Consistency: Difference between revisions

Note that we are looking at the [[direct approximation]] (i.e. the closest approximation) for each interval, and trying to find a [[val]] to line them up.{{clarify}} If there is such a val, then the edo is consistent within that odd-limit, otherwise it is inconsistent.

In general, we can say that some [[equal step tuning]] is '''consistent relative to a [[chord]] ''C''''', or that a '''chord ''C'' is consistent in some equal step tuning''', if its best approximation to all the notes in the chord, relative to the root,<sup style="white-space: nowrap;">{{lbrack}}[[:Category:Todo:clarify|''which?'']]{{rbrack}}</sup> 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 [[Overtone scale|chord 1:3:…:{{nowrap|(''q'' − 2)}}:''q'']]. By convention, when assessing a tuning's '''consistency limit''', this type of odd-integer harmonic series chord (limited to an [[odd limit]]) is used in edos, while in other [[equal step tuning]]s the unmodified harmonic series (limited to an [[integer limit]]) is used instead.

The concept is only defined for [[equal step tuning]]s and not for unequal multirank tunings, since for most choices of generator sizes in these temperaments, you can get any ratio you want to arbitrary precision by piling up a lot of generators (assuming the generator is an irrational fraction of the octave).

Going even further than consistency, an equal step tuning is '''purely consistent'''{{idiosyncratic}} if it approximates all integer harmonics from 1 up to and including ''q'' within one quarter of a step (in other words, maintaining [[relative interval error|relative errors]] of less than 25%). Pure consistency is stronger than consistency but weaker than consistency to distance 2, introduced next.  

A chord is '''consistent to distance''' {{nowrap|''d'' ≥ 1}} or '''consistent to''' ''d'' '''copies''' in an equal step tuning ([[EST]]) {{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.

Note that if the chord comprised of the harmonic series up to ''q'' is "consistent to distance 1", this is equivalent to the [[EST]] being consistent in the [[integer limit|''q''-integer-limit]] (as well as the [[odd limit|<math>2 \lceil \frac{q}{2} \rceil-1</math>-odd-limit]] if it is an edo); more generally, because "consistent to distance 1" means that the direct approximations agree with how the intervals are reached arithmetically, the concept 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) — in this case, intervals between the "basis" harmonics of a truncated harmonic series (an [[integer limit]]).

Formally, for some real {{nowrap| ''d'' > 0 }}, a JI chord ''c'' is consistent to distance ''d'' in an [[equal tuning]] ''T'' if the consistent approximation ''C'' of ''c'' in ''T'' satisfies the property that all intervals in ''C'' are off from their corresponding intervals in ''c'' by less than 1/(2''d'') steps of ''T''.

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.