2edo

If one attempts to use 2edo as an actual scale, it would divide the octave into two equal parts, each of size 600 cents, which is to say sqrt(2) as a frequency ratio. 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 unison, 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. Given this, it's 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.

Factoids about 2EDO

• It is the first zeta integral edo and the first zeta gap edo

• It is also a zeta peak edo, though 2edo is not the first EDO to have this property, with that distinction instead going to 1edo.

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

Music

Title	Composer	Year	Genre	Additional links
Dichotomy	Kaiveran Lugheidh	2017	Classical