Chapter 9 Sets and Tables
A set is a collection of objects. In SAL, we can represent finite sets as lists.
Each object in a set is called an element or a member. In algorithmic compo-sition, sets may be manipulated to achieve a compositional result.
9.1 Introduction to Set Theory
The analysis of atonal music using mathematical set theory was codi-fied by Allen Forte in his landmark book, The Structure of Atonal Music (1973). His theory of atonal music develops a comprehensive framework for the organization of collections of pitches referred to as pitch class sets. We will introduce the concept of pitch class, and then describe Forte’s concept of pitch class set.
Since SAL uses braces {} to denote lists and mathematicians use braces to denote sets, we need to be careful with notation. Here, braces with comma-separated items, all in this font (Times New Ro-man), are used to denote sets. Lists are denoted by parentheses () and comma-separated items, also in this font. If we use SAL’s notation for lists, it will always be with the Lucinda Sans font and without commas:
{this, is, a, set, of, words}
(this, is, a, list, of, words) {this is a list of words too}
A pitch class is a set of pitches, one from each octave, that have the same note names. Informally, we are just saying that the pitch classes are all the A’s, the B-flat’s, the B’s, etc. There are 12 pitch classes. Mathematicians are not happy with informal definitions, so
“all the A’s” is represented more precisely by a set. For example, {A0, A1, A2, …} is a pitch class (here, A4 denotes the A above mid-dle C, A3 is an octave lower, etc.), and {B0, B1, B2, …} is another. In mathematics, these pitch classes are examples of equivalence classes. Members of the classes are equivalent in the sense that they have the same note names. There are other equivalence relations that induce other equivalence classes; for example, the “white keys” and
“black keys” are equivalence classes. We say that equivalence classes form a partition over the set of all pitches, dividing the pitches into subsets.
9.1 Introduction to Set Theory 109
Pitch classes could be named “A,” “B-flat,” “B,” etc., but Forte names them with integers in the range 0 to 11. Pitch class assignment in twelve-tone equal temperament is as follows: C (or its enharmonic equivalent) is 0, C-sharp (or its enharmonic equivalent) is 1, D (or its enharmonic equivalent) is 2, and so on. Notice that in terms of MIDI pitches (key numbers), the pitch class is just the remainder of divid-ing the key number by 12. An aside: the integers are also a set, and the equivalence relation “has the same remainder after division by 12” forms equivalence classes over the integers that correspond to musical pitch classes. We represent pitches as integers rather than symbols to facilitate this kind of pitch manipulation and computa-tion.
A pitch class set, or pc set, is a collection of pitch classes. Thus, a pc set is really a set of sets, but it is less confusing to think of a pc set as all the note names that occur in a collection of pitches. Intuitively, this amounts to removing all the octave names and octave doublings from a collection of pitches. What’s left is a set of pitch classes. How many pitch class sets are there? Consider that there are 12 pitch classes, and each pitch class offers two choices: it can be in the set or not in the set. Thus, there are 2×2×2×2×2×2×2×2×2×2×2×2, or 212, or 4096 possible sets (including the empty set). In keeping with Forte’s naming scheme, we represent pitch class sets as sets of the integers from 0 to 11.
Forte names each pc set based on its prime form. To understand the prime form, think of the pitch classes 0-11 organized around a circle. The pitch class structure is circular because a minor second above 11 (B) is 0 (C). To generate the prime form from a pc set, we list the pitch classes in clockwise order, starting at any pitch class.
For example, the D-minor triad {D, F, A} or {2, 5, 9} in clockwise order generates the lists (2, 5, 9), (5, 9, 2), and (9, 2, 5). Next, we transpose the lists to begin at zero. Remember that these numbers represent pitch classes without octaves, so 1 (D-flat) transposed down by 3 (a minor third) is 10 (B-flat), not −2 (also B-flat, but not in the range 0-11). After transposition, our lists are (0, 3, 7), (0, 4, 9), and (0, 5, 8). Next, we also generate the inversions of these lists by subtracting each pitch class from 12. The effect is to exchange de-scending intervals for ade-scending ones. The results in our example are (0, 9, 5), (0, 8, 3), (0, 7, 4), and these are sorted into increasing order to get (0, 5, 9), (0, 3, 8), (0, 4, 7). Finally, the prime form is the list from among the original and inverted lists that is “most compact,”
meaning that the last element is smallest. In this case, there is a tie between (0, 3, 7) and (0, 4, 7), so we break the tie by taking the list with the smallest next-to-last element, (0, 3, 7).
Notice that two pc sets may have the same form. For example, all transpositions of minor triads have the prime form (0, 3, 7), and be-cause inversions are considered equivalent, all major triads have the same prime form as all minor triads. Thus, prime forms are equiva-lence classes of pitch class sets: two pitch class sets are considered equivalent if they have the same prime form. Forte names prime forms with two numbers. The first number is the length of the prime form, which is also the number of elements in the corresponding pc sets. (The number of elements in a set is called the cardinality of the set.) The second part is derived by listing all prime forms from most to least compact. For example, set 4-1 is comprised of pitch classes (0 1 2 3). The 4 in the pc set name represents the cardinality of the set. The 1 is the unique integer identifier associated with that set.
Only pc-sets with a succession of three minor seconds will have the Forte number 4-1.
The prime form is only one of many equivalence relations that can partition the 4096 pitch class sets. For example, if Forte had not considered inversions to be equivalent, then major and minor triads would not have the same prime form, and a different kind of equiva-lence relation would be obtained. Another interesting equivaequiva-lence relation is obtained from the interval vector, which is a tally of all pair-wise intervals in a pitch class set. For example, the pitch class set {G, B, D, F} or {7, 11, 2, 5} has the following pairs: (7, 11), (7, 2), (7, 5), (11, 2), (11, 5), (2, 5). The corresponding intervals are 4, 5, 2, 3, 6, 3. Note that “interval” is defined as the smallest number of half steps from one pitch class to the other without regard for direc-tion. For example, the interval from 7 to 2 is 5 (not −5), and the inter-val from 11 to 2 (B to D) is 3 because D lies 3 half steps above B.
The interval vector is an ordered list of the number of intervals of size 1, size 2, size 3, and so on up to size 6, the largest interval. This example contains one interval of size 2, two intervals of size 3, and one each of sizes 4, 5, and 6, so the interval vector is 012111. This is the interval vector of all dominant seventh and half diminished chords, corresponding exactly to the prime form 4-27.
Not all prime forms have a unique interval vector. For example, the prime form (0, 1, 3, 5, 6), number 5-12, and prime form (0, 1, 2, 4, 7), number 5-36, have the same interval vector: 222121. Forte calls prime forms with the same interval vector “Z-related sets,” or
“Z correspondents,” and he adds the letter Z to the designation, so these prime forms are actually named 5-Z12 and 5-Z36. You will see the letter Z in the names of some of the prime forms used in exam-ples below.