Distributional evenness: Difference between revisions

In practice, binary DE scales are often referred to as "[[MOS scale]]s", but some consider this usage to be technically incorrect because a MOS as defined by [[Erv Wilson]] was to have ''exactly'' two specific intervals for each class other than multiples of the octave. When Wilson discovered MOS scales and found numerous examples, DE scales with period a fraction of an octave such as [[pajara]], [[augmented]], [[diminished]], etc. were not among them.

Let ''r'' ≥ 2 and let ''S'' be an ''r''-ary [[periodic scale]] with length ''n'' (i.e. ''S''(''kn'') = ''kP'' where ''P'' is the period), with step sizes ''x''₁, ..., ''x''_''r'', i.e. such that Δ''S''(''i'') := ''S''(''i''+1) − ''S''(''i'') ∈ {''x''₁, ..., ''x''_''r''} ∀''i'' ∈ '''Z'''. The scale ''S'' is ''distributionally even'' if for every ''i'' ∈ {1, ..., ''r''},  (Δ''S'')^−1(''x''_''i'') is a [[maximally even]] MOS in '''Z'''/''n'''''Z'''. (For the original definition of DE, simply set ''r'' = 2.)