Trivial EDO - Revision history

Hotcrystal0 at 20:29, 20 August 2026

2026-08-20T20:29:38Z

← Older revision		Revision as of 20:29, 20 August 2026
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	== 3edo ==		== 3edo ==
	The step of 3edo serves as 12edo's major third and also as 15edo's. When interpreted as 5/4, it supports [[Augmented]] temperament, as 3edo is equivalent to an augmented triad in 12edo. It also serves as the period of [[Misty]] temperament, which splits the deficit between three 5/4s and the octave into three. In general, 3edo provides a basic outline for triadic harmony. 3edo may be considered a 2.5.11/7.19/3 temperament, being the first edo to approximate a low prime other than 2.		The step of 3edo serves as 12edo's major third and also as 15edo's. When interpreted as 5/4, it supports [[Augmented (temperament)\|Augmented]] temperament, as 3edo is equivalent to an augmented triad in 12edo. It also serves as the period of [[Misty]] temperament, which splits the deficit between three 5/4s and the octave into three. In general, 3edo provides a basic outline for triadic harmony. 3edo may be considered a 2.5.11/7.19/3 temperament, being the first edo to approximate a low prime other than 2.

	3edo also simplifies 3:4:5-based harmony, reflecting the inaccurate temperament Father which equates 5/4 and 4/3.		3edo also simplifies 3:4:5-based harmony, reflecting the inaccurate temperament Father which equates 5/4 and 4/3.

Hotcrystal0: Links

2026-08-20T20:29:18Z

Links

← Older revision		Revision as of 20:29, 20 August 2026
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	== 3edo ==		== 3edo ==
	The step of 3edo serves as 12edo's major third and also as 15edo's. When interpreted as 5/4, it supports ~~augmented~~ temperament, as 3edo is equivalent to an augmented triad in 12edo. It also serves as the period of ~~misty~~ temperament, which splits the deficit between three 5/4s and the octave into three. In general, 3edo provides a basic outline for triadic harmony. 3edo may be considered a 2.5.11/7.19/3 temperament, being the first edo to approximate a low prime other than 2.		The step of 3edo serves as 12edo's major third and also as 15edo's. When interpreted as 5/4, it supports [[Augmented]] temperament, as 3edo is equivalent to an augmented triad in 12edo. It also serves as the period of [[Misty]] temperament, which splits the deficit between three 5/4s and the octave into three. In general, 3edo provides a basic outline for triadic harmony. 3edo may be considered a 2.5.11/7.19/3 temperament, being the first edo to approximate a low prime other than 2.

	3edo also simplifies 3:4:5-based harmony, reflecting the inaccurate temperament Father which equates 5/4 and 4/3.		3edo also simplifies 3:4:5-based harmony, reflecting the inaccurate temperament Father which equates 5/4 and 4/3.

Hotcrystal0 at 20:48, 1 August 2026

2026-08-01T20:48:07Z

← Older revision		Revision as of 20:48, 1 August 2026
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	== 4edo ==		== 4edo ==
	The step of 4edo serves as 12edo's minor third, which also appears in 16edo, supporting [[Diminished (temperament)\|~~diminished~~]] temperament if interpreted as 6/5. 4edo's step may also be interpreted as 19/16, supporting the temperament wherein four 19/16s are equated to the octave. Additionally, the semioctave retains its interpretations from 2edo, so that 4edo may be considered a 2.5/3.7/3.17/3.27.19.23 temperament. 4edo outlines tetradic harmony, and distinctly (although inaccurately) represents 4:5:6:7.		The step of 4edo serves as 12edo's minor third, which also appears in 16edo, supporting [[Diminished (temperament)\|Diminished]] temperament if interpreted as 6/5. 4edo's step may also be interpreted as 19/16, supporting the temperament wherein four 19/16s are equated to the octave. Additionally, the semioctave retains its interpretations from 2edo, so that 4edo may be considered a 2.5/3.7/3.17/3.27.19.23 temperament. 4edo outlines tetradic harmony, and distinctly (although inaccurately) represents 4:5:6:7.

