Atonality

Pitch Class Sets 1 5/16/2004

Pitch Class Sets Paul Nelson - Revised: 5/16/2004

Pitch Class Sets are a method for describing harmonies in 20th century music. These notations and methods can describe and manipulate any type of chord that can be created within a 12-tone (equally tempered) scale. It is an extremely useful technique for composers to help understand and control the harmonies which make up their music.

Pitch class sets are the chemistry of harmonic color. Modern composers will use pitch class sets like chemistry, to mix and create interesting and vibrantly colorful harmonic sounds, which they then use to create works of music.

1 Basic Definitions 1.1 Pitches

! A "pitch" is any note that we hear. ! The standard piano can play 88 pitches: A0 to C8, where C4 = middle-C. ! For example, the notes above middle-C are as follows:

C4 (B!3), C!4(D"4), D4, D!4(E"4), E4(F"4), F4(E!4), F!4(G"4), G4, G!4(A"4), A4, A!4(B"4), B4(C"5)

1.2 Pitch Classes ! Pitch classes are used to discuss pitches independent of octave displacement and enharmonic spelling. ! Any two pitches which sound the same on an equal tempered scale (for example, C! and B") or are only

different due to octave displacement are said to belong to the same "pitch class". ! For example, the following pitches all belong to Pitch Class C:

B#4, D""4 (enharmonic equivalents), C0, C1, C2, C3, C4, C5, C6 (octave displacements) ! There are only 12 pitch classes in a system where each octave has 12 chromatic notes. ! Pitch classes can also be numbered: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11.

These numbers are sometimes called "Pitch Class Representatives". ! For this tutorial, 0 = Pitch Class C (i.e. "fixed Do"). ! All other pitch classes will by numbered by counting the half steps from pitch-class C. ! Therefore, C = 0, C# = 1, D = 2, D# = 3, E = 4, F = 5, F# = 6, G = 7, G# = 8, A = 9, A# = 10, B = 11 ! Sometimes the letter 'T' (for Ten) or 'A' is used instead of the number 10, and 'E' (for Eleven) or 'B' instead

of 11.

1.3 Pitch Class Sets ! A "Pitch Class Set" is a list of pitch class numbers: [0, 4, 7, 10] (note the square brackets) ! These are also called "PC Sets". ! The PC Set for a C minor triad: [0, 3, 7] ! The PC Set for a G major triad: [7, 11, 2] ! In Pitch Class sets, octave doublings and displacements are ignored:

o [0, 3, 7, 12] => [0, 3, 7] (see the section below on modulo math) o [14, 7, 11] => [2, 7, 11] o For example, all of the following can be described with Pitch Class Set [0, 1,4]. The only

difference in these chords are octave displacements or enharmonic spellings in the pitches.

Piano !! """# """$$"""#

"""# """

#"""#

""""#

%[0,1,4]

&[0,1,4]

&[0,1,4]

&[0,1,4]

&[0,1,4]""

[0,1,4]""##

[0,1,4]

""#

! Oftentimes PC Set notation is shown without the commas: [037]

(here is where A=T=10 and B=E=11 comes in handy, for example: [0,4,7,10] = [047T] = [047A] (note: this is a C dominant 7th chord)


2 Simple Operations on Pitch Classes and Pitch Class Sets 2.1 "Clock" Math or Modulo Math

! When manipulating pitch classes, you will use a special operator, called the "modulo" operator.

! The "modulo" operator takes the remainder of an integer divided by some other integer.

! For example: 19 modulo 12 = 7 (i.e. 12 goes into 19 once, with 7 left over)

! Pitch class sets use "modulo 12". Any number above 12 should be reduced, using "mod 12", to a number from 0 to 11.

! The modulo operator can be visualized using a clock face: ! Some interesting characteristics of the clock face:

o A tritone is made up of two notes which are opposite of each other (for example: C = 0 and F# = 6)

o The notes of a cross make up a doubly-diminished 7th chord (for example: C = 0, D# = 3, F# = 6, A = 9)

o The augmented triad (C = 0, E = 4, G# = 8) is also pleasingly symmetric.

2.2 Transposing Pitch Class Sets ! To transpose a pitch class set, add (or subtract) the same number to all elements of the list:

[0,1,4] => (transpose up a major third) [0+4, 1+4, 4+4] => [4,5,8] In this example, the chord "C D" E" is transposed up to "E F G#".

! Remember to use "Module 12" when numbers are greater than or equal to 12: [0,1,4] => (transpose up a major 7th) [0+11, 1+11, 4+11] => [11, 12, 15] => [11, 0, 3]

2.3 Inverting Pitch Class Sets ! To invert a PC Set, subtract each element of the list from 12:

[0,1,4] => [12 - 0, 12 - 1, 12 - 4] => [12, 11, 8] => [0, 11, 8] (don't forget to use Mod 12 if any of the numbers are greater than 11) For example: The chord "C D" E" becomes "C B A"".

