2edo - Revision history

Overthink: /* Theory */ revamp section

2026-03-15T05:29:10Z

Theory: revamp section

← Older revision		Revision as of 05:29, 15 March 2026
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	== Theory ==		== Theory ==
	The 600{{c}} step of 2edo corresponds to <math>\sqrt{2} \approx 1.414</math> as a frequency ratio. It is the first edo that can be considered to have a [[prime number]] of divisions and the first proper edo, since 1 is not a prime number due to having only itself as a factor and dividing by it returns the same number. It is the first [[The Riemann zeta function and tuning #Zeta EDO lists\|zeta integral edo]] and the first [[The Riemann zeta function and tuning #Zeta EDO lists\|zeta gap edo]], and, in addition, it is also a [[The Riemann zeta function and tuning #Zeta EDO lists\|zeta peak edo]], though 2edo is not the first to have this property, with that distinction instead going to [[1edo]].		The 600{{c}} step of 2edo is the familiar [[tritone]] of [[12edo]], and corresponds to [[Sqrt(2/1)\|<math>\sqrt{2} \approx 1.414</math>]] as a frequency ratio. It is the first [[edo]] that can be considered to have a [[prime number]] of divisions and the first proper edo, since 1 is not a prime number due to having only itself as a factor and dividing by it returns the same number. It is the first [[The Riemann zeta function and tuning #Zeta EDO lists\|zeta integral edo]] and the first [[The Riemann zeta function and tuning #Zeta EDO lists\|zeta gap edo]], and, in addition, it is also a [[The Riemann zeta function and tuning #Zeta EDO lists\|zeta peak edo]], though 2edo is not the first to have this property, with that distinction instead going to [[1edo]].

			=== Structural properties ===

	The harmony that is found in 2edo can be said to revolve around Tonic-Antitonic contrast, with the note at 600{{c}} away from the Tonic having a function akin to [[12edo]]'s diminished fifth. In addition, the full versions of the Antitonic chords of the two possible keys of 2edo are inversions of one another, which can lead to modulations. Furthermore, 2edo can also be used to give a skeletonized version of the 3-limit music such as was used in Medieval Europe, by mapping the fifth and therefore the fourth to 600{{c}}.		The harmony that is found in 2edo can be said to revolve around Tonic-Antitonic contrast, with the note at 600{{c}} away from the Tonic having a function akin to [[12edo]]'s diminished fifth. In addition, the full versions of the Antitonic chords of the two possible keys of 2edo are inversions of one another, which can lead to modulations. Furthermore, 2edo can also be used to give a skeletonized version of the 3-limit music such as was used in Medieval Europe, by mapping the fifth and therefore the fourth to 600{{c}}.

	The mapping of both [[3/2]] and [[4/3]] to the 600-cent tritone, as happens in the [[patent val]], means that 2edo tempers out [[9/8]], and thus supports [[Very low accuracy temperaments #Antitonic\|antitonic]]—an [[exotemperament]] named based on the functionality of the 600{{c}} interval relative to the Tonic. In fact, it even [[support]]s both the 7-limit and 11-limit extensions of antitonic as it also tempers out both [[15/14]] and [[12/11]] respectively. However, the significance of 9/8 in particular being less than half the size of a single step should not be underestimated, as because of this, 2edo is the first edo to demonstrate 3-to-2 [[telicity]]—that is, when not counting the comparatively trivial 1edo. Given this, it is no surprise that 2edo represents the [[3-limit]] [[consistent]]ly. If we treat [[5/4]] the same way as [[81/64]]—which is mapped to the unison courtesy of the tempering of 9/8—we end up with the val {{val\| 2 3 4 }} (2c mapping). This could be used to crush all of the 5 out of 5-limit music, and to then attempt to turn what remains into neo-Medieval harmony.