	== 6edo ==		== 6edo ==
	6edo is large enough to be usable as a melodic system in its own right, and is more familiar as the 12edo wholetone scale. It provides the basic structure of the 2.5.7 subgroup (serving a similar role for it as 9edo does for the 2.3.5 subgroup), and additionally inherits 2edo's tritone representations and 3edo's mappings, making it a 2.9.5.7.11.17/3.19/3.23 subgroup temperament, although the 11 is one of the less accurate intervals along with the 7 and 23, and 13 is actually better approximated than 11. The basic 4:5:7 chord is mapped to 0-2-5 steps of 6edo.		6edo is large enough to be usable as a melodic system in its own right, and is more familiar as the 12edo wholetone scale. It provides the basic structure of the 2.5.7 subgroup (serving a similar role for it as 9edo does for the 2.3.5 subgroup), and additionally inherits 2edo's tritone representations and 3edo's mappings, making it a 2.9.5.7.11.17/3.19/3.23 subgroup temperament, although the 11 is one of the less accurate intervals along with the 7 and 23, and 13 is actually better approximated than 11. The basic 4:5:7 chord is mapped to 0-2-5 steps of 6edo.
	{{Harmonics in ED\|6\|31}}		{{Harmonics in ED\|6\|31}}

Hotcrystal0: Misspelt come on

2026-08-01T20:46:17Z

Misspelt come on

← Older revision		Revision as of 20:46, 1 August 2026
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	== 4edo ==		== 4edo ==
	The step of 4edo serves as 12edo's minor third, which also appears in 16edo, supporting [[~~Dimished~~ (temperament)\|diminished]] temperament if interpreted as 6/5. 4edo's step may also be interpreted as 19/16, supporting the temperament wherein four 19/16s are equated to the octave. Additionally, the semioctave retains its interpretations from 2edo, so that 4edo may be considered a 2.5/3.7/3.17/3.27.19.23 temperament. 4edo outlines tetradic harmony, and distinctly (although inaccurately) represents 4:5:6:7.		The step of 4edo serves as 12edo's minor third, which also appears in 16edo, supporting [[Diminished (temperament)\|diminished]] temperament if interpreted as 6/5. 4edo's step may also be interpreted as 19/16, supporting the temperament wherein four 19/16s are equated to the octave. Additionally, the semioctave retains its interpretations from 2edo, so that 4edo may be considered a 2.5/3.7/3.17/3.27.19.23 temperament. 4edo outlines tetradic harmony, and distinctly (although inaccurately) represents 4:5:6:7.

	== 6edo ==		== 6edo ==
	6edo is large enough to be usable as a melodic system in its own right, and is more familiar as the 12edo wholetone scale. It provides the basic structure of the 2.5.7 subgroup (serving a similar role for it as 9edo does for the 2.3.5 subgroup), and additionally inherits 2edo's tritone representations and 3edo's mappings, making it a 2.9.5.7.11.17/3.19/3.23 subgroup temperament, although the 11 is one of the less accurate intervals along with the 7 and 23, and 13 is actually better approximated than 11. The basic 4:5:7 chord is mapped to 0-2-5 steps of 6edo.		6edo is large enough to be usable as a melodic system in its own right, and is more familiar as the 12edo wholetone scale. It provides the basic structure of the 2.5.7 subgroup (serving a similar role for it as 9edo does for the 2.3.5 subgroup), and additionally inherits 2edo's tritone representations and 3edo's mappings, making it a 2.9.5.7.11.17/3.19/3.23 subgroup temperament, although the 11 is one of the less accurate intervals along with the 7 and 23, and 13 is actually better approximated than 11. The basic 4:5:7 chord is mapped to 0-2-5 steps of 6edo.
	{{Harmonics in ED\|6\|31}}		{{Harmonics in ED\|6\|31}}