! By convention, simple inversion is always around Pitch Class C (0). Therefore, any note of the chord which is N half-steps above C, will be flipped to be come a note N half-steps below C. In the above example, the note "E" (4 half-steps above C) was flipped to become "A"" (4 half-steps below C).

! Very often you will want to invert and transpose at the same time: [0,1,4] => [ (12-0) + 4, (12-1) + 4, (12-4) + 4] => [16, 15, 12] => [4, 3, 0] This has a special notation: T4I (invert and then transpose up 4 half steps)

! Examples of the PC Sets shown above:

Pno. ! ![0,1,4]

"""# """$[0,11,8]inversion

[4,3,0]inversion +

transposition

"""$

0 1

2

3

4

567

8

9

10

11

1213

14

15

16

1718

20

21

22

23

24

19

etc.


Pno. ! !

(note: numbers over twelve shown to demonstrate rotation, these should be reduced to 0-11 with "Modulo 12")

[0,4,6,8]""""$$

[4,6,8,12]

""""$$[6,8,12,16]

""""$$[8,12,16,18]

""""$$

3 The Prime Form 3.1 Similar Pitch Class Sets: Set Classes & Prime Forms

! Some pitch class sets are very similar, for example: [0,1,4] is very similar to [3,4,7] (transposition), [8,11,0] (inversion), [5,8,9] (transposition and inversion), and [8,9,0] (transposition).

! For example, try playing the following chords. Can you hear that they all have something in common?

Pno. ! ! """#[0,1,4]

(original)

"""$[3,4,7]

(transposed)

"""$[8,11,0]

(inverted)

"""$[5,8,9]

(transposed &inverted)

"""$[8,9,0]

(transposed)

! A group of similar PC Sets like these is called a "Pitch Class Set Class", or more simply, a "Set Class". ! If two PC Sets differ only by transposition or inversion, then they belong to the same Set Class. ! There are only 208 different Set Classes! ! Each Set Class is represented by a "Prime Form" PC Set. For example:

[0,1,4]; [3,4,7]; [0,3,4]; [5,8,9]; and [8,9,0] all belong to the Prime Form: (0,1,4) ! Note that parenthesis are used to denote Prime Forms in this tutorial. However, not everybody agrees on

this syntax.

3.2 Uses for The Prime Form ! The prime form is considered to be the "simplest" version of the pitch class set. ! Generally, the "simplest" version of a PC set means that the pitches in the set are packed as tightly together

possible, and as far to the left as possible. ! Once you know the prime form of a PC set, you can look it up in a table of prime forms to get more

information about the PC Set, such as its interval vector and fellow related PC Sets (see appendix). ! You can also use the prime form to search for other, related PC Sets using other software tools.

See http://www.ComposerTools.com . ! If you are a composer, you can use this information to help you better control, understand, and manipulate

the harmonies in your music.

3.3 Determining the Prime Form: The Rigorous Method ! Goal: To identify the prime form for any PC set. ! Example: What is the prime form of [8,0,4,6] ? ! Step 1: Put the Pitch Classes in numerical order => [0,4,6,8] ! Step 2: List all of the rotations of the pitch class set. To rotate a PC Set, simply move the first number to

the end and add 12 to it (i.e. shift it up an octave). For example, the Rotations of [0,4,6,8] are:

[0, 4, 6, 8] [4, 6, 8, 12] [6, 8, 12, 16] [8, 12, 16, 18]

! Step 3: Determine which rotation of the PC Set has the minimum distance between the first and last

numbers in the Set: o [0, 4, 6, 8] => ( 8 - 0) = 8 o [4, 6, 8, 12] => (12 - 4) = 8 o [6, 8, 12, 16] => (16 - 6) = 10 o [8, 12, 16, 18] => (18 - 8) = 10

There is a tie! Versions [0,4,6,8] and [4,6,8,12] both have a minimum distance between first and last of 8

http://www.composertools.com/


! Step 4: If there is a tie, choose the rotation which has a minimum distance between the first and second numbers: Distances between the first and second numbers:

o [0, 4, 6, 8] => ( 4 - 0) = 4 o [4, 6, 8, 12] => ( 6 - 4) = 2

So, in our example, [4,6,8,12] is preferred. ! Step 5: If there is still a tie, then check the first and third numbers, and so on until the tie is resolved.

The PC Set at this point is in "Normal" form. ! Step 6: Transpose the pitch class set so that the first number is zero:

[4 - 4, 6 - 4, 8 - 4, 12 - 4] => [0, 2, 4, 8] ! Step 7: Invert the pitch class set and reduce it using steps 1-5 above.

o Invert [0,2,4,8] => [ 12-0, 12-2, 12-4, 12-8 ] => [12, 10, 8, 4] => [0, 10, 8, 4] o Put in numerical order: [0, 4, 8, 10] o Find the best rotation:

PC Set (last-first) (second-first) [0, 4, 8, 10] 10 4 [4, 8, 10, 12] 8 4 [8, 10, 12, 16] 8 2 << Preferred [10,12, 16, 20] 10 2

o Transpose down: [8 - 8, 10 - 8, 12 - 8, 16 - 8] => [0, 2, 4, 8] ! Step 8: Which form, the original or the inverted, is most packed to the left (has the smallest numbers)?

