Consistency: Difference between revisions
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An [[edo]] represents the q- | 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, if the difference between the closest [[7/4]] and the closest [[5/4]] is also the closest [[7/5]]. An edo is '''distinctly consistent''' (or '''uniquely consistent''') in the ''q''-odd-limit if every interval in that odd limit is consistent and mapped to a distinct edostep. For example, an edo cannot be distinctly consistent in the [[7-odd-limit]] if it maps 7/5 and [[10/7]] to the same step (in this case, the semi-octave of [[2edo]], [[tempering out]] [[50/49]]). | ||
This is equivalent to looking at the [[direct approximation]] (i.e. the closest approximation) for each interval, and trying to find a [[val]] that does the same approximation, so that the intervals are lined up by the val. If there is such a val, then the edo is consistent within that odd limit, otherwise it is inconsistent. | |||
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 [[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 an arbitrarily chosen 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 of nature|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 tunings the unmodified harmonic series (limited to an [[integer limit]]) is used instead. | |||
The concept is only defined for equal-step tunings 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). | |||
The page ''[[Minimal consistent edos]]'' shows the smallest edo that is consistent or distinctly consistent in a given odd limit while the page ''[[Consistency limits of small edos]]'' shows the largest odd limit that a given edo is consistent or distinctly consistent in. | |||
== Mathematical definition == | |||
Formally, if ''T'' is an equal tuning, and if for an interval ''r'', ''T''(''r'') is the closest approximation to ''r'' in ''T'', then ''T'' is '''consistent''' with respect to a set of intervals ''S'' if for any two intervals ''r''<sub>''i''</sub> and ''r''<sub>''j''</sub> in ''S'' where ''r''<sub>''i''</sub>''r''<sub>''j''</sub> is also in ''S'', {{nowrap|''T''(''r''<sub>''i''</sub>''r''<sub>''j''</sub>) {{=}} ''T''(''r''<sub>''i''</sub>) + ''T''(''r''<sub>''j''</sub>).}} | |||
; Alternative formulation using val | |||
If for any interval ''r'', ''T''(''r'') is the closest approximation to ''r'' in ''T'', and if ''V''(''r'') is ''r'' mapped by a val ''V'', then ''T'' is consistent with respect to a set of intervals ''S'' if there exists a val ''V'' such that {{nowrap|''T''(''r'') {{=}} ''V''(''r'')}} for any ''r'' in ''S''. | |||
{{Proof | |||
| title=Proof for equivalence | |||
| contents=Let us denote the monzo of any ratio ''r'' by '''m'''. Due to the linearity of the interval space, for any intervals ''r''<sub>''i''</sub>, ''r''<sub>''j''</sub>, and ''r''<sub>''i''</sub>''r''<sub>''j''</sub> in ''S'', their monzos are '''m'''<sub>''i''</sub>, '''m'''<sub>''j''</sub>, and {{nowrap|'''m'''<sub>''i''</sub> + '''m'''<sub>''j''</sub>}}, respectively. | |||
The ratio ''r'' mapped by the val ''V'' is the tempered step number {{nowrap|''V''(''r'') {{=}} ''V''·'''m'''}}, with the following identity: | |||
<math>V\cdot(\vec {m_i} + \vec {m_j}) = V\cdot\vec {m_i} + V\cdot\vec {m_j}</math> | |||
Hence, | |||
<math>V (r_i r_j) = V (r_i) + V (r_j)</math> | |||
If ''T'' satisfies | |||
<math>T (r_i r_j) = T (r_i) + T (r_j)</math> | |||
then ''T'' is an element of the function space formed by all vals {''V''}. Therefore, there exists a val ''V'' such that {{nowrap|''T''(''r'') {{=}} ''V''(''r'')}} for any ''r'' in ''S''. | |||
}} | |||
Normally, ''S'' is considered to be some set of ''q''-odd-limit intervals, consisting of everything of the form {{nowrap|2<sup>''n''</sup> ''u''/''v''}}, where ''u'' and ''v'' are odd integers less than or equal to ''q''. ''T'' is then said to be ''q-odd-limit consistent''. | |||
If each interval in the ''q''-odd-limit is mapped to a unique value by ''T'', then it is said to be ''uniquely q-odd-limit consistent''. | |||
== Examples == | |||
An example for a system that is ''not'' consistent in a particular odd limit is [[25edo]]: | An example for a system that is ''not'' consistent in a particular odd limit is [[25edo]]: | ||
The | 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 {{nowrap|(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. | ||
As another notable example, [[46edo]] is not consistent in the [[15-odd-limit]]. The 15/13 interval is slightly closer to 9 degrees of 46edo than to 10 degrees, but the ''functional'' [[15/13]] (the difference between 46edo's versions of [[15/8]] and [[13/8]]) is 10 degrees. | |||
