2edo: Difference between revisions

{{EDO intro|2}}

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 [[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]], and, in addition, it is also a [[The Riemann zeta function and tuning #Zeta EDO lists|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 [[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 [[support]]s 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]] [[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.

|+ Intervals of 2edo

 

<nowiki>*</nowiki> 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 of 2edo

|+ Solfege of 2edo