That will be the Prime Form. In our example, both forms produced the same Prime Form (this is because the original PC Set was "inversionally symmetric"), and so the Prime Form is (0, 2, 4, 8)

3.4 Determining the Prime Form: Easier Methods ! Option 1: Use an online tool at http://www.composertools.com . ! Option 2: Figure it out on the piano

o Step 1: Keep rotating your chord until it is as small as possible. o Step 2: If there are ties, then use the rotation that has the notes most packed towards the bottom. o Step 3: Check to see if the inversion is better packed.

! Option 3: Use the "Simplified Set List" at the back of Post Tonal Theory by Joseph N. Straus. ! Option 4: Use a MAX/MSP patch which displays the Prime form of a chord you play on your MIDI

keyboard. See the URL: http://www.euph0r1a.net/projects/?handler=etrof . ! Option 5: Use the table of all prime forms. For example, 1) Find the interval vector first, then look it up in

the table of all PC Sets (see below), or 2) skip steps 6 & 7 above and look up the inversion in the table. ! Option 6: Visualize the Pitch Class Set on a clock face and locate the prime form visually

o Step 1: The shortest distance traveled around the clock. o Step 2: Numbers packed as close to the starting point as possible.

For example, the prime form of [0,8,6,8] is (0,2,4,8); and the prime form of [2,4,8,9] is (01,15,7) :

0 1

2

3

4

567

8

9

10

11

0

7

51

0 1

2

3

4

567

8

9

10

11

0

2

4

8

http://www.euph0r1a.net/projects/?handler=etrof



4 Interval Vectors An "Interval Vector" is a list of six numbers which summarizes the interval content in a PC Set. With a little experience, you will be able to get a sense for how a PC Set sounds when you see its interval vector. Further, once you know the interval content of a PC Set, you will also be able to manipulate the sound of the PC Set by inversion and octave displacement of pitches to emphasize certain intervals over others.

4.1 Pitch Intervals ! The distance between any two pitches is called a "pitch interval". This is the standard definition for an

interval in music. For example: ! Ordered Intervals: A"3 to D"5 = Perfect 11th ascending = +17 half steps D"5 to A"3 = Perfect 11th descending = -17 half steps ! Un-ordered Intervals: A"3 to D"5 = Perfect 11th = 17 half steps D"5 to A"3 = Perfect 11th = 17 half steps

4.2 Interval Classes ! In the same way that many pitches "sound alike" and are therefore put into the same Pitch Class, there are

also many intervals which sound alike and so are put into the same Interval Class. ! There are six different interval classes which are numbered from 1 to 6.

o m2 / M7 => 1 (half-steps) o M2 / m7 => 2 (whole-steps) o m3 / M6 => 3 (minor thirds) o M3 / m6 => 4 (major thirds) o P4 / P5 => 5 (perfect intervals) o A4 / d5 => 6 (tritones)

! The interval class number (1 to 6) is the count of half steps between two pitch classes. In other words, it is the minimum distance between two pitches ignoring the octave displacement of either pitch.

! For example, in the case of A"3 to D"5, if you moved A"3 up an octave to A"4, then the distance between the two is a perfect 4th. And so the interval class is a '5', for a perfect interval.

4.3 Interval Vectors ! An Interval Vector is a summary of all of the intervals between all pairs of pitches in a pitch class set. It is,

essentially, a histogram of all of the interval classes which can be found in a PC Set. ! For the purposes of this tutorial, an interval vector will be represented with angle brackets as follows:

1 2 3 4 5 6

< >

m2/M7(half-step)

M2/m7(whole-step)

m3/M6(minor-3rd)

M3/m6(Major-3rd)

P4/P5(perfect)

A4/d5(tritone)

! For example, a C major chord is represented by the PC Set: [047] and has the interval vector <001110>.

This is because a C major chord contains one minor third (from E to G) one major third (from C to E) and one perfect interval from C to G). Since a major chord contains no half steps, whole steps, or tritones, these entries in the interval vector are all set to zero (0).

! Note that there is no agreed upon standard punctuation for representing an interval vector. The angle brackets appear to be the most common, but there are many other representations being used.


How to compute an interval vector: o Step 1: Go through all pairs of pitches in your PC Set.

