Set Theory Glossary   

 copyright © 1997, rev 2002 by Larry J Solomon  

This glossary contains terms that apply to the two different branches of set theory in music. Those that apply to harmonic theory are listed with an [h] and those that apply to serial theory are listed with an [s]. When no such designation occurs the term applies to both branches.

abdo (absolute-do). 1. [h,s] a system of assigning pitch numbers to a set such that the numbers of a set transposition are the same as the absolute pins (see pin); e.g., DFA would be 279. 2.[s] In abdo Ro and RIo start with the pin that is the last pin of Po. Io starts with the same pin as Po. (see also reldo)

absolute-do. see abdo.

-ad. a suffix that, except for in "triad", has the same meaning as -chord. A triad is 3 pcs arranged in thirds.

aggregate. The collection of all 12 pcs.

all-combinatorial. [s](Babbitt) a set in which any of its transformations (P, I, R, RI and their transpositions), may occur simultaneously with any other transformation without duplicating pitch-classes (pc) before all twelve pc have occurred.

all-interval chord or all-interval set. [h] a chord that contains one of each interval class; e.g., 0146, set name 4-Z15.

all-interval row. [s](a) a twelve tone row that contains all eleven directed intervals (di); e.g., in Berg's Lyric Suite, F E C A G D G# C# D# F# A# B, which contains one of each interval and no duplications of intervals.

array. an arrangement of a series in quantitative values, e.g., as a number series or alphabetical order, etc.

atonality. A misnomer for music without a tonic. The proper term is pantonality.

axis. a point or line used as a divider in a symmetric operation.

basic interval pattern. [h] (Forte) the normal order of an ic set.

best normal order. [h] (Forte), abbrev bno. the most compact of two normal orders, chosen from those of a set and its inverse (see prime form).

cardinality cardinal number. [h] the number of pcs in a pc set, e.g., C,E,G,C,G is a set of cardinality 3 since there are only three different pitch classes.

chord. [h] three or more pcs considered simultaneously or as an unordered set. A chord is a nonlinear pc set with a minimum cardinality of 3 (pc card. 3).

-chord. a suffix used to designate a specific number of pcs considered to be a structural unit; e.g., trichord, hexachord.

chord structure. [h] (Solomon) a cyclically ordered dic set contained in a chord that cycles to the octave; the prime structure of a chord may be determined by first placing it in prime form; e.g., a major chord, C E G, or 047, contains the DIC set 435 (semitones) as a triad, cycling back to the octave (C<4>E<3>G<5>C.). This is its prime structure. In first inversion the chord structure cycles to 354 (E G C E), and in second inversion it is 543 (G C E G). Notice that the complete chord structure fills an octave. It is the chord structure that identifies and defines a major chord (or any other type of chord). The prime structure is unique to a chord type and, therefore, may be used to identify any chord (as opposed to the usual prime form); thus, 435 is the major chord and 345 is the minor chord. See also interval string.

closure property. [h] (Forte) property in which every member of a set complex is a subset or superset of every other member.

combinatorial, combinatoriality. [s] (Babbitt) the special property of combining row forms simultaneously, without duplications of pc (before all twelve pcs have occurred). The primary type of combinatoriality is hexachordal; e.g., the first six notes of P0 of any twelve tone row combined with the first six notes of R0 are combinatorial. (The same is true of the last hexachords of P0 and R0). It follows that one harmonic segment of a row may be mapped onto another by operations of transposition, R, I, or RI. The primary type of combinatoriality is hexachordal; i.e., the first six notes of Po are always different from the first six notes of Ro (abdo) and are, therefore, combinatorial. Retrograde combinatoriality was considered trivial by Babbitt and others. Focus of serial theory has been on other types, such as Inversional and Retrograde-Inversional combinatoriality. Here is an example of Inversional-hexachordal combinatoriality.

complement. (1) all pcs not in a given set. (2) the interval that when added to a given interval will complete an octave (interval inversion).

cyclic permutation. same as rotation; changing an ordered pc set by starting with the second, third, etc. pc and using the whole set as in a circle; e.g., C E G may be rotated to E G C or G C E, but not G E C (out of order).

derived set. [s] a tone row that is constructed by symmetry operations upon a source set; e.g., the trichord G, A#, B can be used to create F E C# by retrograde transposition (r6), and C, A, G# by inversion (i4), and D D# F# by retrograde inversion (ri7), and then combined to form the complete derived set: G A# B F E C# C A G# D D# F#.

di. [s] abbreviation for directed interval.

dic. [s] abbreviation for directed-interval class.

