In this section we introduce a notion of distance that will help us analyze and control discontinuity in clip-sequences. The transformations of an original clip — its permutations, inversion and reflection — alter the visual representation of action and reduce the identification of an action with its outcome. We will use distance as a tool for differentiating sequences constructed from an original clip and for relating clip-sequences from different clips. Eventually distance will be used to define structural equivalence — in effect, synchronicity — among disparate actions.

Definition:

• The metric ‘d’ is a measure of vertical distance between vertices.

This is a mapping on the set of vertices of a clip.

For sake of efficiency, we will often use edges or clip sequences as the inputs of this metric.

Example 1:

d(1,3) = d(3,1) = 2

d({01, 23}) = d(1,2) = 1

d(A) = d({01, 12, 23} = d({01,12}, {12,23}) = d({01,12}) + d({12,23}) = d(1,1) + d(2,2) = 0 + 0 = 0

d(A’) = 4

d(-A) = 4

d(-A’) = 0

Below are graphs of the four sequences of Clip A. Vertical distance is represented by a dashed line.

Graph of A

Graph of A’

Graph of  -A

Graph of -A’

Theorem:  Any clip has distance equal to 0.

Corollary: A = {01, 12, 23} and -A’ = {32, 21, 10} are the only possible sets of edges that comprise Clip A.

• For example, {0 - 1, .5 - 1.5, 2 - 3} are three edges in a clip. When arranged in any sequence, this set of edges will contain six vertices {0, .5, 1, 1.5, 2, 3}. But a clip is a duration and a duration must contains exactly two vertices. So these edges cannot comprise a clip. A similar argument would follow for any other set of three edges, including A’ = {10, 21, 32} and -A = {23, 12, 01}.

Definition:

• A distance table displays the vertical distance of the elements of two sequences.

We call the edges or clips along the top row and the left column the table entries or the factors of the table.We restrict the table entries to the twelve clip-sequence constructed from an original clip.

Example 2:

A distance table for the sequences A = {01, 12, 23} and -A’ = {32, 21, 10}

Example 3:

Below is a table that displays a comprehensive list of the vertical distances between Clip A sequences. Note that a sequence is a set of edges, so

d(A,Ab) = d({01, 12, 23, 01, 12, 23}) = d(01,12) + d(12,23) + d(23, 01) + d(01,12) + d(12,23) = 3

Definition:

• A table diagonalization is a sequencing of the edges or clip-sequences in a table following this order:

Through diagonalization, a distance table produces two pieces of metadata:

• A new sequence of edges or clip-sequences called the product of diagonalization. This product is a binary operation

where S is the set of edge sequences. We will restrict the domain of this binary operation to the twelve clip-sequences constructed from an original clip.

• A total distance.

Example 4:

The diagonalization of the table in Example 2 produces an 18 second sequence:

• A x-A’= {01, 32, 12, 32, 01, 21, 01, 10, 12, 21, 23, 32, 23, 21, 12, 10, 23, 10}

Example 5:

The diagonalization of the table from example 3 produces a 96 second sequence with total distance of 84:

{A,A,A’,A,A,A’,A,-A,A’,A’,-A,A,-A’,A,-A,A’,A’,-A,A,-A’,A’,-A’,-A,-A,-A’,A’,-A’,-A,-A,-A’,-A’,-A’}

Definition:

• Total distance is what we call a global metric: A global metric measures a sum total among edges or sequences; however, it does not account for differences between individual pairs of edges or sequences.
• Distance between edges and points of connection are local metrics, whereas the distance or connection of a clip sequence is a global metric.

Example 6:

The tables below have a total distance of 8, although the distances in the corresponding cells are not equivalent. So in terms of distance, these tables are globally equivalent but locally distinct.

A x-A’= {01, 32, 12, 32, 01, 21, 01, 10, 12, 21, 23, 32, 23, 21, 12, 10, 23, 10}

A x A’ = {01, 10, 12, 10, 01, 21, 01, 32, 12, 21, 23, 10, 23, 21, 12, 32, 23, 32}

Definition:

• A diagonalized distance table displays the distance between elements of two sequences, as well as the distance between the sequences corresponding to the table cells when diagonalized. These distances are displayed in a modified distance table, in which subscript values represents the distance between diagonalized sequences.

Example 7:

For the table below, the diagonalized sequence begins {23, 10, 12, 10…}. The value in the first cell is the measure d(23, 10) = 2. The value in the cell beneath is the measure d(12, 10) = 1. The subscript 2 in the first cell is a measure of the distance between {23, 10} and {12, 10} which equals d(0, 1) = 1.