If your PC Set has: It will contain: 2 pitches 1 interval 3 pitches 3 intervals 4 pitches 6 intervals 5 pitches 10 intervals 6 pitches 15 intervals Overall, the formula for computing the number of intervals (Ni) from the number of pitchs (Pi) is: Number of Intervals = ( N*(N-1) ) / 2

o Step 2: For each pair, subtract the smaller number from the larger number.

o Step 3: Take the result of step 2 and increment the appropriate slot in the interval vector using the chart on the right:

! Example: [0, 2, 7, 8]

o Step 1: 4 pitches in the pitch class set = 6 intervals = 6 pairs of pitches: [0, 2] [0, 7] [0, 8] [2, 7] [2, 8] [7, 8]

o Step 2: For each pair, subtract the smaller number from the larger number: [0, 2] = 2; [0, 7] = 7; [0, 8] = 8; [2, 7] = 5; [2, 8] = 6; [7, 8] = 1

o Step 3: For each difference in Step 2, increment the appropriate slot from the chart above:

11

1

10

2

9 8 7

3 4 5 6

1 1 0 1 2 1

! Therefore, for our example, the interval vector is: <110121> ! This means that the Pitch Class Set [0, 2, 7, 8] contains the following interval classes:

1 half-step, 1 whole-step, 1 major third, 2 perfect intervals, and 1 tritone ! When I listen to this PC Set [0278], what I hear is a triad based on perfect fifths (0,2,7) = <010020> with

an additional pitch (8) that adds some significant 'bite', via the half-step and tritone dissonance.

5 The Table of All Prime Forms of PC Sets Please refer to the Appendix for a two-page table of all possible prime forms of Pitch Class Sets. This table is an indispensable aid for composers, since it is, essentially, a table of all possible types of chords. Not only does it contain all of the standard chords from tonal harmony such as triads (major, minor, diminished, and augmented) and seventh chords (dominant, major-minor sevenths, major-major sevenths, minor sevenths, etc.), but it also contains all chord types used by modern composers as well. Any chord which can be constructed using a 12-tone equal tempered scale is represented in the table.

5.1 The Columns of Data in the Table For each prime form in the table, there are five columns of data:

! Column 1: The interval vector ! Column 2: The count of PC Sets which reduce to the prime form ! Column 3: The Forte code (see below) ! Column 4: The Prime Form PC Set ! Column 5: The inverted form (if different than the Prime Form)

11

1

10

2

9 8 7

3 4 5 6

11

1

10

2

9 8 7

3 4 5 6


5.2 The Layout of the PC Sets in the Table ! The PC Sets are grouped in the table by size, into 13 sections (from 0 pitches to 12 pitches per PC Set). ! Within each group the list is sorted by interval vector. Interval vectors with the most half-step intervals are

listed first, then vectors with the most whole-step intervals, and so on. ! Z-related forms are listed together, one after the other (see section 5.4) ! Commonly known pitch class sets (e.g. well-known chord qualities, types of scales, etc.) are labeled with

{curly braces}. For example, (0, 3, 7) is labeled as {min} because it is a minor triad. ! With the exception of the sets of 6 Pitch Classes, each set is listed opposite of its "complement". For

example, set 4-16, (0,1,5,7) is listed to the left of set 8-16, (0,1,2,3,5,7,8,9). A set and its complement share many similar properties (see below for a discussion of Pitch Class Set complements).

! To conserve space, the table uses the letters A, B, and C for the numbers 10, 11, and 12.

5.3 Forte Names ! Allen Forte's book, The Structure of Atonal Music, published the first version of this table. In his table, he

labeled each prime form of the PC Set with a unique designation, such as 5-20. ! The first number (5-) specifies the number of pitches in the pitch class set. ! The second number (20) is a unique number given to the prime form, which was sequentially assigned by

Dr. Forte when he first created the table. ! When analyzing PC Sets, many music theorists will label them using the Forte designation, although

simply using the prime form (e.g. (0,1,3,7,8) or (01378) ) is becoming more common.

5.4 Z-Related Sets ! When two prime forms produce the same interval vector, and when one can not be reduced to the other (by

inversion or transposition), they are said to be "Z-Related", or "Z Correspondents". ! The Forte Code for all PC Sets which are Z related contains a 'Z' in the PC Set ID. For example, 6-Z25. ! 'Z' doesn't stand for anything, it is just an identifier chosen by Dr. Forte when the table was first created. ! Z-related sets are "close cousins" to one another. They sound similar to each other, but not as similar as sets

related by (say) transposition or inversion. For example, try playing the following PC Sets on the piano. Listen for the intervals they contain. Since the Z-related sets contain the same intervals, do they not sound at least somewhat similar?

Pno. !! """"## """"## "" ""# "" ""$$ "" ""$% &

4-Z15[0,1,4,6]

&4-Z29

[0,1,3,7]

""$$4-Z15

[0,1,4,6]

""$$4-Z15

[1,3,6,7]

""$$4-Z29

[6,7,9,1]

""4-Z15

[0,1,4,6]

""#4-Z29

[9,10,0,4]

""4-Z29

[0,4,6,7]

5.5 Other Comments on the Table ! When I first encountered the table, I was surprised that it contained so few interval vectors (200), prime

forms (208) and chord qualities (351). For some reason, in my mind, I had always thought that the complete list of possible chord types was much much larger.

