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{{interwiki

~~| de = 2edo~~

| en = 2edo

| de = 2-EDO

| es =

| ja =

| ja = 2平均律

| ro = 2DEO

}}

~~'''2 equal divisions of the octave''' ('''2edo''') is the [[tuning system]] derived by dividing the [[octave]] into 2 equal steps of 600 [[cent]]s each.~~

== 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, since 1 is not a prime number due to having only itself as a factor. 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 ~~EDO~~ 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]].

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.

=== Structural properties ===

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's 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 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}}.

== ~~Additional curiosities about 2edo~~ ==

=== Prime harmonics ===

* 99/70 is [[Nearest just interval|~~a good rational representation]] of the square root of~~ 2.

== ~~Music~~ ==

=== In regular temperament theory ===

{| class="wikitable ~~sortable~~"

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]].

!~~Title~~

!~~Composer~~

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.

!~~Year~~

!~~Genre~~

=== Additional curiosities ===

!~~Additional links~~

* [[99/70]] is a [[Nearest just interval|good rational representation]] of <math>\sqrt {2}</math>.

== Intervals ==

{| class="wikitable center-all"

|+ style="font-size: 105%;" | Intervals of 2edo

|-

! [[Degree]]

! [[Cent]]s

! [[Interval region]]

! Approximated [[JI]] intervals ([[error]] in [[¢]])

! Audio

|-

| 0

| Unison (prime)

| [[1/1]] (just)

| [[File:piano_0_1edo.mp3]]

|-

| 1

| 600

| [[Tritone]]

| [[7/5]] (+17.488) [[10/7]] (-17.488) [[17/12]] (-3.000) [[24/17]] (+3.000) [[99/70]] (-0.088)

| [[File:piano_1_2edo.mp3]]

|-

| 2

| 1200

| Octave

| [[2/1]] (just)

| [[File:piano_1_1edo.mp3]]

|}

=== Notation ===

{| class="wikitable center-all"

|+ style="font-size: 105%;" | Notation of 2edo

|-

! rowspan="2" | [[Degree]]

! rowspan="2" | [[Cent]]s

! colspan="2" | [[12edo]] [[subset notation]]

|-

! [[5L 2s|Diatonic]] interval names

! Note names (on D)

|-

| 0

| '''Perfect unison (P1)'''

| '''D'''

|-

| 1

| 600

| Augmented fourth (A4) Diminished fifth (d5)

| G# Ab

|-

| 2

| 1200

| '''Perfect octave (P8)'''

| '''D'''

|}

In 2edo:

* [[ups and downs notation]] is identical to 12edo subset notation;

* mixed [[sagittal notation]] is identical to 12edo subset notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==

{| class="wikitable center-all"

|+ style="font-size: 105%;" | Solfege of 2edo

|-

! [[Degree]]

! [[Cents]]

! 12edo subset standard [[solfege]] (movable do)

! 12edo subset [[uniform solfege]] (2-3 vowels)

|-

|~~[https://soundcloud.com/vale-10/dichotomy ''Dichotomy'']~~

| 0

|~~[[User:Kaiveran|Kaiveran Lugheidh]]~~

| 0

~~|2017~~

| Do (P1)

|~~Classical~~

| Da (P1)

|

|-

|~~[https://www.youtube.com/watch?v=RrqIEYVcqEo "Organized Cacophony"] (from ''[https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn Edolian]'')~~

| 1

|~~NullPointerException Music~~

| 600

|~~2020~~

| Fi (A4) Se (d5)

~~|Classical~~

| Pa (A4) Sha (d5)

|

|-

|~~"[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])~~

| 2

|~~Tancla~~

| 1200

|~~2021~~

| Do (P8)

~~|Metal~~

| Da (P8)

|~~[https://sexytoadsandfrogsfriendcircle.bandcamp.com/album/staffcirc-vol-7-terra-octava Album~~ (~~Bandcamp~~)]

|}

[[~~Category~~:3-~~limit~~]]

== Music ==

[[~~Category~~:~~Equal divisions of the octave~~]]

; [[User:SyntheticThought|Biptunia]]

[[~~Category~~:~~Macrotonal~~]]

* [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)

[[Category:~~Prime EDO~~]]

[[Category:~~Zeta~~]]

; [[User:Kaiveran|Kaiveran Lugheidh]]

* [https://soundcloud.com/vale-10/dichotomy ''Dichotomy''] (2017)

; [[NullPointerException Music]]

* [https://www.youtube.com/watch?v=RrqIEYVcqEo "Organized Cacophony"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[User:Phanomium|Phanomium]]

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

; [[Tancla]]

* "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]]

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

== See also ==

* [[Semioctave]]

[[Category:3-limit record edos|#]]

[[Category:Listen]]