	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|2}}		{{Harmonics in equal\|2}}

			=== In regular temperament theory ===
			The mapping of both [[3/2]] and [[4/3]] to the 600-cent tritone, as occurs in the [[patent val]] {{Val\| 2 3 5 }}, means that 2edo tempers out [[9/8]], and thus [[support]]s the [[Very low accuracy temperaments #Antitonic\|antitonic]] temperament—an [[exotemperament]] named based on the functionality of the 600{{C}} interval relative to the Tonic. In fact, it even supports the canonical [[extension]] of antitonic to the [[7-limit\|7-]] and [[11-limit]]s, since it also tempers out [[15/14]] and [[12/11]]. Since 9/8 is less than half the size of a single step, 2edo is the first [[Trivial temperament\|non-trivial]] edo to demonstrate 3-to-2 [[telicity]], thus giving a "good" representation of the [[3-limit]]. Given this, it is no surprise that 2edo is [[consistent]] to the [[3-odd-limit]]; in fact, every edo is consistent to the 3-odd-limit, since 3/2 and 4/3 are mapped to their nearest steps by patent val in any edo. However, 2edo is not consistent to the [[5-odd-limit]], since [[6/5]] is mapped to the unison by patent val, while it is slightly closer to the 600{{C}} tritone (6/5 ≈ 316{{C}} in [[JI]]). The smallest edo that is consistent to the 5-odd-limit is [[3edo]].

			If we instead temper out [[81/80]] and treat [[5/4]] the same way as [[81/64]], which is mapped to the unison courtesy of the tempering of 9/8, we end up with the [[val]] {{val\| 2 3 4 }} ([[Wart notation\|2c]] mapping). This could be used to crush all of the 5 out of 5-limit music, and to then attempt to turn what remains into neo-Medieval harmony.

	=== Additional curiosities ===		=== Additional curiosities ===
	* [[99/70]] is [[Nearest just interval\|a good rational representation]] of ~~the square root of~~ 2.		* [[99/70]] is a [[Nearest just interval\|good rational representation]] of <math>\sqrt {2}</math>.

	== Intervals ==		== Intervals ==

Overthink: /* Notation */ make this a subsection

2026-01-16T00:13:49Z

Notation: make this a subsection

← Older revision		Revision as of 00:13, 16 January 2026
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	\|}		\|}

	== Notation ==		=== Notation ===
	{\| class="wikitable center-all"		{\| class="wikitable center-all"
	\|+ style="font-size: 105%;" \| Notation of 2edo		\|+ style="font-size: 105%;" \| Notation of 2edo

Overthink: /* Intervals */ simplify things

2026-01-16T00:13:20Z

Intervals: simplify things

← Older revision		Revision as of 00:13, 16 January 2026
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	\|+ style="font-size: 105%;" \| Intervals of 2edo		\|+ style="font-size: 105%;" \| Intervals of 2edo
	\|-		\|-
	! ~~rowspan="2" \|~~ [[Degree]]		! [[Degree]]
	! ~~rowspan="2" \|~~ [[Cent]]s		! [[Cent]]s
	! ~~rowspan="2" \|~~ [[Interval region]]		! [[Interval region]]
	! ~~colspan="4" \|~~ Approximated [[JI]] intervals* ([[error]] in [[¢]])		! Approximated [[JI]]<br>intervals ([[error]] in [[¢]])
	! ~~rowspan="2" \|~~ Audio		! Audio
	\|-
	~~! [[3-limit]]~~
	~~! [[5-limit]]~~
	~~! [[7-limit]]~~
	~~! Other~~
	\|-		\|-
	\| 0		\| 0
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	\| Unison (prime)		\| Unison (prime)
	\| [[1/1]] (just)		\| [[1/1]] (just)
	\|
	\|
	\|
	\| [[File:piano_0_1edo.mp3]]		\| [[File:piano_0_1edo.mp3]]
	\|-		\|-
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	\| 600		\| 600
	\| [[Tritone]]		\| [[Tritone]]
	\|		\| [[7/5]] (+17.488)<br>[[10/7]] (-17.488)<br>[[17/12]] (-3.000)<br>[[24/17]] (+3.000)<br>[[99/70]] (-0.088)
	\|
	\| [[7/5]] (+17.488)<br>[[10/7]] (-17.488)
	\| [[24/17]] (+3.000)<br>[[99/70]] (-0.~~088~~)<br>[[17/12]] (-3.~~000~~)
	\| [[File:piano_1_2edo.mp3]]		\| [[File:piano_1_2edo.mp3]]
	\|-		\|-
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	\| Octave		\| Octave
	\| [[2/1]] (just)		\| [[2/1]] (just)
	\|
	\|
	\|
	\| [[File:piano_1_1edo.mp3]]		\| [[File:piano_1_1edo.mp3]]
	\|}		\|}
	~~<nowiki />* Based on treating 2edo as a subset of [[12edo]], itself treated as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.~~

