Consistency: Difference between revisions

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A chord is '''consistent to distance''' ''d'' ≥ 1 or '''consistent to''' ''d'' '''copies''' in an edo (or other equal division) [[Wikipedia: If and only if|iff]] the following holds: error accrues slowly enough that ''any'' 0 to d intervals can be combined (multiplied or divided) in ''any'' order without accruing 50% (i.e. half a step) or more of [[relative error]], ''as long as all the intervals chosen are ones present in the chord''. (Note that you may use the same interval ''d'' times even if only one instance of that interval is present in the chord.)

For d >= 1, this implies consistency in the ordinary sense.

For ''d'' ≥ 1, this implies consistency in the ordinary sense.

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.

@@ Line 37: / Line 37: @@
 A chord is '''consistent to distance''' ''d'' &ge; 1 or '''consistent to''' ''d'' '''copies''' in an edo (or other equal division) [[Wikipedia: If and only if|iff]] the following holds: error accrues slowly enough that ''any'' 0 to d intervals can be combined (multiplied or divided) in ''any'' order without accruing 50% (i.e. half a step) or more of [[relative error]], ''as long as all the intervals chosen are ones present in the chord''. (Note that you may use the same interval ''d'' times even if only one instance of that interval is present in the chord.)
-For d >= 1, this implies consistency in the ordinary sense.
+For ''d'' &ge; 1, this implies consistency in the ordinary sense.
 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.