SET THEORY, The Diatonic Scale

SET THEORY, Part 1

SET THEORY, Part 1

OVERVIEW Serial composition, a development of the twentieth century, evolved from the use of ideas about the mathematics of sets in musical construction.

Diatonic Sets A set is a collection of like objects, a class of elements. The set can be made up of pitches as easily as any other kind of object. A scale can be seen as a collection of pitch objects that form a pitch set. A key signature (or consistent use of certain accidentals) defines what notes are members of a collection of pitches. This collection can be divided into smaller segments (subsets).

Any segment of a larger collection can be treated as an independent collection of pitches or as a fragment of a larger set. Notes within one of these subsets can be presented in a particular order, given a rhythm, then transposed to another level within the parent set. In other words, a motive can be created from a small group of pitches then transposed to other levels within the key.

A trichord consists of three different pitches. Like other collections of three objects, the notes of a trichord can be presented in six different orders; 123, 231, 312, 321, 213, and 132. The notes E-F-G form a diatonic set because this pattern can be found in a diatonic scale. Because of this, it can also be regarded as a subset of a diatonic scale. In the next example, each permutation was created by shifting the notes to the left. The fourth pattern was created by switching the order of E and F.

Example 1: Permutations of the subset EFG

The interval content of EFG is m2, M2, and m3 does not change no matter how the notes in the pattern are arranged. The same pattern of intervals is present in the notes BCD, the one transposition of EFG possible in the diatonic set of no accidentals.

Example 2: intervals in EFG and BCD

These two patterns and all of their permutations share common traits that distinguish them from other patterns. Thus, both are like objects that belong to the same collection, a set of sets. This set of sets includes all trichords that contain the intervals m2, M2, and m3. Membership in this set can be restricted to the natural pitches (white-keys) or be expanded to include all eleven transpositions of the set.

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SET THEORY, Part 1

The pattern EFG also resides in the key of one flat, BCD in the key of one sharp. Because they can appear in more than one pitch set, these two patterns form an intersection between two diatonic pitch sets.

Example 3: Set intersection

The pitches of a trichord need not be adjacent notes of a scale. Major, minor and diminished triads, for example, are diatonic subsets that contain no steps. The next example contains a diatonic subset that includes both a step and a skip. The first measure can be found in keys from zero to three flats. CDF forms unions (intersects) with four different diatonic scales. This pattern occurs at three other locations in the set of white keys, DEG, GAC, and ABD. Since all of these sets share the same interval content, they and all their permutations are members of a collection of like objects, a set of sets.

Example 4: set membership by interval content

Combinations of Notes in Ordered Subsets The number of possible combinations of notes depends on the number of notes in the pattern. For example, the notes of a particular trichord can be combined 6 ways, expressed mathematically as 3! (three factorial). The arithmetic of three factorial is 1 X 2 X 3. A tetrachord can be presented 24 unique ways (expressed 4!). Four factorial (4!) is 1 X 2 X 3 X 4. Members of a pentachord can be combined 120 ways (expressed 5!). Five factorial (5!) is 1 X 2 X 3 X 4 X 5. The interval content of a particular pattern remains unchanged regardless of how its notes are ordered.

The Diatonic Scale as a Set Key signatures define transpositions of diatonic collections. The traditional modes under one key signature are permutations of a single set of pitches.

The Diatonic Set as a Collection of Intervals Intervals in a diatonic set can be inventoried as illustrated in example 5. The inventory accounts for every interval possible between every pitch in the set. Each interval column includes an interval and its inversion. The octave and unison were not tallied


SET THEORY, Part 1

because the notes in these intervals are pitches of the same name.