@@ Line 1: / Line 1: @@
 {{interwiki
-| de = 2edo
 | en = 2edo
+| de = 2-EDO
 | es =
-| ja =
+| ja = 2平均律
+| ro = 2DEO
 }}
-'''2 equal divisions of the octave''' ('''2edo''') is the [[tuning system]] derived by dividing the [[octave]] into 2 equal steps of 600 [[cent]]s each.
+{{Infobox ET}}
+{{ED intro}}
 == 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, since 1 is not a prime number due to having only itself as a factor.  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 EDO 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]].
-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.
+=== Structural properties ===
-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's 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 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}}.
-== Additional curiosities about 2edo ==
+=== Prime harmonics ===
-* 99/70 is [[Nearest just interval|a good rational representation]] of the square root of 2.
+{{Harmonics in equal|2}}
-== Music ==
+=== In regular temperament theory ===
-{| class="wikitable sortable"
+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]].
-!Title
-!Composer
+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.
-!Year
-!Genre
+=== Additional curiosities ===
-!Additional links
+* [[99/70]] is a [[Nearest just interval|good rational representation]] of <math>\sqrt {2}</math>.
+== Intervals ==
+{| class="wikitable center-all"
+|+ style="font-size: 105%;" | Intervals of 2edo
+|-
+! [[Degree]]
+! [[Cent]]s
+! [[Interval region]]
+! Approximated [[JI]]<br>intervals ([[error]] in [[¢]])
+! Audio
+|-
+| 0
+| 0
+| Unison (prime)
+| [[1/1]] (just)
+| [[File:piano_0_1edo.mp3]]
+|-
+| 1
+| 600
+| [[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)
+| [[File:piano_1_2edo.mp3]]
+|-
+| 2
+| 1200
+| Octave
+| [[2/1]] (just)
+| [[File:piano_1_1edo.mp3]]
+|}
+=== Notation ===
+{| class="wikitable center-all"
+|+ style="font-size: 105%;" | Notation of 2edo
+|-
+! rowspan="2" | [[Degree]]
+! rowspan="2" | [[Cent]]s
+! colspan="2" | [[12edo]] [[subset notation]]
+|-
+! [[5L 2s|Diatonic]] interval names
+! Note names (on D)
+|-
+| 0
+| 0
+| '''Perfect unison (P1)'''
+| '''D'''
+|-
+| 1
+| 600
+| Augmented fourth (A4)<br>Diminished fifth (d5)
+| G#<br>Ab
+|-
+| 2
+| 1200
+| '''Perfect octave (P8)'''
+| '''D'''
+|}
+In 2edo:
+* [[ups and downs notation]] is identical to 12edo subset notation;
+* mixed [[sagittal notation]] is identical to 12edo subset notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.
+== Solfege ==
+{| class="wikitable center-all"
+|+ style="font-size: 105%;" | Solfege of 2edo
+|-
+! [[Degree]]
+! [[Cents]]
+! 12edo subset<br>standard [[solfege]]<br>(movable do)
+! 12edo subset<br>[[uniform solfege]]<br>(2-3 vowels)
 |-
-|[https://soundcloud.com/vale-10/dichotomy ''Dichotomy'']
+| 0
-|[[User:Kaiveran|Kaiveran Lugheidh]]
+| 0
-|2017
+| Do (P1)
-|Classical
+| Da (P1)
-|
 |-
-|[https://www.youtube.com/watch?v=RrqIEYVcqEo "Organized Cacophony"] (from ''[https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn Edolian]'')
+| 1
-|NullPointerException Music
+| 600
-|2020
+| Fi (A4)<br>Se (d5)
-|Classical
+| Pa (A4)<br>Sha (d5)
-|
 |-
-|"[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])
+| 2
-|Tancla
+| 1200
-|2021
+| Do (P8)
-|Metal
+| Da (P8)
-|[https://sexytoadsandfrogsfriendcircle.bandcamp.com/album/staffcirc-vol-7-terra-octava Album (Bandcamp)]
 |}
-[[Category:3-limit]]
+== Music ==
-[[Category:Equal divisions of the octave]]
+; [[User:SyntheticThought|Biptunia]]
-[[Category:Macrotonal]]
+* [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)
-[[Category:Prime EDO]]
-[[Category:Zeta]]
+; [[User:Kaiveran|Kaiveran Lugheidh]]
+* [https://soundcloud.com/vale-10/dichotomy ''Dichotomy''] (2017)
+; [[NullPointerException Music]]
+* [https://www.youtube.com/watch?v=RrqIEYVcqEo "Organized Cacophony"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)
+; [[User:Phanomium|Phanomium]]
+* [https://www.youtube.com/watch?v=AWJn2RlXsNM ''Duotone''] (2024)
+; [[Tancla]]
+* "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]]
+* [https://www.youtube.com/watch?v=X0D9yzSLJrw ''El Bit''] (2023)
+== See also ==
+* [[Semioctave]]
+[[Category:3-limit record edos|#]] <!-- 1-digit number -->
+[[Category:Listen]]