! Along the same lines, the number of chord types used by in common practice music is quite small, as few as a dozen different types chords, perhaps as many as 20 if you include Jazz chords.

o This implies that there is a very number of chords yet to be thoroughly explored! ! The following music shows some very famous chords. With out PC Sets, how could the types of these

chords be specified?

''(''(

)()(

Pno. !! ###****+# ****+ ****+ ****+ **** ****

,---

**** ****, ***$$ *** ***

...****##

. / ****. / / ****##

.****

. /

% ### ****+### ****+

7-32: (0,1,3,4,6,8,9)Stravinsky: Rite of Spring

****+ ****+ **** ****, ---**** ****,*****$#--

7-Z36: (0,1,2,3,5,6,8)Stravinsky: Rite of Spring

********** ***# . / ***

. /4-19: (0,1,4,8)

Bernard Herrmann: "Psycho" Prelude

/ ***# .***

. /


6 Subsets and Supersets Any of the larger PC Sets can be divided into pieces. These pieces are, of course, also PC Sets in their own right. The smaller PC Sets are said to be "subsets" of the larger PC Set, which is the "superset".

! For Example, the superset (0,1,2,6,7,8) is quite dissonant and has the interval vector <420423>. It contains the following subsets: Subsets 1: => [0,2,7] + [1,6,8] = two quintal/quartal triads Subsets 2: => [1,8] + [0,2,6,7] = a simple fifth + a complex chord (a dominant+tonic sound) Subsets 3: => [1,7] + [0, 2, 6, 8] = a tritone + whole-tone-scale fragment Subsets 4: => [6,7,8] + [0,1,2] = two chromatic clusters Subsets 5: => [0,6] + [1,7] + [2,8] = three tritone intervals

!! """"""#$$$"""

$$$ """"$ """"$$ """$$ """

#

%[0,1,2,6,7,8]

& """[0,2,7]+[1,6,8]

""$$[1,8]+[0,2,6,7] [1,7]+[0,2,6,8]

"""$$"""#$

[6,7,8]+[0,1,2]

"""$$

[0,6]+[1,7]+[2,8]

PC subsets and supersets are a very useful compositional technique. Be sure to explore all of the subsets for PC Sets that you use (see ComposerTools.com). This will help you to use, space, and manipulate your harmonies.

! Other things to experiment with: o Use subsets for growth; i.e. restrict sections of your music to use only portions of a larger PC set

and then grow the PC set over time, making your harmonies denser and more complex. o Put the sub-sets in different registers to emphasize their unique sounds (see examples below). o Construct melodies from sub-sets which can be combined together to create

6.1 Definition: Transpositional Combination of Two Common Subsets ! Transpositional combination: When a superset is created from two equal subsets, where one is transposed. ! Example 1: [0,1,2] + [0,1,2]{transposed by 6 halfsteps} => [0,1,2] + [6,7,8] => [0,1,2,6,7,8] ! Example 2: [0,2,7] + [0,2,7]{transposed by 6 halfsteps} => [0,2,7] + [6,8,1] => [0,1,2,6,7,8]

6.2 Definition: Inversional Combination of Two Common Subsets ! Inversional combination: When a superset is created from two equal subsets, where one is inverted (and

possibly transposed) ! Example: [0,1,6] + [0,1,6]{invert and transpose by 8 half steps} => [0,1,6] + [12-0+8, 12-1+8, 12-6+8]

=> [0,1,6] + [8,19,14] => [0,1,6] + [8,7,2] => [0,1,2,6,7,8] ! Note: The result of an inversional combination will always be "inversionally symmetric" (see below for a

discussion of inversional symmetry)

0 1

2

3

4

567

8

9

10

11 0 1

2

3

4

567

8

9

10

11

Transpositional Combination Inversional Combination

0 1

2

3

4

567

8

9

10

11

Transpositional Combination


7 PC Set Complements 7.1 Definitions

! Literal Complement: When one PC Set contains all of the Pitch Classes not in some other PC Set. Example: [0,1,4,7] and [2,3,5,6,8,9,10,11] are literal complements of each other

! Abstract Complement: When two PC Sets would be complements of each other, except that one is transposed or inverted from the other. When someone says that a PC Set is the complement of some other PC Set, it usually means that they are Abstract Complements of each other. Example: (0,1,4,7) and (0,1,2,3,5,6,8,9) are abstract complements of each other

! The prime forms of abstract complements are listed side-by-side in the PC Set table found in the Appendix (except for the sets of 6 pitch classes).

! Note that the Forte designation for a PC Set and it's complement will always have the same PC Set ID number (after the dash). For example, 4-18 and 8-18 are abstract complements of each other.

