2edo

Prime factorization

2 (prime)

Step size

600¢

Fifth

1\2 (600¢)
(convergent)

Semitones (A1:m2)

-1:1 (-600¢ : 600¢)

Consistency limit

3

Distinct consistency limit

1

Special properties

highly composite
zeta peak
zeta peak integer
zeta integral
zeta gap

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 600.000 ¢ each. Each step represents a frequency ratio of 2^1/2, or the 2nd root of 2.

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 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.

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 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 antitonic – a temperament named based on the functionality of the 600 cent interval relative to the Tonic. In fact, it even supports 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 consistently. 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 ⟨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

Approximation of prime harmonics in 2edo
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
Error	relative (%)	+0	-17	+36	+39	+8	-40	-17	-50	-5	+28	+9
Steps (reduced)		2 (0)	3 (1)	5 (1)	6 (0)	7 (1)	7 (1)	8 (0)	8 (0)	9 (1)	10 (0)	10 (0)

Additional curiosities

99/70 is a good rational representation of the square root of 2.

Intervals

Intervals of 2edo
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	600	Tritone			7/5 (+17.488) 10/7 (-17.488)	24/17 (+3.000) 99/70 (-0.088) 17/12 (-3.000)
2	1200	Octave	2/1 (just)

* 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 of 2edo
Degree	Cents	12edo subset notation
Degree	Cents	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.