## Set Theory Information

**Forte Number**(or

**Forte Code**): the pair of numbers, which Allen Forte assigned to the prime form of each pitch-class set of three or more members in

*The Structure of Atonal Music*(1973, ISBN 0300021208), that was subsequently expanded to include all 224 possible set classes within a twelve-tone universe. The first number indicates the number of pitch classes in the set (i.e., the set

**cardinality**) and the second number indicates the set's sequence in Forte's ordering of all pitch-class sets containing that specific cardinality.

**Prime Form**: the best representative of all pitch-class sets that are related by transposition and/or inversion. This can be described as the

**best normal order**transposed to begin with the pitch class "C" or "0" in integer notation. In this database, the prime form algorithm utilized is based on John Rahn's

*Basic Atonal Theory*(1980, ISBN 0028731603), so the following set classes will have a prime form that deviates from Forte's: 5-20, 6-29, 6-31, 7-20, and 8-26. (see Open Music Theory for more details)

**Set-class Membership**: the number of distinct set members contained within a given set class.

**IC Vector**(or

**Interval-class Vector**): a string of six digits wherein each placeholder in the string shows how many of a particular interval class are present within a given set class. < m2/M7, M2/m7, m3/M6, M3/m6, P4/P5, tritone > (see Open Music Theory for more details)

**Stability Index**: a value determined by an algorithm that weights dissonant interval classes more heavily than consonant ones. The interval classes, listed from most consonant to most dissonant, are as follows: IC 5 (P4/P5), IC 4 (M3/m6), IC 3 (m3/M6), IC 6 (tritone), IC 2 (M2/m7), and IC 1 (m2/M7). The current algorithm is set as follows:

- Sum the intervals within the IC Vector with weighted coefficients:

243*(# of IC 1) + 81*(# of IC 2) + 27*(# of IC 6) + 9*(# of IC 3) + 3*(# of IC 4) + (# of IC 5); - Multiply the sum of this weighted IC Vector by the Cardinality of the set class;
- Scale the resulting value to fall within a range of 0 to 12.

**Harmonic Category**: a classification system of harmonies (trichord through enneachord) that conforms to Hindemith's theories of Harmonic Grouping and Harmonic Fluctuation. Categories I, III, and V contain no tritones, while categories II, IV, and VI contain one or more tritones. Categories V and VI have an ambiguous sense of root. Relative chord tension generally increases from I to VI.

**M/MI**: a transformation using the multiplicative operators

**M**(i.e., multiplication by 5) and its inversion

**MI**(i.e., multiplication by 7) that potentially yield a different set class. In general, when elements of a set class are multiplied by 5 or 7, compact set classes expand to become set classes with a more open intervallic structure (and vice versa) or the set class will be invariant.

**Z-relation**(

**Zygotic Relationship**or

**Isomeric Relation**): a relationship between two pitch-class sets in which the two sets have the same intervallic content (i.e., the same interval-class vector) but they are not related by transposition and/or inversion (i.e., they are members of different set classes).

**Degree of Symmetry**: an integer value that indicates the number of invariances within a set class under

**transposition**,

**inversion**,

**M**, and

**MI**.

**Invariance Vector**: a string of eight digits, from Robert Morris's

*Composition With Pitch-Classes: A Theory of Compositional Design*(1987), wherein each placeholder shows set invariance for a set class under

**transposition**,

**inversion**,

**M**,

**MI**and invariance within the set's literal complement under

**transposition**,

**inversion**,

**M**,

**MI**.