7.2 PC Set Complements and Their Interval Vectors ! A Pitch Class Set and its complement will have very similar interval vectors. ! In fact, there is a simple formula for computing the interval vector of a complement:

o How many more pitch classes does the complement have? Call this 'D'. ! Note: If the original PC set has X pitch classes, it's complement will have (12-X) pitch

classes, and the difference between the two will be: D = (12-X)-X = (12-X*2) ! For example, if the original PC set has 5 pitches, the complement will have (12-5) pitches

(i.e. 7 pitch classes) and the difference (D) between 5 and 7 is (12-5*2) = 2. o If the interval vector for the original PC Set is <I1, I2, I3, I4, I5, I6> o Then the interval vector for the complement will be:

<I1+D, I2+D, I3+D, I4+D, I5+D, I6+ (D/2) > o Note that the tritone is special because it divides the 12-tone chromatic scale exactly in half. For

this reason, its interval vector grows by D/2. o Also note that D is always an even number {0, 2, 4, 6, 8, 10}, and so D/2 will always be an integer

number (never a fraction). ! Example:

o 4-18:(0,1,4,7) has 4 pitch classes and an interval vector of <102111> o It's complement is 8-18:(0,1,2,3,5,6,8,9) o 8 - 4 = D = 4 o The complement's interval vector is: <1+4, 0+4, 2+4, 1+4, 1+4, 1+(4/2)> = <546553>

!! """"# """"$$$ & """"$$$

% &(0,1,4,7) (0,1,2,3,5,6,8,9)

""""#$$""""$

(0,1,4,7)""""$

(0,1,2,3,5,6,8,9)

! Some famous complements: o Pentatonic Scale (5 Pitches) : <032140> " Diatonic Scale (7 Pitches) : <254361> o Octatonic Scale (8 Pitches) : <448444> " doubly-diminished 7th chord (4 Pitches) : <004002>

7.3 6-note complements ! The complement of a set with 6 Pitch Classes will itself have 6 Pitch Classes ! Therefore, the difference in number of Pitch Classes is always 0 (zero). ! Therefore, a 6-note complement will always have the same interval vector as it's complement! ! True!


! Since all PC Sets with 6 pitches have a complement with the same interval vector, there are only two ways that one of these PC Sets can be related to its complement:

o The set is "self complementary", that is, the set and it's complement have the same prime form. o The set and its complement are Z-related: Two sets with the same interval vector but which can

not be reduced to the same Prime Form by transposition or inversion.

0 1

2

3

4

567

8

9

10

11

Z-Related Sets with 6 Pitch ClassesSelf-Complementary Set {6-30}with 6 Pitch Classes

6-Z23 6-Z45

0

2

3

4 6

9

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4

567

8

9

10

11

6

3

1

0

7

9

7.4 PC Set Complements Used in Twelve Tone Composition ! PC Set complements are critically important when composing music with 12-tone rows, because:

o If you take any 12-tone row and divide it up into two pieces at any point, then o the two pieces will have similar (or exactly the same) interval content.

! This is one of the reasons why a 12-tone composition has a "built-in" harmonic cohesiveness. ! For example, consider the following 12-tone row:

')(! !Schoenberg, Op. 25

*-3-2:(0,1,3) <111000>

*-

4-12:(0,2,3,6) <112101>

5-28:(0,2,3,6,8) <122212>

*-

6-Z43:(0,1,2,5,6,8) <322332>

*$ *-

Note: This is just a sample, many more combinations are possible

*- *-9-2:(0,1,2,3,4,5,6,7,9) <876663>

*#

8-12:(0,1,3,4,5,6,7,9) <556543>

7-28:(0,1,3,5,6,7,9) <344433>

*#

6-Z17:(0,1,2,4,7,8) <322332>

*# *# *-

! By definition, the last 6 notes of a 12-tone row are the PC Set complement of the first 6 notes ! For more harmonic cohesiveness, make the first and last 6 notes of the row the same PC Set, i.e. self

complementary and possibly inversionally related. ! For more harmonic variety, make the first and last 6 notes of the row Z-related PC-sets. ! This is the first step towards hexachordal combinatoriality: where a 12-tone row is made up of two similar

halves, for example, where the 2nd half is a transposed inversion of the first half (further discussion is beyond the scope of this tutorial). This is a favored technique of late Schoenberg compositions.


8 More Properties of Pitch Class Sets and Interval Vectors 8.1 Common Tones when Transposed

! If two PC Sets contain the same pitches, these are called the "common tones". o For example, the common tones between [0, 3, 4, 7] and [1, 3, 4, 8] are 3 and 4.

! The interval vector can tell you how many common tones you will have after transposing a pitch class set. Simply look up the transposing interval in the interval vector, and the number you find will be the number of common tones after transposing the pitch class set.

! For example, if the interval vector of the PC Set is: <324222> (6-Z13), and if you transpose the PC Set by a minor third, you will have 4 common tones between the original PC Set and the transposed PC Set.

