Our notion of “connection” within a sequence is closely related to distance. In a sense, both distance and connection are levers that register and control the number of breaks in the smooth flow of time. However connection is a more specific measure of continuity: it exclusively measures the number of smooth transitions between durations, whereas distance accounts for both the discontinuity between durations and the extent to which they are discontinuous.  Given this difference, connection will allow us to further differentiate the clip-sequences constructed from an original clip. It will also furnish a table that can be diagonalized to construct sequences with a determined amount of continuity.

Definition:

• Two vertices that have no distance are connectedWe call these pairs of vertices “points of connection”. Connection of a sequence implies
1. coincidence of constituent vertices,
2. duration of constituent edges.

Definition

• The metric “c” is a measure of points of connection in a sequence. This is mapping on the set of vertices of a clip (or a set of clips).

As with the metric ‘d’, for sake of efficiency, we will use edges and sequences as the inputs for this metric.

Example 1:

c(1,1) = 1

c(0,1) = c(1,0) = 0

c({01, 23}) = c(1,2) = 0

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

c(A’) = 0

c(-A ) = 0

c(-A’) = 2

Example 2:

• Graph A has two points of connection: c(A) = 2
• Graph A’ has zero points of connection: c(A’) = 0

Graph of A

Graph of A’

Example 2:

The clip sequences B = {12, 01, 23} and B’ = {21, 10, 32} have the same distance but different connection values.

d(B) = 3      c(B) = 0

d(B’) = 3      c(B’) = 1

Definition:

• A connection table displays the points of connection between elements of two sequences.

Example 3:

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

The diagonalization of this table produced an 18 second sequence A x -A’={32,01,32,12,21,12,10,01,21,12,32,23,21,23,10,23,10,23}

Discussion:

• Connection tables with edge entries (domain) can be derived from the corresponding distance tables. A cell in a distance table with a non-zero value implies that the vertices of the edges that are measured in the cell do not coincide. So, by definition, these vertices are not connected. If the value in the cell is equal to zero, then the vertices coincide; i.e., they have a point of connection. This provides an algorithm for transforming a distance table into a connection table: change the non-zero values to zero and change the zero values to one. However, the converse is not true. If a cell in a connection table is non-zero, then it is uncertain whether the distance between the measured edges is 1 or 2. This ambiguity is compounded in connection tables with clip-sequence entries.

Example 4:

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

Example 5:

A connection table with clip-sequence entries A, A’, -A, -A’. Note that

c(A ) = 2

c(A’) = 0

c(-A) = 0

c(-A’) = 2

c(A,A) = c(01, 12, 23, 01, 12, 23) = c(01,12) + c(12,23) + c(23,01) + c(01,12) + c(12,23) = 4