Consistency: Difference between revisions

For the mathematically/geometrically inclined, you can think of the set of all ''n'' [[Wikipedia: Equality (mathematics)|distinct]] intervals in the chord as forming ''n'' (mutually perpendicular) axes of length 1 that form a (hyper)cubic grid of points (existing in ''n''-dimensional space) representing intervals. Then moving in the direction of one of these axes by 1 unit of distance represents multiplying by the corresponding interval once, and going in the opposite direction represents division by that interval. Then, to be ''consistent to distance d'' means that all points that are a [[Wikipedia: Taxicab geometry|taxicab distance]] of at most ''d'' from the origin (which represents unison) have the [[direct mapping]] of their associated intervals agree with the sum of the steps accumulated through how they were reached in terms of moving along axes, with each axis representing the whole number of steps that closest fits the associated interval present in the chord.

Therefore, consistency to large distances represent very accurate (relative to the step size) [[subgroup]] interpretations because a large "space" of the arithmetic is captured "correctly" (without causing contradictions; consistently). Approximations consistent to some reasonable distance (ideally at least 2) would play more nicely in a regular temperament-style [[subgroup]] context where you might prefer a larger variety of low complexity intervals to be consistent to a lesser degree rather than focusing on long-range consistency of a small number of intervals.

Note that if the chord comprised of all the odd harmonics up to the ''q''-th is "consistent to distance 1", this is equivalent to the edo (or ed''k'') being consistent in the ''q''-[[odd-limit]], and more generally, as "consistent to distance 1" means that the direct mappings agree with how the intervals are reached arithmetically, it is intuitively equivalent to the idea of "consistency" with respect to a set of "basis intervals" (intervals you can combine how you want up to ''d'' times); namely, the ones present in the chord.

Formally, for some real ''d'' > 0, a chord C is consistent to distance ''d'' in ''n''-ed''k'' if the consistent approximation C' of C in ''n''-ed''k'' satisfies the property that all intervals in C' are off from their corresponding intervals in C by less than 1/(2''d'') steps of ''n''-ed''k''.

This more formal definition also provides an interesting generalisation of ''d'' from the naturals to the positive reals, as ''consistency to distance 1/2'' can be interpreted as meaning that all intervals in C are ''at worst'' represented using their second-best mapping, which can be tolerable for some purposes assuming sufficiently small steps. "Consistency to distance 1/2" can be nicknamed "semiconsistency", in which case ''C' '' is said to be a "semiconsistent" representation/approximation of C.

Proof: Consider the union C' = C₁ ∪ C₂ ∪ … ∪ C_''d'' in the equal temperament, where the C_i are copies of the (approximations of) chord C. We need to show that this chord is consistent.

Consider any dyad ''D'' = {''x'', ''y''} consisting of two notes ''x'' and ''y'' that occur in C'. We may assume that the notes ''x'' and ''y'' belong in two different copies of C, C_''i'' and C_{''i'' + ''m''}, where 1 ≤ ''i'' ≤ ''i'' + ''m'' ≤ ''d''. Thus ''x'' and ''y'' are separated by a path of at most ''d'' steps (at most ''d'' − 1 for the different copies of ''C'', and 1 for the additional step within C). By consistency to distance ''d'', each dyad D_''j'' in the path has relative error 1/(2''d''). Hence by the triangle inequality, the total relative error ε on ''D'' is strictly less than 1/2 (50%). Since the adjacent intervals to the approximation of ''D'' must have relative error 1 - ε > 1/2 and 1 + ɛ respectively as approximations to the JI dyad D, the approximation we got must be the best one. Since ''D'' is arbitrary, we have proved chord consistency. QED.