Hotcrystal0: Link to Diminished

2026-08-01T20:45:59Z

Link to Diminished

← Older revision		Revision as of 20:45, 1 August 2026
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	== 4edo ==		== 4edo ==
	The step of 4edo serves as 12edo's minor third, which also appears in 16edo, supporting diminished temperament if interpreted as 6/5. 4edo's step may also be interpreted as 19/16, supporting the temperament wherein four 19/16s are equated to the octave. Additionally, the semioctave retains its interpretations from 2edo, so that 4edo may be considered a 2.5/3.7/3.17/3.27.19.23 temperament. 4edo outlines tetradic harmony, and distinctly (although inaccurately) represents 4:5:6:7.		The step of 4edo serves as 12edo's minor third, which also appears in 16edo, supporting [[Dimished (temperament)\|diminished]] temperament if interpreted as 6/5. 4edo's step may also be interpreted as 19/16, supporting the temperament wherein four 19/16s are equated to the octave. Additionally, the semioctave retains its interpretations from 2edo, so that 4edo may be considered a 2.5/3.7/3.17/3.27.19.23 temperament. 4edo outlines tetradic harmony, and distinctly (although inaccurately) represents 4:5:6:7.

	== 6edo ==		== 6edo ==
	6edo is large enough to be usable as a melodic system in its own right, and is more familiar as the 12edo wholetone scale. It provides the basic structure of the 2.5.7 subgroup (serving a similar role for it as 9edo does for the 2.3.5 subgroup), and additionally inherits 2edo's tritone representations and 3edo's mappings, making it a 2.9.5.7.11.17/3.19/3.23 subgroup temperament, although the 11 is one of the less accurate intervals along with the 7 and 23, and 13 is actually better approximated than 11. The basic 4:5:7 chord is mapped to 0-2-5 steps of 6edo.		6edo is large enough to be usable as a melodic system in its own right, and is more familiar as the 12edo wholetone scale. It provides the basic structure of the 2.5.7 subgroup (serving a similar role for it as 9edo does for the 2.3.5 subgroup), and additionally inherits 2edo's tritone representations and 3edo's mappings, making it a 2.9.5.7.11.17/3.19/3.23 subgroup temperament, although the 11 is one of the less accurate intervals along with the 7 and 23, and 13 is actually better approximated than 11. The basic 4:5:7 chord is mapped to 0-2-5 steps of 6edo.
	{{Harmonics in ED\|6\|31}}		{{Harmonics in ED\|6\|31}}

Hotcrystal0: rewoed beginning

2026-05-12T21:12:56Z

rewoed beginning

← Older revision		Revision as of 21:12, 12 May 2026
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	A '''trivial EDO''' is a small EDO which is not [[xenharmonic]]. ~~The~~ trivial edos ~~are precisely~~ 1edo, 2edo, 3edo, 4edo, and 6edo.		A '''trivial EDO''' is a small EDO which is not [[xenharmonic]] due to being a subset of [[12edo]]. There are five trivial edos: 1edo, 2edo, 3edo, 4edo, and 6edo.

	== 1edo ==		== 1edo ==

Hotcrystal0: fix

2026-05-06T11:27:10Z

fix

← Older revision		Revision as of 11:27, 6 May 2026
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	The step of 3edo serves as 12edo's major third and also as 15edo's. When interpreted as 5/4, it supports augmented temperament, as 3edo is equivalent to an augmented triad in 12edo. It also serves as the period of misty temperament, which splits the deficit between three 5/4s and the octave into three. In general, 3edo provides a basic outline for triadic harmony. 3edo may be considered a 2.5.11/7.19/3 temperament, being the first edo to approximate a low prime other than 2.		The step of 3edo serves as 12edo's major third and also as 15edo's. When interpreted as 5/4, it supports augmented temperament, as 3edo is equivalent to an augmented triad in 12edo. It also serves as the period of misty temperament, which splits the deficit between three 5/4s and the octave into three. In general, 3edo provides a basic outline for triadic harmony. 3edo may be considered a 2.5.11/7.19/3 temperament, being the first edo to approximate a low prime other than 2.

