*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 numbe**r

**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

**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 **dic**s, 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:

QP: t=p, p=t

QR: t=(n-1)-p, p=t

QI: t=p, p=(n-1)-t

QRI: t=(n-1)-p, p=(n-1)-t

As an example, take the four-note set 0,2,3,1. With order numbers this
set is expressed: (0,0), (1,2), (2,3), (3,1). QP would then be: 0,3,1,2;

QR = 2,1,3,0 or (0,2),(1,1),(2,3),(3,0)

QI = 3,0,2,1
or (0,3),(1,0),(2,2),(3,1)

QRI= 1,2,0,3 or (0,1),(1,2),(2,0),(3,3)

**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.