*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 - 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.