Example 5: Interval Inventory (interval vector) in a Diatonic Set

A B C D E F G m2 /M7

M2 /m7

m3 /M6

M3 /m6

P4 /P5

tt

M2 m3 P4 P5 m6 m7 2 1 1 2

B C D E F G

m2 m3 P4 o5 m6 1 1 1 1 1

C D E F G

M2 M3 P4 P5 1 1 2

D E F G

M2 m3 P4 1 1 1

E F G

m2 m3 1 1

F G

M2 1

total 2 5 4 3 6 1

The inventory of all the intervals possible among the white keys remains unchanged regardless of the order of the notes. The same pattern of intervals can be found in any collection of pitches defined by any diatonic key signature. Thus, all diatonic modes in all keys belong to a collection of like objects, a set of diatonic sets.

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SET THEORY, Part 2

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).


SET THEORY, Part 2

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


SET THEORY, Part 2

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


SET THEORY, Part 2

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:


SET THEORY, Part 2

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


SET THEORY, Part 2

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 THEORY, Part 2

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.


SET THEORY, Part 2

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.


SET THEORY, Part 2

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

Go to Terms and Problems for Part 2

Back to Set Theory, Part 1


SET THEORY, Part 2






SET THEORY, Part 1 Problems

SET THEORY Part 1 Terms and Problems, Diatonic Sets

1. Terms:

setsubset diatonic set interval set interval inventory (interval vector) permutations and combinations trichord chromatic set pentachord tetrachord

2. How many ways can the notes of a particular trichord be combined? a tetrachord? a pentachord? What are the arithmetical expressions of the possible combinations of notes in three, four, five, and six-note patterns?

3. Self-help problems. Solutions follow.

Problem 1: Write all the permutations of this tetrachord (Hint: rotate the pattern three times. Next, switch a pair of notes. Write the resulting pattern at each *. Rotate each new combination three times.). Test the following statement with a spot check interval inventory: all 24 patterns belong to the same class of tetrachord because all have the same interval content.

1* E F G A 9* 17*

2 10 18

3 11 19

4 12 20

5* 13* 21*

6 14 22

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7 15 23

8 16 24

Problem 2: Complete an interval inventory (interval vector) of this arrangement of white keys to see if the interval content changes or remains the same as other versions of the set of white keys. Use example 5 as a model.

F G A B C D E m2 /M7

M2 /m7

m3 /M6

M3 /m6

P4 /P5

tt

G A B C D E

A B C D E

B C D E

C D E

M2 M3 1 1

D E

total

Problem 3: Complete an interval vector of this pentachord. Use example 5 as a model.

D E G A C m2 /M7

M2 /m7

m3 /M6

M3 /m6

P4 /P5

tt

E



G

A

total

Solution 1: Answers will be in this approximate order if the hint was followed. The arrows indicate which note was shifted create a new permutation of the basic tetrachord. Any solution pattern is acceptable so long as it is efficient and organized.

1* E F G A 9* E G F A 17* A F G E

2 F G A E 10 G F A E 18 F G E A

3 G A E F 11 F A E G 19 G E A F

4 A E F G 12 A E G F 20 E A F G

5* F E G A 13* E F A G 21* F E A G

6 E G A F 14 F A G E 22 E A G F

7 G A F E 15 A G E F 23 A G F E

8 A F E G 16 G E F A 24 G F E A

Solution 2: The interval content matches that of CDEFGAB. In fact, this inventory remains unchanged regardless of the order in which the white keys are presented. This remains true under every diatonic key signature.

F G A B C D E m2 /M7

M2 /m7

m3 /M6

M3 /m6

P4 /P5

tt

M2 M3 tt P5 M6 M7 1 1 1 1 1 1

G A B C D E

M2 M3 P4 P5 M6 1 1 1 2



A B C D E

M2 m3 P4 P5 1 1 2

B C D E

m2 m3 P4 1 1 1

C D E

M2 M3 1 1

D E

M2 1

total 2 5 4 3 6 1

Solution 3:

D E G A C m2 /M7

M2 /m7

m3 /M6

M3 /m6

P4 /P5

tt

M2 P4 P5 m7 2 2

E G A C

m3 P4 m6 1 1 1

G A C

M2 P4 1 1

A C

m3 1

total 3 2 1 4

Problem 4: Complete an interval vector of this pentachord. Use example 5 as a model.

