Consistency: Difference between revisions

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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 copies'' 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 many copies 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 ~~high~~-~~span~~ consistency of a small number of intervals.

Therefore, consistency to many copies 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.

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'''Theorem:''' A (1/(2''d'') steps of ''n'' ED''k'') threshold 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 will always satisfy condition #2 of chord consistency.

Proof: Consider a dyad D = {''x'', ''y''} consisting of two notes ''x'' and ''y'' that occur in the resulting chord C' = C1 ∪ C2 ∪ … ∪ C''d'' in the equal temperament, where the Ci are copies of the (approximations of) chord 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''. Suppose C''i'', C''i'' + 1, …, C''i'' + ''m'' are separated by dyads D1, D2, …, D''m'' that occur in C. Let ''x' '' be the interval ''x'' + D1 + D2 + … + D''m'', the counterpart of ''x'' in C''i'' + ''m''. Since ''m'' ≤ ''d'' - 1, by consistency to ~~span~~ ''d'', each dyad D''j'' have relative error 1/(2''d'') since D''i'' occurs in C. The relative error on D1 + D2 + … + D''m'' compared to its just counterpart is < (''d'' - 1)/(2''d''), and again by assumption of consistency to ~~span~~ ''d'', the dyad ''y'' - ''x' '' has error 1/(2''d''). Hence the total relative error on D is strictly less than 1/2. Since D is arbitrary, we have proved condition #2 of chord consistency. QED.

Proof: Consider a dyad D = {''x'', ''y''} consisting of two notes ''x'' and ''y'' that occur in the resulting chord C' = C1 ∪ C2 ∪ … ∪ C''d'' in the equal temperament, where the Ci are copies of the (approximations of) chord 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''. Suppose C''i'', C''i'' + 1, …, C''i'' + ''m'' are separated by dyads D1, D2, …, D''m'' that occur in C. Let ''x' '' be the interval ''x'' + D1 + D2 + … + D''m'', the counterpart of ''x'' in C''i'' + ''m''. Since ''m'' ≤ ''d'' - 1, by consistency to ''d'' copies, each dyad D''j'' have relative error 1/(2''d'') since D''i'' occurs in C. The relative error on D1 + D2 + … + D''m'' compared to its just counterpart is < (''d'' - 1)/(2''d''), and again by assumption of consistency to ''d'' copies, the dyad ''y'' - ''x' '' has error 1/(2''d''). Hence the total relative error on D is strictly less than 1/2. Since D is arbitrary, we have proved condition #2 of chord consistency. QED.

Question: Will the resulting chord always satisfy condition #1 as well?

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 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 copies'' 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 many copies 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 high-span consistency of a small number of intervals.
+Therefore, consistency to many copies 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.
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 '''Theorem:''' A (1/(2''d'') steps of ''n'' ED''k'') threshold 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 will always satisfy condition #2 of chord consistency.
-Proof: Consider a dyad D = {''x'', ''y''} consisting of two notes ''x'' and ''y'' that occur in the resulting chord 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 may 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''. Suppose C<sub>''i''</sub>, C<sub>''i'' + 1</sub>, …, C<sub>''i'' + ''m''</sub> are separated by dyads D<sub>1</sub>, D<sub>2</sub>, …, D<sub>''m''</sub> that occur in C. Let ''x' '' be the interval ''x'' + D<sub>1</sub> + D<sub>2</sub> + … + D<sub>''m''</sub>, the counterpart of ''x'' in C<sub>''i'' + ''m''</sub>. Since ''m'' ≤ ''d'' - 1, by consistency to span ''d'', each dyad D<sub>''j''</sub> have relative error 1/(2''d'') since D<sub>''i''</sub> occurs in C. The relative error on D<sub>1</sub> + D<sub>2</sub> + … + D<sub>''m''</sub> compared to its just counterpart is < (''d'' - 1)/(2''d''), and again by assumption of consistency to span ''d'', the dyad ''y'' - ''x' '' has error 1/(2''d''). Hence the total relative error on D is strictly less than 1/2. Since D is arbitrary, we have proved condition #2 of chord consistency. QED.
+Proof: Consider a dyad D = {''x'', ''y''} consisting of two notes ''x'' and ''y'' that occur in the resulting chord 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 may 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''. Suppose C<sub>''i''</sub>, C<sub>''i'' + 1</sub>, …, C<sub>''i'' + ''m''</sub> are separated by dyads D<sub>1</sub>, D<sub>2</sub>, …, D<sub>''m''</sub> that occur in C. Let ''x' '' be the interval ''x'' + D<sub>1</sub> + D<sub>2</sub> + … + D<sub>''m''</sub>, the counterpart of ''x'' in C<sub>''i'' + ''m''</sub>. Since ''m'' ≤ ''d'' - 1, by consistency to ''d'' copies, each dyad D<sub>''j''</sub> have relative error 1/(2''d'') since D<sub>''i''</sub> occurs in C. The relative error on D<sub>1</sub> + D<sub>2</sub> + … + D<sub>''m''</sub> compared to its just counterpart is < (''d'' - 1)/(2''d''), and again by assumption of consistency to ''d'' copies, the dyad ''y'' - ''x' '' has error 1/(2''d''). Hence the total relative error on D is strictly less than 1/2. Since D is arbitrary, we have proved condition #2 of chord consistency. QED.
 Question: Will the resulting chord always satisfy condition #1 as well?