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 scale\|MOS scales]] and [[regular temperament theory]].

← Older revision		Revision as of 04:29, 20 December 2024
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	== See also ==		== See also ==
	* [[MOS Scale Family Tree\|MOS family tree]], a similar tree for organizing MOS scales by production rules.		* [[MOS Scale Family Tree\|MOS family tree]], a similar tree for organizing MOS scales by production rules.
			* [[Sharpness]]

	[[Category:Scale]]		[[Category:Scale]]