A C D E G m2 /M7

M2 /m7

m3 /M6

M3 /m6

P4 /P5

tt



C

D

E

total

Problem 5: This exercise will produce one example of each type of trichord inherent in a diatonic mode. Each is a one-of-a-kind series of intervals. Each can be transposed at least once without adding sharps or flats. Using only natural pitches (white key notes), write one example of each type of trichord having a range of P5 or less. Do not duplicate an interval sequence (i.e. once M2-m2 has been used, do not use it again). Label the intervals in each trichord. When writing these patterns, consider the melodic potential of each pattern. Each pattern can be rotated, reordered, and transposed to produce a melody based on varied repetitions of a thematic cell.

Problem 6:

Write four different versions of the interval sequence P4-M2 using only natural pitches. Measure 1 is the original pattern.

Solution 4: Compare the results of Solution 3 and Solution 4.



A C D E G m2 /M7

M2 /m7

m3 /M6

M3 /m6

P4 /P5

tt

m3 P4 P5 m7 1 1 2

C D E G

M2 M3 P5 1 1 1

D E G

M2 P4 1 1

E G

m3 1

total 3 2 1 4

Solution 5: Sample solutions. Each of these patterns is a one-of-a-kind sequence of intervals. All can be transposed at least one time without adding any sharps or flats. For example, the pattern in measure 4 occurs at two different levels among the white keys. The pattern in measure 7 occurs at four different levels. All patterns with the same combination of intervals are like objects and thus belong to the same set of sets.

Any of these melodic cells can be transposed and reordered to create a cohesive thematic flow in a composition or an improvisation. Experiment with a few of these cells. Improvise a melody in which the original idea is followed by a retrograde transposition, a transposed original by another retrograde transposition, etc. Try a series of other kinds of variants. Later in this chapter, you will be asked to compose a melody based upon trichord cells. This method of composing can be used to create diatonic, chromatic, or fully twelve-tone melodies.



Solution 6: All P4-M2 inherent in the natural pitches.

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Set Theory Part 2 Terms and Problems

1. Terms:

nondiatonic subset octave equivalence pitch class enharmonic equivalence pitch class numbers interval numbers interval class interval class numbers interval vector dyad trichord tetrachord octachord undecachord quintad tetrad hexachord heptichord hexad septad pentachord normal form inversion(mod 12) transposition(mod 12) prime form set class mapping onto mapping onto self unordered collection aggregate simultaneity ordered collection

2. Which of the twelve trichord set classes (012 , 013, 014, 015, 016, 024, 025, 026, 027, 036, 037, and 048) cannot be found in adiatonic scale?

3. Make a trichord dictation exercise for use in class. Using the trichords in exercise 2, let 0 be any pitch except C. Let the notesbe in any order except that in exercise 2. Displace the middle note up or down an octave.

4. Compose a melody based on one of the trichords in exercise 2. Use transpositions and a new order of pitches each time thetrichord occurs.

5. Self-help problems and solutions

Problem 1: Identify the intervals according to the information needed to fill in the blanks in the grid. Use the first column of the gridas a sample solution.