	== Notation ==		== Notation ==

FloraC at 04:04, 5 June 2025

2025-06-05T04:04:31Z

← Older revision		Revision as of 04:04, 5 June 2025
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	* [[Semioctave]]		* [[Semioctave]]

	[[Category:3-limit record edos]]		[[Category:3-limit record edos\|#]] <!-- 1-digit number -->
	[[Category:Listen]]		[[Category:Listen]]

FloraC: Recategorize

2025-06-05T04:02:57Z

Recategorize

← Older revision		Revision as of 04:02, 5 June 2025
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	* [[Semioctave]]		* [[Semioctave]]

	[[Category:3-limit]]		[[Category:3-limit record edos]]
	[[Category:Listen]]		[[Category:Listen]]

Lériendil at 02:22, 26 March 2025

2025-03-26T02:22:08Z

← Older revision		Revision as of 02:22, 26 March 2025
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	; [[Ulbass]]		; [[Ulbass]]
	* [https://www.youtube.com/watch?v=X0D9yzSLJrw ''El Bit''] (2023)		* [https://www.youtube.com/watch?v=X0D9yzSLJrw ''El Bit''] (2023)

			== See also ==
			* [[Semioctave]]

	[[Category:3-limit]]		[[Category:3-limit]]
	[[Category:Listen]]		[[Category:Listen]]

FloraC: -redundant category

2025-02-23T13:25:06Z

-redundant category

← Older revision		Revision as of 13:25, 23 February 2025
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	[[Category:3-limit]]		[[Category:3-limit]]
	~~[[Category:Macrotonal]]~~
	[[Category:Listen]]		[[Category:Listen]]

ArrowHead294 at 14:33, 20 February 2025

2025-02-20T14:33:23Z

← Older revision		Revision as of 14:33, 20 February 2025
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	}}		}}
	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro\|2}}		{{ED intro}}

	== Theory ==		== Theory ==
	The 600 ~~cents~~ step of 2edo corresponds to <math>\sqrt{2} \approx 1.414</math> as a frequency ratio. It is the first edo that can be considered to have a [[prime number]] of divisions and the first proper edo, since 1 is not a prime number due to having only itself as a factor and dividing by it returns the same number. It is the first [[The Riemann zeta function and tuning #Zeta EDO lists\|zeta integral edo]] and the first [[The Riemann zeta function and tuning #Zeta EDO lists\|zeta gap edo]], and, in addition, it is also a [[The Riemann zeta function and tuning #Zeta EDO lists\|zeta peak edo]], though 2edo is not the first to have this property, with that distinction instead going to [[1edo]].		The 600{{c}} step of 2edo corresponds to <math>\sqrt{2} \approx 1.414</math> as a frequency ratio. It is the first edo that can be considered to have a [[prime number]] of divisions and the first proper edo, since 1 is not a prime number due to having only itself as a factor and dividing by it returns the same number. It is the first [[The Riemann zeta function and tuning #Zeta EDO lists\|zeta integral edo]] and the first [[The Riemann zeta function and tuning #Zeta EDO lists\|zeta gap edo]], and, in addition, it is also a [[The Riemann zeta function and tuning #Zeta EDO lists\|zeta peak edo]], though 2edo is not the first to have this property, with that distinction instead going to [[1edo]].

