The following study is based on a general definition of a string of any objects such as that of Eric Weisstein: “A string of length n on an alphabet l of m objects is an arrangement of n not necessarily distinct symbols from l . There are mn such distinct strings.” We will begin by defining the alphabet as a set of intervals between pitches in order to introduce six string transformations. We will then begin to redefine the alphabet in a series of studies taking us from a traditional pitch-centric viewpoint toward basic digital manipulations of electroacoustic source material.

10.1 Interval Strings

A generalized interval string a is an ordered n-tuple of integers which sum to m = km:

Here n is the length of the string, m is the chromatic span of the string, m is its modulus, and k is an integer. When k = 1 (i.e., m = m), the string is said to be “compact,” or in its “minimal configuration.” We choose m > 1 to avoid the trivial case (mod1) in the present context.

Interval strings are circular in that element a1 is considered the successor of an . Thus an interval string can be represented as a polygon inscribed in a circle such that its n points coincide with n of the m equally spaced sites around the circle's circumference. Figure 10.1a shows the interval string 3, 4, 5 represented as a triangle inscribed in a 12-circle. Note that the sites on the circle's circumference are not (yet) labeled, so the rotational orientation of the triangle is undefined in strictly intervallic terms. Note also that the 12 equal intervals between contiguous sites around the circumference represent the chromatic span, which should not be confused with the modulus. The modulus in this case might be 12 (figure 10.1a), 6 (figure 10.1b), 4 (figure 10.1c), 3 (figure 10.1d), 2 or, trivially, 1 (corresponding to the values 1,2,3,4,6, or 12 for k ). This will be further clarified when pitch structures are introduced in the next section.

Repetitions of elements are allowed, and therefore any given interval string may be treated as a multiset. Thus, 2, 2, 3, 4 and 1,1, 2,1, 2, 6, 2 are valid interval strings.