interval number 4 7 2 9 5 11 8 1 3 10 6



interval name M3

interval class 4


interval number 10 6 3 1 8 11 5 9 2 7 4

interval name m7

interval class 2


interval number 9 2 7 4 1 8 11 5 6 10 3

interval name M6

interval class 3


interval number 3 10 1 6 8 11 5 2 7 4 9

interval name m3

interval class 3

Solution 1:

interval number 4 7 2 9 5 11 8 1 3 10 6

interval name M3 P5 M2 M6 P4 M7 m6 m2 m3 m7 tt

interval class 4 5 2 3 5 1 4 1 3 2 6

Solution 2:

interval number 10 6 3 1 8 11 5 9 2 7 4

interval name m7 tt m3 m2 m6 M7 P4 M6 M2 P5 M3

interval class 2 6 3 1 4 1 5 3 2 5 4

Solution 3:

interval number 9 2 7 4 1 8 11 5 6 10 3

interval name M6 M2 P5 M3 m2 m6 M7 P4 tt m7 m3



interval class 3 2 5 4 1 4 1 5 6 2 3

Solution 4:

interval number 3 10 1 6 8 11 5 2 7 4 9

interval name m3 m7 m2 tt m6 M7 P4 M2 P5 M3 M6

interval class m3 2 1 6 4 1 5 2 5 4 3

Problem 5:

1. Write the trichord, C D F, in the first row of the 3 X 3 grid (a123).

2. Write the melodic inversion of the trichord in the first column (1abc), using "C" as a common tone.

3. Use 1b and 1c as the starting notes of transpositions of the original trichord (1abc).

4. Reading from top to bottom, what version of the original pattern is present in columns 2 and 3?

5. Reading from right to left, what version of the original pattern is present in rows a, b, and c?

6. Reading from bottom to top, what version of the original pattern is present in columns 1, 2, and 3?

Problem 6:

1. Write the pitch class numbers 0 3 4 of the quadrants (pie wedges) of row a.

2. Write the pitch class inversion numbers in the wedges of column 1 (12 - pitch class number).

3. Write the transpositions of row a in rows b and c. Add the value in 1b to the original pitch class number. If the result is greaterthan 11, subtract it from 12. Repeat the operation operation by adding the value of 1c to the original pitch class number.

4. Assuming that pitch class 0 = C , use the pitch class number to determine which pitch class should be written in each box.

Sample of Interval Vector Procedure

pitch class in hexachord Interval Classes

0 1 3 4 5 8 1 2 3 4 5 6

C D E E F A 1 1 2 1

D E E F A 1 1 1 1

E F 1 1 1



E A

E F A 1 1

F A 1

total ICs 3 2 3 4 3 0

solution 5:

C D F

B C E

G A C

Reading from top to bottom, the patterns in columns 2 and 3 are transposed inversions of the original set, each started on a pitchin the original set. Reading from right to left, the patterns in rows b and c are transposed retrogrades of the original. Reading frombottom to top, the patterns are retrograde inversions of the original.

solution 6:

0C

3E

4F

9B

0D

1D

8A

11C

0D

Problem 7: Complete an interval vector using the sample solution on the previous page as a guide. Derive the pitch names fromthe pitch class numbers.


0 1 4 5 8 9 1 2 3 4 5 6

F F A B D D

total ICs

Problem 8: Complete an interval vector using the sample solution as a guide. Derive pitch names from the pitch class numbers.


0 1 2 6 7 8 1 2 3 4 5 6



A

total ICs

Solution 7:


0 1 4 5 8 9 1 2 3 4 5 6

F F A B D D 1 1 2 1

F A B D D 1 2 1

A B D D 1 1 1

B D D 1 1

D D 1

total ICs 3 0 3 6 3 0

Solution 8:


0 1 2 6 7 8 1 2 3 4 5 6

A B B E E F 1 1 1 1 1

B B E E F 1 2 1

B E E F 1 1 1

E E F 1 1

E F 1

total ICs 4 2 0 2 4 3

Problem 9: Complete an interval vector using the sample solution as a guide. Use the pitch class numbers to determine pitchnames.


0 2 4 5 7 9 1 2 3 4 5 6



B

total ICs

Sample of Inversion Procedure

1 2 1 1 3 Interval number (note to note)

C D E E F A note names of Pø

12 12 12 12 12 12 12

0 1 3 4 5 8 original hexachord - pitch classnumber

0 11 9 8 7 4 pitch class number of I

C B A A G E note names of I

- 1 - 2 - 1 - 1 - 3 Interval number (note to note, adouble check, should be same astop line.)

