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.
[[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 were invented by [[Kite Giedraitis]] in April 2014.
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


== Spirals of twelve fifths ==
== 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 ===
=== Gallery ===
Line 14: Line 20:
File:31-edo spiral.png|31edo spiral chart
File:31-edo spiral.png|31edo spiral chart
File:41-edo spiral.png|41edo 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
File:53-edo spiral.png|53edo spiral chart
</gallery>
</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.
{| class="wikitable"
|+41edo intervals as a circle of 5ths, grouped into 12 categories
!Tritones
|
|vd5 = ~4 = 556¢
|d5 = vA4 = 585¢
|^d5 = A4 = 615¢
|~5 = ^A4 = 644¢
|-
!Minorish 2nds
|
|vm2 = 59¢
|m2 = 88¢
|^m2 = 117¢
|~2 = 146¢
|-
!Minorish 6ths
|
|vm6 = 761¢
|m6 = 790¢
|^m6 = 820¢
|~6 = 849¢
|-
!Minorish 3rds
|
|vm3 = 263¢
|m3 = 293¢
|^m3 = 322¢
|(~3)
|-
!Minorish 7ths
|
|vm7 = 966¢
|m7 = 995¢
|^m7 = 1024¢
|(~7)
|-
!Perfectish 4ths
|
|v4 = 468¢
|P4 = 498¢
|^4 = 527¢
|
|-
!Perfectish 1sns / 8ves
|
|v8 = 1171¢
|P1 = 0¢
|^1 = 29¢
|
|-
!Perfectish 5ths
|
|v5 = 673¢
|P5 = 702¢
|^5 = 732¢
|
|-
!Majorish 2nds
|(~2)
|vM2 =176 ¢
|M2 = 205¢
|^M2 = 234¢
|
|-
!Majorish 6ths
|(~6)
|vM6 = 878¢
|M6 = 907¢
|^M6 = 937¢
|
|-
!Majorish 3rds
|~3 = 351¢
|vM3 = 380¢
|M3 = 410¢
|^M3 = 439¢
|
|-
!Majorish 7ths
|~7 = 1054¢
|vM7 = 1083¢
|M7 = 1112¢
|^M7 = 1141¢
|
|-
!Tritones
|~4 = vd5 = 556¢
|vA4 = d5 = 585¢
|A4 = ^d5 = 615¢
|^A4 = ~5 = 644¢
|
|}


== Spirals of other amounts, other intervals ==
== 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.
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.
{| class="wikitable"
|+27edo intervals as a circle of major 3rds, grouped into 8 categories
!Tritones
|
|A4 = v5
|vA4 = vv5
|^^4 = ^d5
|^4 = d5
|-
!Middish 7ths
|
|vM7
|~7
|^m7
|m7
|-
!Minorish 3rds
|
|~3
|^m3
|m3
|(M2)
|-
!Minorish 6ths
|
|^m6
|m6
|vm6
|
|-
!Perfectish 1sns / 8ves
|
|^1
|P1
|v8
|
|-
!Majorish 3rds
|
|^M3
|M3
|vM3
|
|-
!Majorish 6ths
|(m7)
|M6
|vM6
|~6
|
|-
!Middish 2nds
|M2
|vM2
|~2
|^m2
|
|-
!Tritones
|A4 = v5
|vA4 = vv5
|^^4 = ^d5
|^4 = d5
|
|}


== Relationship to the scale tree ==
== Relationship to the scale tree ==
Line 30: Line 193:
* [[Scale tree]]
* [[Scale tree]]


== References ==
<references />
[[Category:EDO theory pages]]
[[Category:EDO theory pages]]

Latest revision as of 21:35, 20 August 2025

A 12-spoke spiral chart showing the chain of fifths of 41edo, with note names in Kite's ups and downs notation.
English Wikipedia has an article on:

A spiral chart is a 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 classes), or alternatively the note names (i.e. pitch classes), 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,[1] 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 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. 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 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

  • 31edo spiral chart
    31edo spiral chart
  • 41edo spiral chart
    41edo spiral chart
  • 53edo spiral chart
    53edo spiral chart

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.

41edo intervals as a circle of 5ths, grouped into 12 categories
Tritones vd5 = ~4 = 556¢ d5 = vA4 = 585¢ ^d5 = A4 = 615¢ ~5 = ^A4 = 644¢
Minorish 2nds vm2 = 59¢ m2 = 88¢ ^m2 = 117¢ ~2 = 146¢
Minorish 6ths vm6 = 761¢ m6 = 790¢ ^m6 = 820¢ ~6 = 849¢
Minorish 3rds vm3 = 263¢ m3 = 293¢ ^m3 = 322¢ (~3)
Minorish 7ths vm7 = 966¢ m7 = 995¢ ^m7 = 1024¢ (~7)
Perfectish 4ths v4 = 468¢ P4 = 498¢ ^4 = 527¢
Perfectish 1sns / 8ves v8 = 1171¢ P1 = 0¢ ^1 = 29¢
Perfectish 5ths v5 = 673¢ P5 = 702¢ ^5 = 732¢
Majorish 2nds (~2) vM2 =176 ¢ M2 = 205¢ ^M2 = 234¢
Majorish 6ths (~6) vM6 = 878¢ M6 = 907¢ ^M6 = 937¢
Majorish 3rds ~3 = 351¢ vM3 = 380¢ M3 = 410¢ ^M3 = 439¢
Majorish 7ths ~7 = 1054¢ vM7 = 1083¢ M7 = 1112¢ ^M7 = 1141¢
Tritones ~4 = vd5 = 556¢ vA4 = d5 = 585¢ A4 = ^d5 = 615¢ ^A4 = ~5 = 644¢

Spirals of other amounts, other intervals

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.

27edo intervals as a circle of major 3rds, grouped into 8 categories
Tritones A4 = v5 vA4 = vv5 ^^4 = ^d5 ^4 = d5
Middish 7ths vM7 ~7 ^m7 m7
Minorish 3rds ~3 ^m3 m3 (M2)
Minorish 6ths ^m6 m6 vm6
Perfectish 1sns / 8ves ^1 P1 v8
Majorish 3rds ^M3 M3 vM3
Majorish 6ths (m7) M6 vM6 ~6
Middish 2nds M2 vM2 ~2 ^m2
Tritones A4 = v5 vA4 = vv5 ^^4 = ^d5 ^4 = d5

Relationship to the scale tree

Todo: complete section

See also

References

  1. ↑ https://jjensen.org/spiral5ths/Spiral5ths.html
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