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{{interwiki | |||
| en = 2edo | |||
| de = 2-EDO | |||
| es = | |||
| ja = 2平均律 | |||
| ro = 2DEO | |||
}} | |||
{{Infobox ET}} | |||
{{ED intro}} | |||
== | == Theory == | ||
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}}. | |||
=== Prime harmonics === | |||
{{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 === | |||
* [[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) | |||
|- | |||
| 0 | |||
| 0 | |||
| Do (P1) | |||
| Da (P1) | |||
|- | |||
| 1 | |||
| 600 | |||
| Fi (A4)<br>Se (d5) | |||
| Pa (A4)<br>Sha (d5) | |||
|- | |||
| 2 | |||
| 1200 | |||
| Do (P8) | |||
| Da (P8) | |||
|} | |||
== Music == | |||
; [[User:SyntheticThought|Biptunia]] | |||
* [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]] | |||
* [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]] | |||
Latest revision as of 05:29, 15 March 2026
| ← 1edo | 2edo | 3edo → |
(convergent)
2 equal divisions of the octave (abbreviated 2edo or 2ed2), also called 2-tone equal temperament (2tet) or 2 equal temperament (2et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 2 equal parts of exactly 600 ¢ each. Each step represents a frequency ratio of 21/2, or the square root of 2.
Theory
The 600 ¢ step of 2edo is the familiar tritone of 12edo, and corresponds to [math]\displaystyle{ \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 zeta integral edo and the first zeta gap edo, and, in addition, it is also a 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 ¢ 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 ¢.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0 | -102 | +214 | +231 | +49 | -241 | -105 | -298 | -28 | +170 | +55 |
| Relative (%) | +0.0 | -17.0 | +35.6 | +38.5 | +8.1 | -40.1 | -17.5 | -49.6 | -4.7 | +28.4 | +9.2 | |
| Steps (reduced) |
2 (0) |
3 (1) |
5 (1) |
6 (0) |
7 (1) |
7 (1) |
8 (0) |
8 (0) |
9 (1) |
10 (0) |
10 (0) | |
In regular temperament theory
The mapping of both 3/2 and 4/3 to the 600-cent tritone, as occurs in the patent val ⟨2 3 5], means that 2edo tempers out 9/8, and thus supports the antitonic temperament—an exotemperament named based on the functionality of the 600 ¢ interval relative to the Tonic. In fact, it even supports the canonical extension of antitonic to the 7- and 11-limits, 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 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 ¢ tritone (6/5 ≈ 316 ¢ 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 ⟨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.
Additional curiosities
- 99/70 is a good rational representation of [math]\displaystyle{ \sqrt {2} }[/math].
Intervals
| Degree | Cents | Interval region | Approximated JI intervals (error in ¢) |
Audio |
|---|---|---|---|---|
| 0 | 0 | Unison (prime) | 1/1 (just) | |
| 1 | 600 | Tritone | 7/5 (+17.488) 10/7 (-17.488) 17/12 (-3.000) 24/17 (+3.000) 99/70 (-0.088) |
|
| 2 | 1200 | Octave | 2/1 (just) |
Notation
| Degree | Cents | 12edo subset notation | |
|---|---|---|---|
| Diatonic interval names | Note names (on D) | ||
| 0 | 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 (
) and sagittal flat (
) respectively.
Solfege
| Degree | Cents | 12edo subset standard solfege (movable do) |
12edo subset uniform solfege (2-3 vowels) |
|---|---|---|---|
| 0 | 0 | Do (P1) | Da (P1) |
| 1 | 600 | Fi (A4) Se (d5) |
Pa (A4) Sha (d5) |
| 2 | 1200 | Do (P8) | Da (P8) |
Music
- "Tritones All the Way Down", from FreakBone 9000 (2023)
- Dichotomy (2017)
- "Organized Cacophony", from Edolian (2020)
- Duotone (2024)
- "domestic use guillotine", from STAFFcirc vol. 7 (2021) – SoundCloud | Bandcamp
- El Bit (2023)