Consistency: Difference between revisions

An [[edo]] represents the [[odd limit|''q''-odd-limit]] '''consistently''' if the closest approximations of the odd harmonics of the ''q''-odd-limit in that edo also give the closest approximations of all the differences between these odd harmonics; for example, the difference between the closest [[7/4]] and the closest [[5/4]] is also the closest [[7/5]]. An [[equal-step tuning]] is '''distinctly consistent''' (or '''uniquely consistent''') in the ''q''-odd-limit if every interval in that odd limit is mapped to a distinct/unique step. So for example, an equal-step tuning cannot be distinctly consistent in the [[7-odd-limit]] if it maps 7/5 and [[10/7]] to the same step—this would correspond to [[tempering out]] [[50/49]], and in the case of edos, would mean the edo must be a multiple, or superset, of 2edo.

While the term '''consistency''' is most frequently used to refer to some odd limit, sometimes one may only care about 'some' of the intervals in some odd limit; this situation often arises when working in [[JI subgroup]]s. We can also skip certain intervals when evaluating consistency. For instance, [[12edo]] is consistent in the no-11's, no-13's [[21-odd-limit]], meaning the set of the odd harmonics 1, 3, 5, 7, 9, 15, 17, 19, and 21, where we deliberately skip 11 and 13.  

In general, we can say that some edo is '''consistent relative to a chord ''C''''', or that '''chord ''C'' is consistent in some edo''', if its best approximation to all the notes in the chord, relative to the root, also gives the best approximation to all of the intervals between the pairs of notes in the chord. In particular, an edo is consistent in the ''q''-odd-limit if and only if it is consistent relative to the chord 1:3:…:{{nowrap|(''q'' − 2)}}:''q''.

A chord is '''consistent to distance''' {{nowrap|''d'' ≥ 1}} or '''consistent to''' ''d'' '''copies''' in an edo (or other equal division) {{w|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 {{nowrap|''d'' ≥ 1}}, this implies consistency in the ordinary sense.

For the mathematically/geometrically inclined, you can think of the set of all ''n'' {{w|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 [[taxicab distance]] of at most ''d'' from the origin (which represents unison) have the [[direct approximation]] 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.

Formally, for some real {{nowrap| ''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''.

| title=Consistency to distance ''d'' can be interpreted as allowing stacking ''d'' copies of a chord ''C'', including the original chord, via intervals that occur in the chord, so that the resulting chord (the union of the ''d'' copies) will always be consistent in the temperament (no matter which intervals are used to stack the ''d'' copies).

| contents=Consider the union {{nowrap|''C'' {{=}} ''C''₁ ∪ ''C''₂ ∪ … ∪ ''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 interval {{nowrap|''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 {{nowrap|C_{''i'' + ''m''}}}, where {{nowrap|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 interval ''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 {{nowrap| 1 − ''ε'' > 1/2 }} and {{nowrap| 1 + ε }} respectively as approximations to the JI interval ''D'', the approximation we got must be the best one. Since ''D'' is arbitrary, we have proved chord consistency.

In non-edo [[equal-step tuning]]s, octaves are not perfectly tuned, and thus an infinite odd limit cannot fully be consistently represented. Instead, we measure consistency in the [[integer limit|''q''-integer-limit]], which is simply the set ''S'' consisting of all intervals ''u''/''v'' where {{nowrap|''u'' ≤ ''q''}} and {{nowrap|''v'' ≤ ''q''}} (and ''q'' is the largest integer harmonic in ''S''). Accordingly, the '''consistency limit''' of an edo describes the highest odd limit it represents consistently, while the consistency limit of any other equal-step tuning (or [[equal temperament]] without an exact octave) instead describes the highest integer limit it represents consistently.

It is possible to extend the concept of odd limits to other equaves, such as the "''q''-throdd-limit" with 3/1 (tritave) equivalence, but because an [[edt]] that is consistent to a certain throdd limit will also be consistent to the corresponding integer-limit, there is little reason to complicate the analysis with additional types of infinite interval sets. This wiki measures consistency in the special case of edos with odd limits instead of integer limits for ease of explanation, but the two types of consistency are effectively equivalent for edos anyways (an edo that is consistent to the ''q''-odd-limit will be consistent to the {{nowrap|(''q'' + 1)}}-integer-limit and vice versa) unless intervals or primes are skipped or if a [[JI subgroup]] is used.