### SET THEORY, Part 2

Nondiatonic Subsets   A trichord or larger pitch set can include any pitch of the chromatic scale, a collection of all twelve pitches within the octave. The following trichord does not exist in any diatonic collection but can be found in the chromatic and the F harmonic minor scales.

Example 6: a nondiatonic subset

Terms Used in The Set Theory of Pitches

Octave Equivalence   Pitches at the octave or octaves are considered to have the same function. Thus, every C has equal significance regardless of register.

Pitch Class   A pitch class is the collection of pitches of the same name. For example, the note A in any audible octave belongs to pitch class A. There are twelve pitch classes, one for each note in the chromatic octave.

Enharmonic Equivalence   Any enharmonic spelling of a pitch class is considered to be equivalent; that is,

B = C, E = F, or C = D. Thus, C and D are members of the same pitch class.

Pitch Class Numbers   In the dodecatonic scale (the chromatic scale), each pitch class can be assigned a number from 0 through 11 to indicate its position in the chromatic octave. This numbering is relative to the initial pitch. The initial pitch is numbered 0 and serves as a reference for the numbering of the other pitches. Some individuals prefer to always number C as 0, a practice similar to "fixed do." This approach is quite convenient under most circumstances. However, relative numbering (first note is always 0) was adopted in this text in order to demonstrate the direct relationship between set transformation and mod 12 arithmetic. This is like using a "moveable do."

Both approaches are useful but one should be careful not to mix the two in the same analysis.

Pitch class numbers are integer expressions in modulo or base 12 arithmetic (0-11). In twelve-tone music, numbers are added or subtracted only. Any number larger than 11 should be subtracted from 12 (12-12 = 0). If one counts downwards in half-steps from 0, a negative number will result. 12 must added to a negative number to convert it to a mod 12 integer. For example -2 (the note 2 half steps below PCø) is PC10 (-2 + 12 = 10).

Example 7: Pitch Class numbers of the C chromatic scale in mod 12 integers

Interval Numbers   Intervals are numbered according to number of half-steps they contain, counting the bottom note as 0. The first two columns in the following table include the interval and its interval number. The third column includes the mod 12 operation used to create the inversion. The last two columns contain the results of the operation, the numbers and inversions names of the intervals.

Table 1: dodecimal (mod 12) numbering of intervals

 INTERVAL Interval number OPERATION Interval number INTERVAL INVERSION m2(+U) 1 (12-1) 11 M7 (o8) M2 (o3) 2 (12-2) 10 m7 (+6) m3 (+2) 3 (12-3) 9 M6 (o7) M3 (o4) 4 (12-4) 8 m6 (+5) P4 (+3) 5 (12-5) 7 P5 (o6) +4 (o5) 6 (12-6) 6 o5 (+4) P5 (o6) 7 (12-7) 5 P4 (+3) m6 (+5) 8 (12-8) 4 M3 (o4) M6 (o7) 9 (12-9) 3 m3 (+2) m7 (+6) 10 (12-10) 2 M2 (o3) M7 (o8) 11 (12-11) 1 m2 (+u)

Interval Class Numbers An interval class (IC) contains the interval and its inversion. All compound and enharmonic versions of the interval and its inversion belong in the same interval class. The sum of the intervals in an interval class is always 12.

Example 8: Interval numbers, inversion arithmetic

Interval Vector An interval vector is the inventory of all interval classes possible within a collection of pitches. The interval content of a pattern helps to define it as a unique pattern. For example, a major triad (C-E-G) includes the intervals M3, m3, and P5. These intervals when converted to interval classes (m3=IC3, M3=IC4, and P5=IC5) produce the unique interval vector for the major triad.

Example 9: Interval vector of the major triad

 IC (interval class) 1 2 3 (m3) 4 (M3) 5 (P5) 6 no. of occurrences 0 0 1 1 1 0

The interval vector (interval inventory) of C-A-G is given in example 10. It includes one m2 (A-G), one m3 (C-A), and one M3 (C-G).

Example 10: Interval vector of CAG

 IC (interval class) 1 (m2) 2 3 (m3) 4 (M3) 5 6 no. of occurrences 1 0 1 1 0 0

The interval vector for A-B-E is given in Example 11. It includes one M2 (A-B), and two P4/P5 (A-E, B-E).

Example 11: Interval Vector of ABE

 IC (interval class) 1 2 (M2) 3 4 5 (P5) 6 no. of occurrences 0 1 0 0 2 0

The same procedure is used to determine the interval vector of a larger pattern such as a major scale (see example 12).

Example 12: Interval Vector of any major scale

 Interval Classes 1 2 3 4 5 6 Starting on:first step second step third step fourth step fifth step sixth step 1 1 1 1 1 1 1 1 2 1 1   1 1 2 2 2 1 totals 2 5 4 3 6 1

The interval vector of any major scale contains two m2/M7 (IC1), five M2/m7 (IC2), four m3/M6 (IC3), three M3/m6 (IC4), six P4/5 (IC5), and one tritone (IC6).

Pitch Group Terminology The pitches of the dodecatonic scale can be arranged in groups of from two to twelve pitch classes as follows:

 dyad, dichord: two-note pattern (an interval) triad, trichord: three-note pattern quartad, tetrachord: four-note pattern quintad, pentachord: five-note pattern hexachord: six-note pattern Less common terms are heptichord (7), octachord (8), nonachord (9), and decachord (10) Terms possible but not used are undecachord (11) and dodecachord (12). A simultaneity is any group of notes that sound at the same time.