di or directed interval. [s](Babbitt) The distance between two linear pcs (see linear set) always measured in an upward direction; e.g., A to G is 10 semitones, even if the G is below the A.

dic or directed-interval class. [s](Solomon) In an ordered set, the distance between any two adjacent pcs measured as the shortest distance (in the number of semitones from 0 to 6, which can be positive or negative). The dic from C to B is -1 as opposed to 11 for the directed interval (di). Notice that it does not matter whether the B is "above" or "below" the C since they are pitch classes. The dic from C to F is 5 and from C to G it is -5.

dcv or directed-class vector. [s](Solomon) A series of di representing the distance between ordered pins as ics 1to 6 with a minus sign for a downward direction when necessary.

div or directed-interval vector. [s] (Solomon) A series of directed intervals representing the di between the pcs of an linear (ordered) set.

dyad. (1) a pc set whose cardinality is 2. (2) same as interval.

hexadecimal. [abbrev hex] a number system of base sixteen (16), often used in computer programming. A=10, B=11, C=12, D=13, E=14, F=15, 10=16. Hexadecimal notation in set theory provides a compact single digit system for representing pc sets; e.g., 2, 5, 7, 9, 11 in hexidecimal is 2579B, which is more concise and does not need separators.

I. abbreviation for inversion; e.g., Io is the inversion of a row form at zero transposition, having the same initial pc as Po. Lower case i may be used for subset operations.

ic or IC. abbreviation for interval class. the distance between two pcs measured as the shortest distance in semitones; e.g., C G is ic5, G E is ic3. In set theory the word "interval" is often informally used to mean interval class, the latter of which is of greater generality. Notice that there is no ic larger than a tritone (ic6) since larger intervals invert to smaller ones.

inclusion relation. Two sets so related that one is included in the other (also called the subset relation).

index number. The transposition number, in semitones, above a reference pc. P5 would be a transposition up 5 semitones from P0.

intersection. elements in common; e.g., E G B D and G D F have an intersection of two pcs: G D.

interval. (1) the distance between two pitches or notes; in set theory this is measured in semitones; (2) sometimes used for interval class (see ic), (3) a pc set whose cardinality is 2.

interval class (ic). (Babbitt) The distance between two pitch classes, measured by the shortest distance. C to G may be the interval of 7, but its interval class is 5. Thus, the largest ic is the tritone (6).

interval content. see interval vector.

interval string. [h] a series of intervals completing an octave, usually expressed in semitones; e.g., 435 (meaning 4 semitones, then 3, then 5) is the interval string for a major chord such as CEGC. C to E is 4 semitones, E to G is 3, and G back to C is 5.

interval vector. [h] (Forte) abbrev iv. an array of six digits representing the ic content of a chord, where the first digit indicates the number of ic 1, the second= ic 2, third= ic 3, fourth= ic 4, fifth= ic 5, and sixth= ic 6. E.g., 001110 is the iv for a major chord, showing that it contains zero semitones (ic 1), no ic 2 (wholetones), one ic 3 (=minor 3rd), one ic 4 (=major 3rd), one ic 5 (=perfect 4th) and no tritones (ic 6).

invariant. anything remaining unchanged after an oparation; e.g., a div remains invariant after a transposition.

inversion or inverse. (1) the mirroring of the intervals of a pc set; also: "inversion". (2) pins or div which are complements of 12.

iv. [h] abbreviation for interval vector.

linear set. [s] A set ordered in time, as a temporal array, e.g., a tone row.

matrix. [s] a two-dimensional array of numbers, which in music, often represents the ordered set of a twelve-tone row in a twelve by twelve square arranged in such a way that all 48 transformations may be read in one direction or another.

mirroring. reflecting a set around an axis of time or pitch.

mirror or mirror set. a pc set that is symmetric by reflection around a pc axis. A minor-seventh chord is a mirror because if its intervals are projected in the opposite direction the same chord results. In Solomon's Table of Set Classes all mirror sets are indicated with an asterisk next to the set name.

modulo 12 (mod12). An arithmetic system nearly identical to that of a clock, where 13=1, 14=2 etc. However, in modulo 12 the number 12=0. If we want to know what 2 hours past 11 is (11+2), we say it is one o'clock (1). Thus, in mod12, 11+2=1, and there is no number greater than 11.

nexus set. [h] a set that is used as a reference for a set complex.

nonlinear set. [h] A set in which linear order is irrelevant, as in chords; see also: unordered set.

normal order. [h](Howe) a cyclic permutation of the pitch numbers of a pc set arranged in ascending order as compactly as possible with respect to the smallest pitch number. Each pc is designated as a pitch number in the absolute system. The normal order for a G major chord would be 7B2. [See also prime form and best normal order].