	The harmony that is found in 2edo can be said to revolve around Tonic-Antitonic contrast, with the note at 600 ~~cents~~ away from the Tonic having a function akin to [[12edo]]'s diminished fifth. In addition, the full versions of the Antitonic chords of the two possible keys of 2edo are inversions of one another, which can lead to modulations. Furthermore, 2edo can also be used to give a skeletonized version of the 3-limit music such as was used in Medieval Europe, by mapping the fifth and therefore the fourth to 600 ~~cents~~.		The harmony that is found in 2edo can be said to revolve around Tonic-Antitonic contrast, with the note at 600{{c}} away from the Tonic having a function akin to [[12edo]]'s diminished fifth. In addition, the full versions of the Antitonic chords of the two possible keys of 2edo are inversions of one another, which can lead to modulations. Furthermore, 2edo can also be used to give a skeletonized version of the 3-limit music such as was used in Medieval Europe, by mapping the fifth and therefore the fourth to 600{{c}}.

	The mapping of both [[3/2]] and [[4/3]] to the 600-cent tritone, as happens in the [[patent val]], means that 2edo tempers out [[9/8]], and thus supports [[Very low accuracy temperaments #Antitonic\|antitonic]] ~~– a temperament~~ named based on the functionality of the 600 ~~cent~~ interval relative to the Tonic. In fact, it even [[support]]s both the 7-limit and 11-limit extensions of antitonic as it also tempers out both [[15/14]] and [[12/11]] respectively. However, the significance of 9/8 in particular being less than half the size of a single step should not be underestimated, as because of this, 2edo is the first edo to demonstrate 3-to-2 [[telicity]] ~~– that~~ is, when not counting the comparatively trivial 1edo. Given this, it is no surprise that 2edo represents the [[3-limit]] [[consistent]]ly. If we treat [[5/4]] the same way as [[81/64]] ~~– which~~ is mapped to the unison courtesy of the tempering of 9/~~8 – we~~ end up with the val {{val\| 2 3 4 }} (2c mapping). This could be used to crush all of the 5 out of 5-limit music, and to then attempt to turn what remains into neo-Medieval harmony.		The mapping of both [[3/2]] and [[4/3]] to the 600-cent tritone, as happens in the [[patent val]], means that 2edo tempers out [[9/8]], and thus supports [[Very low accuracy temperaments #Antitonic\|antitonic]]—an [[exotemperament]] named based on the functionality of the 600{{c}} interval relative to the Tonic. In fact, it even [[support]]s both the 7-limit and 11-limit extensions of antitonic as it also tempers out both [[15/14]] and [[12/11]] respectively. However, the significance of 9/8 in particular being less than half the size of a single step should not be underestimated, as because of this, 2edo is the first edo to demonstrate 3-to-2 [[telicity]]—that is, when not counting the comparatively trivial 1edo. Given this, it is no surprise that 2edo represents the [[3-limit]] [[consistent]]ly. If we treat [[5/4]] the same way as [[81/64]]—which is mapped to the unison courtesy of the tempering of 9/8—we end up with the val {{val\| 2 3 4 }} (2c mapping). This could be used to crush all of the 5 out of 5-limit music, and to then attempt to turn what remains into neo-Medieval harmony.

	=== Prime harmonics ===		=== Prime harmonics ===
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	== Intervals ==		== Intervals ==
	{\| class="wikitable center-all"		{\| class="wikitable center-all"
	\|+ Intervals of 2edo		\|+ style="font-size: 105%;" \| Intervals of 2edo
			\|-
	! rowspan="2" \| [[Degree]]		! rowspan="2" \| [[Degree]]
	! rowspan="2" \| [[Cent]]s		! rowspan="2" \| [[Cent]]s
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	\| [[File:piano_1_1edo.mp3]]		\| [[File:piano_1_1edo.mp3]]
	\|}		\|}
			<nowiki />* Based on treating 2edo as a subset of [[12edo]], itself treated as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.
	<nowiki>*~~</nowiki> based~~ on treating 2edo as a subset of [[12edo]], itself treated as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

