### Introduction to Post-Functional Music Analysis: Post-Functional Theory Terminology

Some important terms and concepts in atonal theory that are not explicitly mentioned in the Introduction to Post-Functional Music Analysis are briefly defined in the table below. The chart is given here for reference when one or more of these terms is mentioned elsewhere in the Introduction to Atonal Music or in other writings. However, some of the terms specifically refer to set relations that appear in the Set Analyzer Applet (subset/superset, literal/abstract complement, Z-relation).

TermDefinition
Ordered Pitch IntervalThe distance between two pitches, measured by the number of semitones between them. Ordered pitch intervals are determined by subtracting the pitch number (use MIDI note numbers (C4 = 60) or choose your own value for C4) of the first pitch from the pitch number of the second pitch. This produces both positive and negative numbers, indicating ascending and descending intervals, respectively.
Unordered Pitch IntervalThe absolute value of the ordered pitch interval, used to determine absolute size of an interval without regard to the order in which the pitches are presented.
MOD-12The operation of dividing a number by 12 and taking the remainder as the result. For example, 23 mod 12 = 11. (23 ÷ 12 = 1 with a remainder of 11.) In this way, any number can be reduced to between 0 and 11. The mod-12 group is a mathematical group where a mod-12 operation is imposed on the result of every arithmetic operation (it may also be necessary to add 12 to negative numbers until the number is positive between 0 and 11). This group is useful as an abstraction of the properties of pitches in 12-tone equal temperament with the assumption of octave equivalence. Addition and subtraction mod 12 are equivalent to transposition; subtraction from 12 is equivalent to inversion; multiplication mod 12 of the ordered pitch class interval by consecutive mod 12 group members gives interval series (circle of fifths, circle of fourths, augmented triad, diminished seventh chord, wholetone scale, chromatic scale, etc.) Mod-12 complements are the result of subtraction from twelve, such that complement-related pairs are 0/0, 1/11, 2/10, 3/9, 4/8, 5/7, and 6/6.
Ordered PC IntervalThe distance between two pitch classes, expressed as an integer from 0 to 11. To determine an ordered pc interval, subtract the pc number of the first pc from the number of the second pc, then add 12 (to eliminate negative numbers) and reduce the number mod-12 (divide by 12 and take the remainder as the result). It is easiest to envision ordered pc intervals as the distance from one pc to the next around a clock face (with 0 at the top, rather than 12) always measured clockwise.
Unordered PC Interval (Interval Class)The absolute value of the shortest distance between two pitch classes on the clock face. To obtain the IC of an ordered pc interval, take any ordered pc interval larger than 6 and reduce it to its mod-12 complement by subtracting it from 12. The concept of interval class assumes interval complement equivalence. For example, a perfect fourth is equivalent in sound to its inversion, a perfect fifth. Thus, all sixths can be seen as inverted thirds, and all sevenths can be seen as inverted seconds. The expression "IC3", then, subsumes all types of minor third, including ascending and descending minor thirds, major sixths, minor tenths, etc.
INT1A list of the successive ordered pc intervals between the pcs of a given ordered pc set. The INT1 for set B,D,E,A,G#,F is [325E9(6)]. Technically, the "wrapping around" interval between the last pc and the first is not part of the INT1, but its inclusion can often be useful (the INT1 can be used to determine normal order and prime form).
IC Vector (or Interval Vector)An array of six integers representing the ic (interval class) content of a chord, where the first digit indicates the number instances of IC1 in the set, the second, IC2, third, IC3, fourth, IC4, fifth, IC5, and sixth, IC6. For example, <001110> is the ic vector for a major triad, showing that it contains zero IC1 (semitones), zero IC2 (wholetones), one IC3 (minor 3rd), one IC4 (major 3rd), one IC5 (perfect 4th), and zero IC6 (tritones). Identity of interval vectors (when two sets have the same IC vector) is the determinant Forte used for his equivalence relations between pc sets (transposition, inversion, Z-relation).
Literal ComplementThe relationship between two pc sets such that one set contains all of the pitch classes that are not contained in the other. Such complement-related sets always form aggregates when combined.
Abstract ComplementThe relationship between two pc sets such that one set, when transposed and/or inverted, contains all of the pitch classes that are not contained in the other. With the exception of 6-note sets (see Z-Relation), any complement-related sets (literal or abstract) will have the same ordinal numbers on the Forte list, and their cardinalities will be mod-12 complements.
Z-RelationA pair of pc sets that have the same ic vector (interval vector) but are not reducible to the same prime form. In the case of 6-note sets (hexachords), any Z-related sets are also abstract complements. All 6-note sets that do not have a Z-related set are self-complementary (see Hexachordal Combinatoriality).
Hexachordal CombinatorialityUse of a 12-tone row where the first hexachord is self-complementary. (see Z-Relation) Because the first hexachord is the literal complement of one of its own transpositions or inversions, the prime form of the row can be combined with at least one other row form to create two aggregates. For a more complete description and an example, see Advanced Techniques: Combinatoriality
Subset (or Inclusion Relation)A relationship between two sets where all of the pcs in the 'subset', when it is transposed and/or inverted, can be found in the 'superset'. A literal subset relation requires no transposition or inversion. Any set is its own literal subset, but this is considered a trivial subset relation.
SupersetA relationship between two sets where some or all of the pcs in the 'superset' make up the entire 'subset', when the subset is transposed and/or inverted. A literal superset relation requires no transposition or inversion.
SymmetryAny temporal and/or pitch structure that displays some kind of symmetrical structuring (inversional, retrograde, retrograde-inversional (rotational), etc.). In pc sets, a collection that is symmetrical has at least one 'axis' of symmetry, and both the set and its inversion reduce to the same normal order (when transposed to zero). (Any set and its inversion reduce to the same prime form, but only a symmetrical set and its inversion reduce to the same normal form.)
AggregateAny collection of all 12 pitch classes, especially one that has a different ordering from the main tone-row used in a composition (or any of its permutations).
CardinalityThe number of distinct pcs in a set (discounting repetitions). This number is used as the index (or first number) in the Forte set-class numbering system. Thus, the set G,B,C#,F,G has 4 distinct pcs (cardinality 4), and has Forte number 4-25 (0268).
Ordinality1. The second number in the Forte number of a set class, indicating its position in the list.
2. The order number (beginning with 0), or position, of a pitch in an ordered set or tone row. Thus the ordinality of G in the ordered set C#,F,G,A# is 2.

All explanations given here are based on my own understanding of atonal theory and its terminology. Certain variations on the terminology and its use may exist among atonal theorists, but the mathematical concepts of the mod-12 group and the identities of the pitch class sets remain the same. Please with any questions, suggestions, comments, or corrections with regard to my definitions.