! How does the interval vector help? Take any Pitch Class Set ! Algorithm:

o Step 1: Find the Prime Form of your PC Set. Suppose it is (0,3,4,7) o Step 2: Lookup the PC Set, (0,3,4,7), in the Prime Forms table and find its interval vector,

<102210>. o Step 3: The elements of the PC Set will tell you how many common tones to expect as you

transpose the PC Set. ! For example, if [0,3,4,7] is transposed by a half-step it becomes [1,4,5,8]. The original

PC Set and the transposed PC Set have one common tone (4 = E). ! As a second example, if [0,3,4,7] is transposed by a major third it becomes [4,7,8,11],

which has two common tones (4 and 7). o Based on the interval vector <102210>, here is a complete list of how many common tones to

expect when the PC Set is transposed: ! a half-step => 1 pitch class remains the same ! a whole-step => All new pitch classes ! a minor-third => 2 pitch classes remains the same ! a major third => 2 pitch classes remains the same ! a perfect 4th => 1 pitch class remains the same ! a tritone => All new pitch classes

((! ![0,3,4,7]

""""$ "*""$$m2 M2

""""#m3

*"*"$$$

Shows the common tones when [0,3,4,7] is transposed up by various intervals.

**""#m3 P4

"""*$Tritone

""""$$# """$#[0,1,6] Transpose by

a Tritone

*"*$

! Except: For tritones (it would figure). When transposing by a tritone, you get double the number of common pitches as specified in the interval vector. For example, if you transpose [0,1,6] by a tritone, you would get two common pitch class sets, rather than one (see above for an example).

! You can use this fact for composition to either make transitions smoother or more abrupt. If two adjacent harmonies in your music have many common tones, they will transition smoothly from one to the other. If they have few common tones, then the transition will be less smooth. For example:

o Use for common tone transposition / modulation: Transpose a PC Set around a common tone for smoother transitions.

01! ! * * *$ *+ * * *$ *Using common tone transposition to smoothly transpose a [0,2,6,7] ( 4-16:(0,1,5,7) ) figure

*. * * *$ *+ * * *# *- *.

((! ! 2222$ 222 2222$ 222Transposing up using two common tones

2222$ 222 2222$ 222


o Alternatively, transpose with all new notes to emphasize the difference. ! Also, the interval vector can be used to help identify when a PC Set can be combined with itself to make

larger PC Sets with all unique pitches. o For example, the PC Set 6-8 has 6 pitches and the interval vector <343230>. This PC Set can be

combined with itself by transposing it a tritone to make up a complete twelve tone row. 8.1.1 More Examples and Transpositional Symmetry

Let us consider two interesting PC Sets: The diatonic scale and the whole tone scale. Both of these scales have some rather interesting properties when they are transposed. ! Diatonic Scale, 7-35:(0,1,3,5,6,8,10) which has the interval vector <254361>

o Transpose the scale by a fifth or fourth (i.e. modulate to the dominant or the sub-dominant) and there will be 6 common pitch classes, and 1 new pitch class.

o Transpose the scale by a half step, and there will be only 2 common pitches and 5 new ones. For example: C Major to C# Major, or C Major to B major.

o This gives rise, in tonal music, to the notion of "near" and "distant" keys. ! The whole tone scale: 6-35:(0,2,4,6,8,10) has interval vector <060603>

o Transpose this scale by any interval and either 1) all the pitch classes will be new or 2) all the pitch classes will be different.

o Remember to double the value of the tritones entry (from 3 to 6). ! If any entry of the interval vector is equal to the number of pitch classes in the set, then the PC Set can be

transposed to itself with all pitch classes in common. This is called "Transpositional Symmetry." ! In the Pitch Class Set Table, any PC Set with a "Count" column smaller than 12 has some Transpositional

Symmetry.

8.2 Inversional Symmetry ! The best way to see if (and how) a PC Set contains common pitch classes when inverted is to visualize the

PC set on a clock face, and then look for one or more axis of symmetry. ! OR: when looking at the table of all prime forms in the Appendix, if a PC Set has no entry in the

"inversion" column, then it is inversionally symmetric on at least one axis. ! Otherwise, there is no special math involved to determine inversional symmetry. ! A PC Set which can invert to itself (on some axis of inversion) is said to be "Inversionally Symmetric". ! Looking for inversions and inversional symmetry is another way to manipulate PC Sets to get new sounds. ! In the following examples, the first PC Set is inversionally symmetric, and the second is not.

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11 0 1

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567

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two commontwo new

one commonthree new

inversionally symmetricalong two axis

!! **$ . **-# ++ **++ **$- . **$ . **-# ++ **++ **$- .

% 222 222Alternating inversions of (0,1,6,7) with two common tones

222 222


9 Other PC Set Similarity Relations This section covers other ways in which two PC Sets can be related. Again, this can be a useful compositional technique. For example, you could choose a PC Set and compose a work which is made up of just the original PC Set plus other, closely related sets. Such a composition should have a fairly consistent harmonic color throughout. Similarly, if you are looking for dramatic color contrasts, you will likely want to avoid similarly related PC Sets. Note that you can explore many of these similarity relations at http://www.ComposerTools.com .