Solution 9:


0 2 4 5 7 9 1 2 3 4 5 6

B C D E F G 1 1 1 2

C D E F G 1 1 2

D E F G 1 1 1

E F G 1 1

F G 1

total ICs 1 4 3 2 5 0

Problem 10: Work out an inversion of the original hexachord. Use the sample procedure on the previous page as a guide. Fill in allblanks.



Interval number (note to note)

D note names of Pø (use pitch classnumbers)

12 12 12 12 12 12 12

0 2 4 6 8 10 original hexachord

- pitch class number

pitch class number of I

note names of I

Interval number (note to note) (adouble check, should be same astop line.)

Problem 11: Work out an inversion of the original hexachord. Use the sample solution as a guide. Fill in all blanks.


C note names of Pø (use pitch classnumbers)

12 12 12 12 12 12 12



pitch class number of I

note names of I


Solution 10:


D E F G A C note names of Pø (use pitchclass numbers)

12 12 12 12 12 12 12





0

(12=0)

10 8 6 4 2 pitch class number of I

D C A G F E note names of I

-2 -2 -2 -2 -2 Interval number (note to note) (adouble check, should be same astop line.)

Solution 11:


C D E F G A note names of Pø (use pitchclass numbers)

12 12 12 12 12 12 12



0

(12=0)

11 9 6 5 3 pitch class number of I

C B A F F E note names of I

-1 -2 -3 -1 -2 Interval number (note to note) (adouble check, should be same astop line.)

Sample of Transposition Procedure


C D E E F A note names of Pø

5 5 5 5 5 5 T interval ( 5 )


+ pitch class number

5 6 8 9 10 1* *(13 - 12 = 1)

pitch class number ofTransposition (Pn)

F F G A B D note names of Pn

1 2 1 1 3 Interval number (note to note) (a



double check, should be same astop line.)

Problem 12: Work out a transposition of the original hexachord. Use the sample solution as a guide. Fill in all blanks.


G note names of Pø (use pitch classnumbers)

8 8 8 8 8 8 T interval (8)

0 1 2 6 7 8 + pitch class numbers in originalhexachord


note names of Pn


Problem 13: Work out a transposition of the original hexachord. Use the sample solution as a guide. Fill in all blanks.


B note names of Pø (use pitch classnumbers)

6 6 6 6 6 6 T interval (6)

0 2 3 5 6 9+ pitch class numbers in originalhexachord


note names of Pn


Solution 12:


G A A D D E note names of Pø (use pitch classnumbers)



8 8 8 8 8 8 T interval (8)


8 9 10 2

(14-12)

3 4 pitch class number ofTransposition (Pn)

E E F A B B note names of Pn

1 1 4 1 1 Interval number (note to note) (adouble check, should be same astop line.)

Solution 13:


B C C E E G note names of Pø (use pitchclass numbers)

6 6 6 6 6 6 T interval (6)


6 8 9 11 0

(12-12)

3 pitch class number ofTransposition (Pn)

E F G A B C note names of Pn

2 1 2 1 3 Interval number (note to note) (adouble check, should be sameas top line.)

Problem 14: Find and label the prime form of these tetrachords (C E F G , C D E A , C E A B)

[Hint: Prime form is the most compact version of normal form and its inversion. Normal form is the most compact version of allrotations of a pitch set. The first step in determining these forms is to arrange the PCs in low-to-high order. ] Rotate the notes ofthe original pattern. Let the lowest note of each rotation be PC#0. The inversion of normal form is the same as the normal form ofthe inversions. Arrange it in lowest-to-highest order, assigning Ø to the lowest note. As a shortcut to determining prime form, readnormal form top-to-bottom, making the highest note Ø.



Solution 14:



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