Consistency: Difference between revisions
m →Consistency to d copies: Need to revise the statement of the theorem. Problem is only "consistency to 2d copies" (i.e. rel error < 25%) actually guarantees that an approximation is the best approximation in the edo |
|||
| Line 34: | Line 34: | ||
An example of the difference between consistency vs. unique 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 uniquely consistent only up to the [[5-odd-limit]]. Another example or non-unique 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 uniquely consistent only up to the [[11-odd-limit]]. | An example of the difference between consistency vs. unique 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 uniquely consistent only up to the [[5-odd-limit]]. Another example or non-unique 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 uniquely consistent only up to the [[11-odd-limit]]. | ||
== Consistency to ''d'' | == Consistency to distance ''d'' == | ||
A chord is '''consistent to''' ''d | A chord is '''consistent to distance''' ''d'' in an edo (or other equal division) [[Wikipedia: If and only if|iff]] all of the following are true: | ||
* The chord is consistent in the ordinary sense, and | * The chord is consistent in the ordinary sense, and | ||
* 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.) | * 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 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 d | 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. | ||
Therefore, consistency to | 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 all the odd harmonics up to the ''q''-th is "consistent to 1 copy", this is equivalent to the EDO (or ED''k'') being consistent in the ''q''-[[odd-limit]], and more generally, as "consistent to 1 copy" means that the direct mappings agree with how the intervals are reached arithmetically, it 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); namely, the ones present in the chord. | Note that if the chord comprised of all the odd harmonics up to the ''q''-th is "consistent to 1 copy", this is equivalent to the EDO (or ED''k'') being consistent in the ''q''-[[odd-limit]], and more generally, as "consistent to 1 copy" means that the direct mappings agree with how the intervals are reached arithmetically, it 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); namely, the ones present in the chord. | ||
For example, 4:5:6:7 is consistent to 3 | For example, 4:5:6:7 is consistent to distance 3 in [[31edo]]. However, 4:5:6:7:11 is only consistent to distance 1 because 11/5 is mapped too inaccurately (relative error 26.2%). This shows that 31edo is especially strong in the 2.3.5.7 subgroup and weaker in 2.3.5.7.11. | ||
Formally, for some real ''d'' > 0, a chord C is consistent to ''d'' | Formally, for some real ''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''. | ||
This more formal definition also provides an interesting generalisation of ''d'' from the naturals to the positive reals, as ''consistency to 1/2 | 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. | ||
'''Theorem:''' Consistency to '' | '''Theorem:''' Consistency to distance ''2d+2'' can be interpreted as allowing stacking ''d'' copies of a chord C, including the original chord, via dyads 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). | ||
Proof: Consider a dyad D = {''x'', ''y''} consisting of two notes ''x'' and ''y'' that occur in the union C' = C<sub>1</sub> ∪ C<sub>2</sub> ∪ … ∪ C<sub>''d''</sub> in the equal temperament, where the C<sub>i</sub> are copies of the (approximations of) chord C. We need to show that this chord is consistent. | Proof: Consider a dyad D = {''x'', ''y''} consisting of two notes ''x'' and ''y'' that occur in the union C' = C<sub>1</sub> ∪ C<sub>2</sub> ∪ … ∪ C<sub>''d''</sub> in the equal temperament, where the C<sub>i</sub> are copies of the (approximations of) chord C. We need to show that this chord is consistent. | ||
| Line 57: | Line 57: | ||
We may choose an arbitrary root ''r'' ∈ C' and assume that the notes ''x'' and ''y'' belong in two different copies of C, C<sub>''i''</sub> and C<sub>''i'' + ''m''</sub>, where 1 ≤ ''i'' ≤ ''i'' + ''m'' ≤ ''d''. | We may choose an arbitrary root ''r'' ∈ C' and assume that the notes ''x'' and ''y'' belong in two different copies of C, C<sub>''i''</sub> and C<sub>''i'' + ''m''</sub>, where 1 ≤ ''i'' ≤ ''i'' + ''m'' ≤ ''d''. | ||
Thus ''x'' and ''y'' are separated by at most ''d+1'' steps. By consistency to distance ''2d+2'', each dyad D<sub>''j''</sub> in the path has relative error 1/(4''d''+4). Hence the total relative error on D is strictly less than 1/4. Since the adjacent intervals to the approximation of D must have relative error >75% and > 25% respectively as apporixmations to the JI dyad D, the approximation we got must be the best one. Since D is arbitrary, we have proved chord consistency. QED. | |||
Examples of more advanced concepts that build on this are [[telicity]] and [[#Maximal consistent set|maximal consistent set]]s. | Examples of more advanced concepts that build on this are [[telicity]] and [[#Maximal consistent set|maximal consistent set]]s. | ||