octave displacement. the presentation of a pitch at a different octave register.

ordered set. [s](1) a pc set arranged in a temporal order. (2) a set, normally a pc set, that is arranged in an array. Note that an ordered set may be arranged in a non-temporal dimension, e.g., an alphabetical pitch order (e.g., A# B C# D). Such an ordered set is not necessarily a pitch (pc) set, e.g., it may be a set of durational values, hence rhythm.

ordering. [s](1) a set placed in a temporal order. (2) a set placed into some logical order; e.g., the set played as D B G F may be placed into different logical orderings: 1. alphabetically as B D F G, 2. temporally as it is played, D B G F, 3. tertially as G B D F, or in normal order as 0 3 6 8.

order number. [s] a number assigned to a pc in a linear set to indicate its order in the series. The first order number is zero (0).

order-number, pitch-number couple. [s] two numbers that identify a pc in a series by its order number, beginning with zero, and the di above the first pc of the prime set; e.g., C, D#, B, G would be (0,0) (1,3) (2,11) (3,7).

ordinal number. a catalog number following a cardinal number in the Table of Set Classes; e.g., 6-34 means that 6 is the cardinality and 34 is the catalog number, which is normally in alphanumeric order.

P. abbreviation for the prime or prime set. Po indicates the prime at zero transposition, which is normally assigned to the first occurrence of a row.

pantonality. A musical organization where all pcs are treated with a degree of equanimity and which diverges from traditional means of tonal organization.

parent set. a set that includes all other given sets.

pc. abbreviation for pitch class, or pitch classes= pcs. (Babbitt) a pitch and all of its octave equivalents, including enharmonic equivalents.

pc set. abbreviation for pitch class set.

permutation. any possible ordering of a set; e.g., the set G, E, C has 3 factorial (3!=6) different permutations: 1. C E G , 2. E G C, 3. G C E , 4. G E C, 5. C G E, or 6. E C G.

pin. see pitch number

pitch. (a) In set theory (unfortunately?) the same as pc. (b) (Solomon) the predominant frequency in a sound.

pitch class (pc). (Babbitt) All pitches with the same name plus their enharmonic equivalents; e.g. all C#s make up a single pitch class. But, Db and Bx are also in the same class.

pitch-class number. see pitch number

pitch-class set. A group of pitch classes; compare to chord.

pitch number (pin). (Solomon) Each pc can be represented by a number from 0 to 11 in the twelve-tone system.

C C# D D# E F F# G G# A Bb B
0 1 2 3 4 5 6 7 8 9 10 11
0 1 2 3 4 5 6 7 8 9 A B

The first row of numbers in this table indicates the decimal notation for each pc. The last row shows the same pcs in hexadecimal (base 16) notation. The table shows what is known as absolute-do (abdo) notation, where C is always zero (0). In the relative-do (reldo) notation, any pc may be set to zero, usually the first. Thus, FAC is represented as 590 in abdo, but as 047 in reldo.

pitch set. (a) a set of pitches; (b) sometimes used to mean pc set.

prime. see prime set.

prime structure. [h] (Solomon) the dic set of the prime form of a chord, or pc set, that cycles to the octave. To find the prime structure of a dominant-seventh chord, first place it in its prime form, 0368. Cycling to the octave it becomes 03680. The intervals between these would be: 3324, which is the prime structure for the chord.

prime form. [s] (1) a prime set or the presumed original order of a tone row. (2) (Forte) the most compact form chosen from a set and its inverse. (3) (Solomon) a transposition of the normal order of a set such that the initial pitch number (relative) is zero. Note the conflict with Forte's meaning. The prime form of a pc set is used to catalog the set in the Table of Set Classes.

prime set. [s] a row taken as a reference, symbolized P; compare to prime form.

primordial-set structure. [s](Solomon) abbrev ps. The absolute values of the dics, or directed-interval classes, whose order is maintained either forward or backward (retrograde) in all 48 transformations of twelve tone rows. For example, in Milton Babbitt's Composition for Guitar (1984), the row is C G G# F# D C# A# D# A B E F, whose di analysis is 71A8B956251. The dic analysis is -5 1 -2 -4 -1 -3 5 6 2 5 1. From the latter we derive the ps as the absolute values: 51241356251 and its retrograde, 152465314215. See twelve-tone row.

quadrate. [s] a 90 degree transformation or set form of a linear set. Traditionally, the set forms were restricted to P, R, I, and RI. By switching the time and pitch axes an additional 48 transformations, the quadrate transformations, are added (see Solomon, "New Symmetric Transformations", Perspectives of New Music, 1973). Quadrates are restricted to rows where the number of pcs equal the number of ordinal numbers, such as in twelve-tone rows. They may be formed by the following formulae, where p=pitch number or pin, t=(temporal) order number, and n=number of pcs:

quartal. [h] anything (usually a chord) that can be spelled in fourths; e.g., CFG can be arranged GCF.

quintal. [h] an inverted quartal chord.

