2edo: Difference between revisions

'''2 equal divisions of the octave''' ('''2EDO''') is the [[tuning system]] derived by dividing the [[octave]] into 2 equal steps of 600 [[cent]]s each.

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 [[Very low accuracy temperaments#Antitonic|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|1EDO]].  Given this, it's no surprise that 2EDO represents the [[3-limit]] [[consistent]]ly. 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 {{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 [[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]]

:* It is also a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak EDO]], though 2EDO is not the first EDO to have this property, with that distinction instead going to 1EDO.