Normal form To produce normal form, the notes of a pitch set are written in the most compact, compressed form. Root position triads in close spacing are in normal form. To find the normal form of a pitch set, exclude all doubling then rotate the notes (one-note shifts to the left) to find the rotation of least range.

Example 13: Rotations of D F C to find normal form

The third rotation above is the most compact version. Pitch class numbers were assigned to the notes, using C as Ø. The pitch class numbers are used identify the trichord class (026).

If two rotations have the same smallest range between the first and last note, chose the version that has the smaller interval between the first and next to the last note. In the next example, both second and third versions of G B D E span a minor sixth. When comparing these two version, the interval between notes 1 and 3 of the third pattern was smaller. Thus, D E G B (0246) is the normal form of the pattern.

Example 14: determination of normal form in case of a tie

The next example contains an unusual example of a pattern in which two of its rotations can be selected as the normal form. The intervals of both G A D E and D E G A are M2, M3, m6.

Example 15: a pattern with two normal forms (maps onto itself at T6, see example 17)

Transposition Transposition means to raise or lower all the notes in a pattern by the same interval. To complete a transposition when using pitch class numbers, add the transposition interval number to the original pitch class number. For example, T6 indicates the transposing interval is a tritone, T4 a M3. PC8 becomes PC2 when transposed a diminished fifth (8 + 6 - 12 = 2). Remember to subtract 12 from any number larger than 11.

Inversion The sum of a pitch class and its inversion is always 12. To complete an inversion, subtract the pitch class number from 12 (12 - pitch class). Thus the inversion of PC7 is PC5 (12 - 7 = 5).

Example 16 : inversion

Mapping Mapping is the transformation of pitches in a pattern by an operation such as transposition or inversion. "Map onto" means that the original pitch classes become other specific pitch classes in the chromatic scale as a result of the operation. One can say that "C maps onto E at T4." "Map onto itself" means that the pattern, when transformed, uses the pitch classes of the original pattern arranged a different order.

Example 17: A hexachord that maps onto itself at T6 (see example 15)

Set Class A set class includes a pitch set of a particular type, its inversion, the notes in any ordered combination, in all eleven transpositions. According to these criteria, the eleven intervals can be grouped in six interval classes (any transposition of an interval, its inversion, in any octave, any compound version). In the table that follows, notice that six possible set classes are listed for the dyad, a two-note set of pitches.

Table 2: Set Class possibilities

 kind of pitch set no. of PC no. of possible set classes Dyad 2 6 Trichord 3 12 Tetrachord 4 29 Pentachord 5 38 Hexachord 6 50 Heptichord 7 38 see Pentachord Octachord 8 29 see Tetrachord Nonachord 9 12 see Trichord Decachord 10 6 see Dyad

More About Table 2 The decachord has the same number of set class possibilities as the dyad. In fact, each decachord class contains the omission of one of the interval classes. Thus, each interval class (dyad ) has a decachord class that complements it (dyad set + decachord set = all 12 pitch classes). Every set class has a complementary set class (i.e. pentachord/heptichord, tetrachord/octachord, and trichord/nonachord).

All sets can be placed into set class families, reducing the number of distinctly different patterns to remember. Altogether, there are 135 complementary pairs of unique pitch sets. A dyad and the decachord that omits the dyad is such a complementary pair.

Trichord Set Classes

Prime Form Prime form is the most compact version of a pattern and its inversion. To determine prime form, invert the normal form (also the normal form of the inversions)(1). Arrange the notes of the inversion in low-to-high order and make the lowest note PCØ (2). This makes comparison easier. As illustrated in the next example, the major triad 047 is not included in the basic list of trichord set classes because the prime form of this pattern is 037.

Example 18: prime form of the major and minor triad

Note: as a shortcut to prime form, read the normal form top-to-bottom, using mod12 numbers to indicate the relative size of intervals. D=Ø, B=-3, G=-7.

Unordered Collections; An unordered collection is a set of pitches without regard for order. When transposed, an unordered collection can be melodic, harmonic, or a harmonic/melodic mixture, the notes in any order. In normal form, the interval vectors of the original and the transposed unordered collections are identical. An aggregate is an unordered collection of all twelve tones.

Example 19: unordered collections

About example 20 The next example is from Schoenberg's last pre-twelve tone composition. This excerpt has some of the features of later works such as economy of idea, use of pungent dissonance, and use of pitch cells or modules. It also includes conventional patterns such as accompanied melody, phrasing, closure, rhythmic flow, contour, and conversation-like exchange among components in the texture.

When studying the example, first become familiar with the phrasing. Note that the phrase in mm. 1-3 is an accompanied melody. A version of this idea is repeated in mm. 9-11.

A contrasting phrase consists of rhythmic variations on a pair of motives (mm. 4-8). These ideas become progressively longer (seven, nine and eleven eighth notes in duration respectively). This is done by elongating the first motive (treble clef) and delaying the second motive (bass clef). These two ideas overlap at first.

Schoenberg's thoughts are presented in short pitch and rhythm cells. The marked cells that have special significance to the listener and performer. Awareness of these cells and how they go together helps the performer to control accenting and phrasing with greater acuity.

Like others in its genre, this composition carries a rich, expressive message. The examination of cells and of mathematic-like composition method reveals a great deal about a composer's thought process. One should never conclude that music organized in this manner is automatically abstract, devoid of feeling and humanity.

Example 20: Drei Klavierstüke, no. 1 (mm1-11) Arnold Schoenberg, Op. 11, Nr. 1.

© Universal Edition renewed 1938 Arnold Schoenberg USA Belmont Music Publ. LA

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