	== Notation ==		== Notation ==
	{\| class="wikitable center-all"		{\| class="wikitable center-all"
	\|+ Notation of 2edo		\|+ style="font-size: 105%;" \| Notation of 2edo
			\|-
	! rowspan="2" \| [[Degree]]		! rowspan="2" \| [[Degree]]
	! rowspan="2" \| [[Cent]]s		! rowspan="2" \| [[Cent]]s
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	== Solfege ==		== Solfege ==
	{\| class="wikitable center-all"		{\| class="wikitable center-all"
	\|+ Solfege of 2edo		\|+ style="font-size: 105%;" \| Solfege of 2edo
			\|-
	! [[Degree]]		! [[Degree]]
	! [[Cents]]		! [[Cents]]

FloraC: Interwiki

2024-11-23T12:07:18Z

Interwiki

← Older revision		Revision as of 12:07, 23 November 2024
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	{{interwiki		{{interwiki
			\| en = 2edo
	\| de = 2-EDO		\| de = 2-EDO
	~~\| en = 2edo~~
	\| es =		\| es =
	\| ja = 2平均律		\| ja = 2平均律
			\| ro = 2DEO
	}}		}}
	{{Infobox ET}}		{{Infobox ET}}

FloraC: /* Music */ -bold, +italic; misc. cleanup

2024-10-16T09:28:36Z

Music: -bold, +italic; misc. cleanup

← Older revision		Revision as of 09:28, 16 October 2024
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	== Music ==		== Music ==
	; [[User:SyntheticThought\|Biptunia]]		; [[User:SyntheticThought\|Biptunia]]
	* [https://biptunia.bandcamp.com/track/2-tet-tritones-all-the-way-down Tritones All the Way Down] (2023)		* [https://biptunia.bandcamp.com/track/2-tet-tritones-all-the-way-down "Tritones All the Way Down"], from [https://biptunia.bandcamp.com/album/freakbone-9000 ''FreakBone 9000''] (2023)

	; [[User:Kaiveran\|Kaiveran Lugheidh]]		; [[User:Kaiveran\|Kaiveran Lugheidh]]
Line 130:		Line 130:

	; [[User:Phanomium\|Phanomium]]		; [[User:Phanomium\|Phanomium]]
	;* [https://www.youtube.com/watch?v=AWJn2RlXsNM Duotone] (2024)		* [https://www.youtube.com/watch?v=AWJn2RlXsNM ''Duotone''] (2024)

	; [[Tancla]]		; [[Tancla]]
	* ~~[https://soundcloud.com/sexytoadsandfrogsfriendcircle/2-clown-core-domestic-use~~ "domestic use guillotine"], from [https://soundcloud.com/sexytoadsandfrogsfriendcircle/sets/staffcirc-vol-7-terra-octava ''STAFFcirc vol. 7''] (2021) ([https://sexytoadsandfrogsfriendcircle.bandcamp.com/track/2-domestic-use-guillotine Bandcamp])		* "domestic use guillotine", from [https://soundcloud.com/sexytoadsandfrogsfriendcircle/sets/staffcirc-vol-7-terra-octava ''STAFFcirc vol. 7''] (2021) – [https://soundcloud.com/sexytoadsandfrogsfriendcircle/2-clown-core-domestic-use SoundCloud] \| [https://sexytoadsandfrogsfriendcircle.bandcamp.com/track/2-domestic-use-guillotine Bandcamp]

	; [[Ulbass]]		; [[Ulbass]]
	* [https://www.youtube.com/watch?v=X0D9yzSLJrw ~~"Ulbass -~~ El Bit"] (2023)		* [https://www.youtube.com/watch?v=X0D9yzSLJrw ''El Bit''] (2023)


	[[Category:3-limit]]		[[Category:3-limit]]
	[[Category:Macrotonal]]		[[Category:Macrotonal]]
	[[Category:Listen]]		[[Category:Listen]]