Scale tree: Difference between revisions

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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.

~~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 ==

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]]

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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.
-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 ==
-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]]