	3edo also simplifies 3:4:5-based harmony, reflecting the inaccurate ~~"father"~~ temperament ~~equating~~ 5/4 and 4/3.		3edo also simplifies 3:4:5-based harmony, reflecting the inaccurate temperament Father which equates 5/4 and 4/3.

	== 4edo ==		== 4edo ==

Vector at 06:34, 6 May 2026

2026-05-06T06:34:47Z

← Older revision		Revision as of 06:34, 6 May 2026
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	A '''trivial EDO''' is a small EDO which is not [[xenharmonic]]. The trivial edos are precisely 1edo, 2edo, 3edo, 4edo, and 6edo.		A '''trivial EDO''' is a small EDO which is not [[xenharmonic]]. The trivial edos are precisely 1edo, 2edo, 3edo, 4edo, and 6edo.

			== 1edo ==
			1edo is equivalent to the 2-limit (the system consisting of only octaves) and the 1-odd-limit, and of the primes only meaningfully represents 2/1. One step of 1edo (that is, the octave itself) serves as the period of most octave-equivalent temperaments.

			== 2edo ==
			2edo may be considered to represent the 2.17/3.7/5.45.23 subgroup, or any other subgroup consisting of only octaves and tritones as generators. Its step is the semioctave, which appears as a critical structural interval in temperaments like pajara (separating 5/4 from 7/4), and it additionally contains the simplest form of functional contrast (between tonic and antitonic).

			2edo represents a simplification of the structure of a [[trine]], and counts as a trine itself.

			== 3edo ==
			The step of 3edo serves as 12edo's major third and also as 15edo's. When interpreted as 5/4, it supports augmented temperament, as 3edo is equivalent to an augmented triad in 12edo. It also serves as the period of misty temperament, which splits the deficit between three 5/4s and the octave into three. In general, 3edo provides a basic outline for triadic harmony. 3edo may be considered a 2.5.11/7.19/3 temperament, being the first edo to approximate a low prime other than 2.

			3edo also simplifies 3:4:5-based harmony, reflecting the inaccurate "father" temperament equating 5/4 and 4/3.

			== 4edo ==
			The step of 4edo serves as 12edo's minor third, which also appears in 16edo, supporting diminished temperament if interpreted as 6/5. 4edo's step may also be interpreted as 19/16, supporting the temperament wherein four 19/16s are equated to the octave. Additionally, the semioctave retains its interpretations from 2edo, so that 4edo may be considered a 2.5/3.7/3.17/3.27.19.23 temperament. 4edo outlines tetradic harmony, and distinctly (although inaccurately) represents 4:5:6:7.

			== 6edo ==
			6edo is large enough to be usable as a melodic system in its own right, and is more familiar as the 12edo wholetone scale. It provides the basic structure of the 2.5.7 subgroup (serving a similar role for it as 9edo does for the 2.3.5 subgroup), and additionally inherits 2edo's tritone representations and 3edo's mappings, making it a 2.9.5.7.11.17/3.19/3.23 subgroup temperament, although the 11 is one of the less accurate intervals along with the 7 and 23, and 13 is actually better approximated than 11. The basic 4:5:7 chord is mapped to 0-2-5 steps of 6edo.
			{{Harmonics in ED\|6\|31}}

Vector: Created page with "A '''trivial EDO''' is a small EDO which is not xenharmonic. The trivial edos are precisely 1edo, 2edo, 3edo, 4edo, and 6edo."

2026-05-06T06:03:57Z

Created page with "A '''trivial EDO''' is a small EDO which is not xenharmonic. The trivial edos are precisely 1edo, 2edo, 3edo, 4edo, and 6edo."

New page

A '''trivial EDO''' is a small EDO which is not [[xenharmonic]]. The trivial edos are precisely 1edo, 2edo, 3edo, 4edo, and 6edo.