View source for 3edo

{{interwiki
| de = 3-EDO
| en = 3edo
| es = 
| ja = 3平均律
| ro = 3DEO
}}
{{Infobox ET}}
{{ED intro}}

== Theory ==
Some would argue that 3edo is more chord than scale on the basis that as it is identical for what [[12edo]] uses as an augmented triad, it is familiar in that capacity. However, it is nonetheless possible to make music in it. Specifically, tonality is actually established with only two notes at a time as opposed to three—aside from octave reduplications of the Tonic that is—and the two notes directly above the Tonic are used together as a contrast chord. However, it also has real significance in the theoretical realm as well. Its associated 5-limit [[mapping]], or [[val]], is {{val| 3 5 7 }}, meaning octaves are mapped to 3 steps (hence 3edo), the 3rd harmonic to 5 steps and the fifth harmonic to 7 steps. It represents the [[5-limit]] [[consistent]]ly and modulo 3, the mappings of 2, 3, and 5 are distinct, and both of [[4:5:6|1–5/4–3/2]] and [[10:12:15|1–6/5–3/2]] are mapped to 0–1–2. 3edo therefore erases the distinction between major and minor, and between the root, third and fifth of the chord, while keeping the basic outline of the triad.

It also erases leading tones in the sense that 10/9, 16/15 and 25/24 are all mapped to the unison, or in other words to 0 steps. It can therefore be seen as related to 19th century musical theories such as those of {{w|Carl Friedrich Weitzmann}}, who classified triads in terms of the associated augmented triad of 12edo as semitonal displacements. Moreover if we encode the kind of triad (major or minor, or even including augmented and diminished) along with the 3edo note, we can reconstruct a 5-limit or even 7-limit version of the music. Changing the code will change the music, but preserve its 3edo skeleton. And because of the way 3edo relates to 10/9, 16/15 and 25/24, there is a tendency for voice-leading to work itself out correctly when we do.

3edo can be seen as a trivial tuning of the [[augmented (temperament)|augmented]] temperament, since it tempers [[128/125]] (the diesis) by equating three major thirds to an octave.

=== Prime harmonics ===
{{Harmonics in equal|3}}

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 3edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)<br />[[9/7]] (-35.084)
| [[34/27]] (+0.910)
| [[File:piano_1_3edo.mp3]]
|-
| 2
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)<br />[[100/63]] (+0.108)
| [[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 3
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}
<nowiki />* Based on treating 3edo as a subset of [[12edo]], itself treated as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 3edo
|-
! 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
| 400
| Major third (M3)<br />Diminished fourth (d4)
| F#<br />Gb
|-
| 2
| 800
| Augmented fifth (A5)<br />Minor sixth (m6)
| A#<br />Bb
|-
| 3
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 3edo:
* [[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 3edo
|-
! [[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
| 400
| Mi (M3)
| Ma (M3)<br />Fo (d4)
|-
| 2
| 800
| Si (A5)<br />Le (m6)
| Su (A5)<br />Fla (m6)
|-
| 3
| 1200
| Do (P8)
| Da (P8)
|}

== Scales ==
=== MOS ===
This edo is the first to have an actual valid [[MOS]] scale. Specifically, it has the basic form of the [[1L 1s|monowood]] MOS scale.

== Music ==
; [[No Clue Music]]
* [https://www.youtube.com/watch?v=mBxNwsCka3o ''AugTri''] (2024)

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=kMEJ2UG95sY "Augmented"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=6fmcrcQ-lGA ''Augmented''] (2024)

; [[STC_1002]]
* "Neainaz Antithetica, Variation I", from [https://soundcloud.com/sexytoadsandfrogsfriendcircle/sets/staffcirc-vol-7-terra-octava ''STAFFcirc vol. 7''] (2021) – [https://soundcloud.com/sexytoadsandfrogsfriendcircle/3-stc-s1002-neainaz SoundCloud] | [https://sexytoadsandfrogsfriendcircle.bandcamp.com/track/3-neainaz-antithetica-variation-i Bandcamp]

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