R. [s] abbreviation for retrograde. R0 indicates the retrograde at zero transposition, e.g., starting with the same pc as the prime set.

R relation. [h](Solomon) a maximal similarity relation in which two sets are equal excepting one pc pair that are a semitone from a match, and their interval vectors have a minimum of interval correspondence, T, where T equals the number of ic common to both sets. In order to satisfy R: T must be equal or greater than XY/8 where X=sum of all ic in either set and Y=the cardinality. This is a statistical relation based upon a perception model in which R similarity is increasingly more difficult to hear as the sets get larger. Sets must be smaller than cardinality 8 to have the R relation. See the Set Theory Primer for examples.

reflection. (Solomon) the result of turning a figure about an axis; e.g., to invert a theme it is turned around a pitch axis. To retrograde a theme it is turned around a time axis.

relative-do. see reldo.

reldo (relative-do). 1. a system of assigning pitch numbers to a set such that the first pin of the prime is always zero (0). 2. [s] In reldo Io, Ro, RIo start with a pin of zero (0). (See also abdo.)

retrograde. [s] the reverse order of a prime set, symbolized R.

retrograde inversion (abbrev RI). [s] the retrograde of the inversion of a row form.

rhythmic series. [s] a durational series created by mapping pc numbers in an row into a series of attack points in time, taking some note value as a temporal measure; i.e., a row is equated to a series of durations. For example, if the di set were 5 4 2 3 and the reference time value is taken to be x (where x might be an eighth note), then the durational series would also consist of a pattern of 5 4 2 3; i.e., the second attack in a rhythmic series would come 5x after the initial attack. The third event would come 4x later, and the fourth would come 2x after the third, and the final event in the series will come 3x later.

RI. [s] abbreviation for retrograde-inversion.

rotation. (Solomon) starting a set with a different pin and cycling it; e.g., CEGB rotates to EGBC, GBCE, or BCEG.

row. [s] (1) A group of pcs (usually the 12 chromatic pcs) placed in a particular order to be so used in a composition. (2) a linear ordering; syn. series. Normally a row is thought of as a twelve-tone row, i.e., a linear ordering of the twelve pitch classes. However, a row may be a linear ordering of less or more than twelve notes, or it may be an ordering of rhythmic values, etc.

secondary set. [s] (Babbitt) a twelve-tone set formed from the first hexachord of one transformation appended to either hexachord of another transformation where the two are combinatorial.

secundal. [h] anything that can be spelled or arranged in seconds; e.g. a series of sevenths is secundal, since they can be rearranged in seconds.

segment. [s] a contiguous part of a series.

segmental invariance. [s] a contiguous part of a linear set that remains unchanged after a transformation.

semi-combinatorial set. [s](Babbitt) a twelve-tone row so constructed that one its transformations other than the retrograde can be transposed so that its first six pcs (unordered) are equivalent to the last six of the original set.

serial. [s] (1) organization of music by means of linear sets. (2) Music organized by means of pc orderings.

series. [s] a linear ordering; syn. row.

set. (1) any group of things, (2) (Babbitt) an ordering of the twelve tones, (3) a group od pcs.

set complex. [h] (Forte) a group of sets having some common property.

set complex K. [h] (Forte) All sets that are in a subset relation with a given set or its complement.

set complex Kh. [h] (Forte) All sets that are in a subset relation with a given set and its complement.

set complex relation. [h] (Forte) property of a group of sets possessing the subset, or inclusion, relation.

set form. [s] any of the four standard forms of an ordered set, prime, retrograde, inversion, or retrograde inversion. A new set of forms are the Quadrates.

set list. [h] The list of the possible unordered sets in the twelve-tone system. Allen Forte's original list contains 208 sets in The Structure of Atonal Music (Yale, 1973), Appendix I. Larry Solomon's revised list contains all 352 sets in Interface (V11/2, 1982) and the Music Analysis System, including the 0,1,2,10, 11, and 12 note chords.

set name. [h] (Forte) two numbers adjoined but separated by a dash, the first representing cardinality, and the second representing their order of compactness (normal order) in a numerical array containing all possible sets within that cardinality; e.g., 3-11 is a three-note set that is number 11 in compactness in the list of all three-note sets. Solomon adds the inverse sets to Forte's list with an appended B to each set name.

