3edo

Prime factorization

3 (prime)

Step size

400¢

Fifth

2\3 (800¢)
(semiconvergent)

Semitones (A1:m2)

2:-1 (800¢ : -400¢)

Consistency limit

5

Distinct consistency limit

3

Special properties

zeta peak
zeta peak integer
zeta gap

3 equal divisions of the octave (abbreviated 3edo or 3ed2), also called 3-tone equal temperament (3tet) or 3 equal temperament (3et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 3 equal parts of exactly 400 ¢ each. Each step represents a frequency ratio of 2^1/3, or the 3rd root of 2.

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 ⟨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 consistently and modulo three, the mappings of 2, 3, and 5 are distinct, and both of 1-5/4-3/2 and 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 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.

When viewed from a regular temperament perspective, 3edo can be seen as a tuning of the augmented temperament, since it tempers 128/125 (the diesis) by equating three major thirds to an octave.

Harmonics

Approximation of prime harmonics in 3edo
Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	absolute (¢)	+0	+98	+14	-169	-151	-41	-105	+102	+172	+170	+55
Error	relative (%)	+0	+25	+3	-42	-38	-10	-26	+26	+43	+43	+14
Steps (reduced)		3 (0)	5 (2)	7 (1)	8 (2)	10 (1)	11 (2)	12 (0)	13 (1)	14 (2)	15 (0)	15 (0)

Intervals

Intervals of 3edo
Degree	Cents	Interval region	Approximated JI intervals* (error in ¢)				Audio
Degree	Cents	Interval region	3-limit	5-limit	7-limit	Other	Audio
0	0	Unison (prime)	1/1 (just)
1	400	Major third	81/64 (-7.820)	5/4 (+13.686)	63/50 (-0.108) 9/7 (-35.084)	34/27 (+0.910)
2	800	Minor sixth	128/81 (+7.820)	8/5 (-13.686)	14/9 (+35.084) 100/63 (+0.108)	27/17 (-0.910)
3	1200	Octave	2/1 (just)

* 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

Notation of 3edo
Degree	Cents	12edo subset notation
Degree	Cents	Diatonic interval names	Note names (on D)
0	0	Perfect unison (P1)	D
1	400	Major third (M3) Diminished fourth (d4)	F# Gb
2	800	Augmented fifth (A5) Minor sixth (m6)	A# 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 () and sagittal flat () respectively.