Spiral chart: Difference between revisions

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A '''spiral chart''' is ~~an illustration which converts~~ a ~~circle~~ of ~~repeats~~ of an interval in ~~one~~ [[~~edo~~]] (~~or simply~~ a ~~temperament~~-~~agnostic~~ chain of that interval~~) into~~ a ~~self~~-~~similar~~ spiral ~~shape, so~~ that ~~it may be compared with~~ a ~~circle~~ of the ~~same~~ interval in a ~~related edo~~.

[[File:41-edo spiral with notes.png|thumb|A 12-spoke spiral chart showing the [[chain of fifths]] of [[41edo]], with note names in [[Kite's ups and downs notation]].]]

{{Wikipedia|Circle of fifths#Enharmonic equivalents, theoretical keys, and the spiral of fifths}}

A '''spiral chart''' is a {{w|spiral}}-shaped visualization of an [[interval chain]], generated by [[stacking]] the same [[interval]] repeatedly above or below itself. A series of points along the spiral show the intervals, usually [[octave-reduced]] (i.e. [[interval class]]es), or alternatively the note names (i.e. [[pitch class]]es), that make up the interval chain.

Spiral charts can be used to visualize open interval chains, which never return to the starting interval class exactly, notably in [[just intonation]]. They can also be used to visualize closed interval chains, especially when they contain an interval class that is very close in [[pitch]] to the starting interval class (i.e. a [[comma]]). Using a spiral chart for closed chains emphasizes the parts of the chain which are slightly pitch-shifted copies of another part of the chain, although as a side effect, equivalent pitches can be found at multiple points along the spiral. In both cases (open and closed chains), the spacing of notes around the spiral can be adjusted so that comma-separated interval classes are aligned on the same ''spoke'', a straight or slightly curved line radiating from the center. By extension, spiral charts can be used to compare an [[edo]] to a smaller [[coprime]] edo by choosing the number of spokes accordingly.

For example, whereas the [[circle of fifths]] of [[12edo]] closes after 12 fifths, as it comes back exactly to the starting interval class, the corresponding 12-spoke spiral of fifths instead starts a new loop at that point. If the spiral of fifths is used to represent a chain of pure fifths ([[3/2]]), then the first interval class on the second loop is a [[Pythagorean comma]] away from the starting interval class, and so on.

Spiral charts were first known to be used by Jeff Jensen in 2004,<ref>https://jjensen.org/spiral5ths/Spiral5ths.html</ref> to describe the chain of fifths as it relates to 12edo. Much of the theory on this page, however, comes from [[Kite Giedraitis]], no later than April 2014

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File:31-edo spiral.png|31edo spiral chart

File:41-edo spiral.png|41edo spiral chart

~~File:41-edo spiral with notes.png|41edo spiral chart (notation)~~

File:53-edo spiral.png|53edo spiral chart

</gallery>The same information can be presented as a table. To follow the circle of 5ths, read the columns left to right, and within each column read top to bottom.

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-A '''spiral chart''' is an illustration which converts a circle of repeats of an interval in one [[edo]] (or simply a temperament-agnostic chain of that interval) into a self-similar spiral shape, so that it may be compared with a circle of the same interval in a related edo.
+[[File:41-edo spiral with notes.png|thumb|A 12-spoke spiral chart showing the [[chain of fifths]] of [[41edo]], with note names in [[Kite's ups and downs notation]].]]
+{{Wikipedia|Circle of fifths#Enharmonic equivalents, theoretical keys, and the spiral of fifths}}
+A '''spiral chart''' is a {{w|spiral}}-shaped visualization of an [[interval chain]], generated by [[stacking]] the same [[interval]] repeatedly above or below itself. A series of points along the spiral show the intervals, usually [[octave-reduced]] (i.e. [[interval class]]es), or alternatively the note names (i.e. [[pitch class]]es), that make up the interval chain.
+Spiral charts can be used to visualize open interval chains, which never return to the starting interval class exactly, notably in [[just intonation]]. They can also be used to visualize closed interval chains, especially when they contain an interval class that is very close in [[pitch]] to the starting interval class (i.e. a [[comma]]). Using a spiral chart for closed chains emphasizes the parts of the chain which are slightly pitch-shifted copies of another part of the chain, although as a side effect, equivalent pitches can be found at multiple points along the spiral. In both cases (open and closed chains), the spacing of notes around the spiral can be adjusted so that comma-separated interval classes are aligned on the same ''spoke'', a straight or slightly curved line radiating from the center. By extension, spiral charts can be used to compare an [[edo]] to a smaller [[coprime]] edo by choosing the number of spokes accordingly.
+For example, whereas the [[circle of fifths]] of [[12edo]] closes after 12 fifths, as it comes back exactly to the starting interval class, the corresponding 12-spoke spiral of fifths instead starts a new loop at that point. If the spiral of fifths is used to represent a chain of pure fifths ([[3/2]]), then the first interval class on the second loop is a [[Pythagorean comma]] away from the starting interval class, and so on.
 Spiral charts were first known to be used by Jeff Jensen in 2004,<ref>https://jjensen.org/spiral5ths/Spiral5ths.html</ref> to describe the chain of fifths as it relates to 12edo. Much of the theory on this page, however, comes from [[Kite Giedraitis]], no later than April 2014
@@ Line 14: / Line 20: @@
 File:31-edo spiral.png|31edo spiral chart
 File:41-edo spiral.png|41edo spiral chart
-File:41-edo spiral with notes.png|41edo spiral chart (notation)
 File:53-edo spiral.png|53edo spiral chart
 </gallery>The same information can be presented as a table. To follow the circle of 5ths, read the columns left to right, and within each column read top to bottom.