set theory.(ST) in music, a study primarily concerned with describing the relationships between pc sets, either ordered or unordered. Unordered (nonlinear) ST tends to focus on harmony, while ordered (linear) ST tends to focus on melodies or lines (series).

similarity relations. [h] (Forte) ways in which two non-equivalent sets of equal cardinality may be compared. Forte describes three basic types of "maximum" similarity relations, but regards the first type (Rp, sets having a common subset of cardinality C-1)) to be less significant than the other two. Both the latter are defined by the same interval content (iv) in four out of the six iv positions. The remaining two numbers may (Forte's R1) or may not (Forte's R2) be interchanged, thus their distinction. In the Music Analysis System R1=X and R2=O. see also the R relation.

simultaneity. all pitches sounding at one moment.

source set. [s] a subset of a tone row that is used to generate the entire row through symmetric transformation, e.g., the G A# B example under derived set.

subset. a set that is contained within a larger set.

subset relation. the property that two sets have when one is contained in the other. One set is either a subset or a superset. The sets must be of differing cardinalities. syn. inclusion relation.

superset. a set that contains other set/s and whose cardinalty is greater.

symmetry. (Solomon) a congruence resulting from an operation of translation, reflection, or rotation. see Solomon's dissertation: Symmetry as a Compositional Determinant (1973, 2002).

symmetrical set. see mirror set.

time-point set. [s] a rhythmic set of note attacks that is determined by metric placement and a some reference note value (such as a ). The notes are mapped into their respective positions in the meter, which is divided up according to the reference note value. For example, if an eighth-note is taken as the reference, then a meter such as 3/4 divides into three groups of two eighth notes apiece; the meter is then accordingly divided and can be represented 0 1 2 3 4 5. A series such as 0 3 5 2 translates into this time-point set with the first note occurring on the first beat. The next note would occur on the second half of the second beat, the third note would occur on the last eighth of the measure, and the following note would occur on the second beat of the next measure.

tonal. Adjective for tonality.

tonality. Musical organization around a tonic or a pc hierarchy; e.g., I   IV  V  I  etc.

tone row. [s] a fixed, linear ordering of pcs used as an organizational feature of a serial composition. A tone row is normally a twelve-tone row.

tonic. the predominant pc.

transformation. [s] (Babbitt) in a row, P, R, I or RI and any of their transpositions. A transformation of a row is normally indicated with the transposition number, e.g., R7. There are 48 traditional transformations. A new set of transformations are called quadrates, introduced by Solomon.

translation. (Solomon) moving a figure through the dimension of time, pitch, space, etc; e.g., a recurring Alberti bass figure is a time translation. Transposition is a translation in pitch.

transposition number. [s] the measure of transposition indicated as the di from a reference, usually Po (see transformation). P5 would be P transposed up 5 semitones or down 7 from Po.

twelve-tone row. [s] a tone row that uses all twelve pcs in a specific, fixed linear order. IIt is used to generate a twelve-tone composition through the use of transformations and other variation techniques. It has been claimed that the primary law governing the use of this row is that the order of the pcs is maintained. However, each such row has four set forms and 48 transformations, and these do not sustain the original pc order; thus, it is not possible to maintain the original pc order in row composition. It has also been claimed that the interval order is maintained, but this is not so either since any transformation may occur with differing melodic contours and registrations; besides, all the set forms cannot and do not have the same interval order. Therefore, interval order cannot be claimed as the basic principal of row composition. One will find the same problems with claims for constancy of ic ordering or di ordering. The only constant principal that can be and is held in row composition is that of the absolute values of the dics, or directed interval classes, whose order is maintained either forward or backward (retrograde) in all 48 transformations. This is the primordial set structure, or ps.

twelve-tone music. [s] Serial music which orders and uses all twelve of the chromatic pcs available in the twelve-tone system.

unordered set. [h] (Forte) a pc set in which the linear order of pcs is irrelevant, as in chords.

Z-related pair. [h] (Forte) a pair of sets with the same iv (interval vector) but are not reducible to the same prime form. Note that inversely related sets always have the same iv. In Forte (SAM, 1973) inverse sets are reduced to the same prime form; i.e., are not Z related. However, in Solomon (Interface, 1984) inverse sets are not reduced to the same prime form; i.e., are Z-related.

Z-symmetric set. [h] (Solomon) a set whose prime form is equivalent to its inversion. All mirror sets are Z-symmetric, but not vice versa.