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]] into a self-similar spiral shape, so that it may be compared with a circle of the same interval in a smaller coprime edo.

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 smaller coprime edo.

Spiral charts ~~in the form referenced on this page were described by [[Kite Giedraitis]] no later than April 2014, though very similar spiral-based visualizations of notes~~ were ~~also~~ used by Jeff Jensen ~~as early as~~ 2004.<ref>https://jjensen.org/spiral5ths/Spiral5ths.html</ref>

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

== Spirals of twelve fifths ==

The spiral charts for [[31edo]], [[41edo]] and [[53edo]] relate each of those edos to [[12edo]]. ~~Each~~ chart has 12 ~~'''wheel-~~spokes~~'''~~.

The spiral charts for [[31edo]], [[41edo]] and [[53edo]] relate each of those edos to [[12edo]] via their chains of fifths. Thus, each chart has 12 spokes.

The larger edo's spiral of fifths is not really a spiral, it's a larger [[circle of fifths]] that is broken into a chain to make several smaller 12-note loops. Then a few duplicates are added at each end of the chain, so that one can reconnect the ends mentally to get the original larger circle.

The larger edo's spiral of fifths is not really a spiral, it's a larger [[circle of fifths]] that is broken into a chain to make several smaller 12-note loops. Then a few duplicates are added at each end of the chain, so that one can reconnect the ends mentally to get the original larger circle. For the general case of fifth tunings that are not edos, the spiral is a true infinite spiral.

A 12-spoke spiral chart of fifths ~~is only possible~~ if the ~~[[sharpness#dodeca-sharpness|dodeca-sharpness]] (edosteps per [[Pythagorean~~ comma~~]]) of the larger edo is 1~~ or -1. A 12-spoke spiral of ''semitones'' ~~is possible~~ for edos of the form 12n+1 or 12n-1, but those spirals are less interesting because they convey very little info that isn't already in the table of edosteps.

A 12-spoke spiral chart of fifths makes the most sense if, after going around by 12 fifths, the resulting interval differs by a very small amount, such as a comma or a single edostep. A 12-spoke spiral of ''semitones'' works for edos of the form 12n+1 or 12n-1, but those spirals are less interesting because they convey very little info that isn't already in the table of edosteps.

=== Gallery ===

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== Spirals of other amounts, other intervals ==

Such a spiral chart can be made for any two edos, as long as they are coprime~~. It's often~~ a ~~spiral of something other than fifths. In fact, it~~'s ~~a spiral of~~ the ~~[[User:TallKite/The delta method|nearest miss]]~~.

Such a spiral chart can be made for any two edos, as long as they are coprime; the interval that makes the most sense to choose in this case is the closest interval between the two edos, called the "near miss", because that results in a small comma alteration once you go around by the smaller edo's size in the larger edo (like fifths in 31edo).

For example, consider [[8edo]] and [[27edo]]. The near misses are 3\8 and 10\27. You get an 8-spoke spiral of 27edo major 3rds. This might be useful for someone researching [[octatonic]] scales in 27edo. To follow the circle of 3rds, 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]] into a self-similar spiral shape, so that it may be compared with a circle of the same interval in a smaller coprime edo.
+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 smaller coprime edo.
-Spiral charts in the form referenced on this page were described by [[Kite Giedraitis]] no later than April 2014, though very similar spiral-based visualizations of notes were also used by Jeff Jensen as early as 2004.<ref>https://jjensen.org/spiral5ths/Spiral5ths.html</ref>
+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
 == Spirals of twelve fifths ==
-The spiral charts for [[31edo]], [[41edo]] and [[53edo]] relate each of those edos to [[12edo]]. Each chart has 12 '''wheel-spokes'''.
+The spiral charts for [[31edo]], [[41edo]] and [[53edo]] relate each of those edos to [[12edo]] via their chains of fifths. Thus, each chart has 12 spokes.
-The larger edo's spiral of fifths is not really a spiral, it's a larger [[circle of fifths]] that is broken into a chain to make several smaller 12-note loops. Then a few duplicates are added at each end of the chain, so that one can reconnect the ends mentally to get the original larger circle.
+The larger edo's spiral of fifths is not really a spiral, it's a larger [[circle of fifths]] that is broken into a chain to make several smaller 12-note loops. Then a few duplicates are added at each end of the chain, so that one can reconnect the ends mentally to get the original larger circle. For the general case of fifth tunings that are not edos, the spiral is a true infinite spiral.
-A 12-spoke spiral chart of fifths is only possible if the [[sharpness#dodeca-sharpness|dodeca-sharpness]] (edosteps per [[Pythagorean comma]]) of the larger edo is 1 or -1. A 12-spoke spiral of ''semitones'' is possible for edos of the form 12n+1 or 12n-1, but those spirals are less interesting because they convey very little info that isn't already in the table of edosteps.
+A 12-spoke spiral chart of fifths makes the most sense if, after going around by 12 fifths, the resulting interval differs by a very small amount, such as a comma or a single edostep. A 12-spoke spiral of ''semitones'' works for edos of the form 12n+1 or 12n-1, but those spirals are less interesting because they convey very little info that isn't already in the table of edosteps.
 === Gallery ===
@@ Line 112: / Line 112: @@
 == Spirals of other amounts, other intervals ==
-Such a spiral chart can be made for any two edos, as long as they are coprime. It's often a spiral of something other than fifths. In fact, it's a spiral of the [[User:TallKite/The delta method|nearest miss]].
+Such a spiral chart can be made for any two edos, as long as they are coprime; the interval that makes the most sense to choose in this case is the closest interval between the two edos, called the "near miss", because that results in a small comma alteration once you go around by the smaller edo's size in the larger edo (like fifths in 31edo).
 For example, consider [[8edo]] and [[27edo]]. The near misses are 3\8 and 10\27. You get an 8-spoke spiral of 27edo major 3rds. This might be useful for someone researching [[octatonic]] scales in 27edo. To follow the circle of 3rds, read the columns left to right, and within each column read top to bottom.