An example for a system that ''is'' consistent in the [[7-odd-limit]] is [[12edo]]: 3/2 maps to 7\12, 7/6 maps to 3\12, and 7/4 maps to 10\12, which equals 7\12 plus 3\12. 12edo is also consistent in the [[9-odd-limit]], but not in the [[11-odd-limit]]. | |||
An example of the difference between consistency vs distinct consistency: In 12edo the [[7-odd-limit]] intervals 6/5 and 7/6 are both consistently mapped to 3 steps, and although 12edo is consistent up to the [[9-odd-limit]], it is distinctly consistent only up to the [[5-odd-limit]]. Another example of non-distinct consistency is given by the intervals [[14/13]] and [[13/12]] in [[72edo]] where they are both mapped to 8 steps. Although 72edo is consistent up to the [[17-odd-limit]], it is distinctly consistent only up to the [[11-odd-limit]]. | |||
== Generalizations == | |||
=== Pure consistency === | |||
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]]s of no greater than than 25%). Pure consistency is stronger than consistency but weaker than consistency to distance 2, introduced next. | |||
=== Consistency to distance ''d'' === | |||
A chord is '''consistent to distance''' {{nowrap| ''d'' ≥ 1 }} or '''consistent to''' ''d'' '''copies''' in an equal-step tuning {{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 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. | |||
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 the harmonic series up to ''q'' is "consistent to distance 1", this is equivalent to the tuning being consistent in the [[integer limit|''q''-integer-limit]] (as well as the {{nowrap|(2{{ceil|''q''/2}} − 1)}}-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]]). | |||
For example, 4:5:7 is consistent to distance 10 in [[31edo]]. However, 4:5:7:11 is only consistent to distance 1 because 11/5 is mapped too inaccurately (relative error 26.2%). This shows that 31edo is extremely strong in the 2.5.7 subgroup and much weaker in 2.5.7.11. | |||
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''. | |||
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 | |||
| 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''<sub>1</sub> ∪ ''C''<sub>2</sub> ∪ … ∪ ''C''<sub>''d''</sub>}} in the equal tuning, where the ''C''<sub>''i''</sub> 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''<sub>''i''</sub> and {{nowrap|C<sub>''i'' + ''m''</sub>}}, 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''<sub>''j''</sub> 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. | |||
}} | |||
Examples of more advanced concepts that build on this are [[telicity]] and [[#Maximal consistent set|maximal consistent set]]s. | |||
== Maximal consistent set == | |||
Non-technically, a '''maximal consistent set''' (MCS) is a piece of a [[JI subgroup]] such that when you add another interval which is adjacent to the piece (viewed as a chord), then the piece becomes inconsistent in the edo. | |||
Formally, given ''N''-edo, a consistent chord ''C'' and a [[JI subgroup]] ''G'' [[generator|generated]] by the octave and the intervals in ''C'', a ''maximal consistent set'' is a connected set ''S''(connected via intervals that occur in C) such that adding another interval adjacent to ''S'' via an interval in ''C'' results in a chord that is inconsistent in ''N''-edo. The ''maximal connected neighborhood'' (MCN) of ''C'' is a maximal consistent set containing ''C''. | |||
== | == For non-octave tunings == | ||
In non-octave equal-step tunings, 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. | |||
The concept of integer limits means that octave inversion and octave equivalence no longer apply: for example, [[13/10]] and [[11/7]] are in the 16-integer-limit, but [[20/13]] and [[22/7]] are not. | |||
As a result, [[stretched and compressed tuning|octave stretch and compression]] can be employed to improve an equal tuning's consistency limits: if we compress the octave of 46edo slightly (by about a cent), we end up with an 18-''integer''-limit consistent system, which makes it ideal for approximating Mode 8 of the harmonic series. | |||
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. | |||
== External links == | |||
[ | * [http://www.tonalsoft.com/enc/c/consistent.aspx TonalSoft encyclopedia | Consistency / consistent] | ||
[ | * [https://docs.google.com/spreadsheets/d/1yt239Aeh26RwktiI9Nkli87A0nmBcRri1MXrFb4hE-g/edit?usp=sharing Consistency and relative error of edo] | ||
[[Category: | [[Category:EDO theory pages]] | ||
[[Category: | [[Category:Terms]] | ||
[[Category:Consistency| ]] <!-- Main article --> | |||
[[Category:Odd limit]] | |||
[[Category:Pages with proofs]] | |||