Consistency: Difference between revisions

An [[edo]] (or other [[equal-step tuning]]) represents the ''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''' (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 [[19-odd-limit]], meaning for the set of the odd harmonics 1, 3, 5, 7, 9, 15, 17, and 19, 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:…:(''q'' - 2):''q''.

Formally, if ''N''-edo is an [[equal division of the octave]], and if for any interval ''r'', ''N''-edo (''r'') is the closest ''N''-edo approximation to ''r'', then ''N'' is '''consistent''' with respect to a set of intervals S if for any two intervals ''r''_''i'' and ''r''_''j'' in S where ''r''_''i''''r''_''j'' is also in S, ''N''-edo (''r''_''i''''r''_''j'') = ''N''-edo (''r''_''i'') + ''N''-edo (''r''_''j'').

If for any interval ''r'', ''N''-edo (''r'') is the closest ''N''-edo approximation to ''r'', and if V (''r'') is ''r'' mapped by a val V, then ''N'' is consistent with respect to a set of intervals S if there exists a val V such that ''N''-edo (''r'') = V (''r'') for any ''r'' in S.  

contents=Let us denote the monzo of any ratio ''r'' by m. Due to the linearity of the interval space, for any intervals ''r''_''i'', ''r''_''j'', and ''r''_''i''''r''_''j'' in S, their monzos are m_''i'', m_''j'', and m_''i'' + m_''j'', respectively.  

The ratio ''r'' mapped by the val V is the tempered step number V (''r'') = V·m. There is the following identity:

then ''N''-edo is a member of the function space formed by all vals. Therefore, there exists a val V such that ''N''-edo (''r'') = V (''r'') for any ''r'' in S.  

Normally, S is considered to be some set of [[odd limit|''q''-odd-limit intervals]], consisting of everything of the form 2^''n'' ''u''/''v'', where ''u'' and ''v'' are odd integers less than or equal to ''q''. ''N'' is then said to be ''q-odd-limit consistent''.

The closest approximation for the interval of [[7/6]] (the septimal subminor third) in 25edo is 6 steps, and the closest approximation for the just perfect fifth ([[3/2]]) is 15 steps. Adding the two just intervals gives 3/2 × 7/6 = [[7/4]], the harmonic seventh, for which the closest approximation in 25edo is 20 steps. Adding the two approximated intervals, however, gives 21 steps. This means that 25edo is not consistent in 7-odd-limit. The 4:6:7 triad cannot be mapped to 25edo without one of its three component intervals being inaccurately mapped.

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.

It is possible to generalize the concept of consistency to non-edo equal-step tunings. Because octaves are no longer equivalent, instead of an odd limit we might use an integer limit, and the term 2^''n'' in the above equation{{clarify}} <!-- which? --> is no longer present. Instead, the set S consists of all intervals ''u''/''v'' where ''u'' ≤ ''q'' and ''v'' ≤ ''q'' (''q'' is the largest integer harmonic in S).

This also means that the concept of octave inversion no longer applies: in this example, [[13/9]] is in S, but [[18/13]] is not.