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 scales the resulting value to fall within a range of 0 to 12. (N.B. the lower the stability index, the more stable the set class)
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.
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 scales the resulting value to fall within a range of 0 to 12. (N.B. the lower the stability index, the more stable the set class)
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.
About |
Publications |
|