9.1 Special Purpose Relations: Rp, R0, R1, R2 ! Rp => When two PC Sets are the same except for one different pitch class, i.e. one note different

o Very useful for composers, this is one way to "morph" PC sets. For example, you can go from PC Set 1 to PC Set 2 by changing a single note, as long as the two sets are related by Rp.

o But not too useful for analysis, since this relates many PC sets to many many other PC sets ! R0 => When two PC Sets have the same number of pitch classes, but no interval vector entries in

common, for example: o 4-2:(0,1,2,4) has interval vector <221100> o 4-13:(0,1,3,6) has interval vector <112011> o There is no interval which has the same count in both interval vectors. o Not a very useful measure, since it has to do with the relative strengths of the intervals, rather than

the presence or total absence of intervals. ! R1 => When two PC Sets have the same number of pitch classes, and their interval vectors are as similar

as they can be without being equal o This will be the case when the 4 of the 6 entries in the interval vector are the same, and the

remaining two entries are simply exchanged, for example: o 4-2: (0,1,2,4) has interval vector <221100> o 4-3: (0,1,3,4) has interval vector <212100> o Note the highlighted entries in the interval vector are the only ones which are different, and the

two entries are merely exchanged from one to the other. ! R2 => Just like R1, except that the two different entries are not merely an exchange of numbers. For

example: o 5-10: (0,1,3,4,6) has interval vector <223111> o 5-Z12: (0,1,3,5,6) has interval vector <222121>

! Note that R1 and R2 are also Rp.

9.2 Other techniques for generating related PC Sets ! Rotational arrays: Used by Oliver Knussen and Igor Stravinsky ! Intervallic projection to relate subsets and supersets:

o Add notes to a PC Set by projecting up from the top note by a certain interval o For example: Quartal / Quintal harmony is created by projecting by adding a note to a PC set

which is a perfect 4th or 5th above the last note added o Or this can be done with alternating intervals (i.e. first add a 5th, then a tritone, etc)



Appendix - Table of All PC Set Prime Forms

interval Forte prime inverted vector count code form form

Sets of 0 pitch classes, 0 intervals (1 vector, 1 quality, 1 total)

<000000> (1) (){silence}


<000000> (12) (0){single-note}

Sets of 2 pitch classes, 1 intervals (6 vectors, 6 qualities, 66 total)

<100000> (12) (0,1){half-step} <010000> (12) (0,2){whole-step} <001000> (12) (0,3){minor-third} <000100> (12) (0,4){major-third} <000010> (12) (0,5){perfect} <000001> (6) (0,6){tritone}

Sets of 3 pitch classes, 3 intvls (12 vectors, 19 qualities, 220 total)

<210000> (12) 3-1: (0,1,2) <111000> (24) 3-2: (0,1,3) [0,2,3] <101100> (24) 3-3: (0,1,4) [0,3,4] <100110> (24) 3-4: (0,1,5) [0,4,5] <100011> (24) 3-5: (0,1,6) [0,5,6] <020100> (12) 3-6: (0,2,4) <011010> (24) 3-7: (0,2,5) [0,3,5] <010101> (24) 3-8: (0,2,6){It.} [0,4,6] <010020> (12) 3-9: (0,2,7){quar-3} <002001> (12) 3-10: (0,3,6){dim} <001110> (24) 3-11: (0,3,7){min} [0,4,7]{maj} <000300> (4) 3-12: (0,4,8){aug}


<321000> (12) 4-1: (0,1,2,3) <221100> (24) 4-2: (0,1,2,4) [0,2,3,4] <212100> (12) 4-3: (0,1,3,4) <211110> (24) 4-4: (0,1,2,5) [0,3,4,5] <210111> (24) 4-5: (0,1,2,6) [0,4,5,6] <210021> (12) 4-6: (0,1,2,7) <201210> (12) 4-7: (0,1,4,5) <200121> (12) 4-8: (0,1,5,6) <200022> (6) 4-9: (0,1,6,7) <122010> (12) 4-10: (0,2,3,5) <121110> (24) 4-11: (0,1,3,5) [0,2,4,5] <112101> (24) 4-12: (0,2,3,6) [0,3,4,6] <112011> (24) 4-13: (0,1,3,6) [0,3,5,6] <111120> (24) 4-14: (0,2,3,7) [0,4,5,7] <111111> (48) 4-Z15: (0,1,4,6) [0,2,5,6] 4-Z29: (0,1,3,7) [0,4,6,7] <110121> (24) 4-16: (0,1,5,7) [0,2,6,7] <102210> (12) 4-17: (0,3,4,7) <102111> (24) 4-18: (0,1,4,7) [0,3,6,7] <101310> (24) 4-19: (0,1,4,8){mM7} [0,3,4,8] <101220> (12) 4-20: (0,1,5,8){maj7} <030201> (12) 4-21: (0,2,4,6) <021120> (24) 4-22: (0,2,4,7) [0,3,5,7] <021030> (12) 4-23: (0,2,5,7){quar-4} <020301> (12) 4-24: (0,2,4,8){7+5} <020202> (6) 4-25: (0,2,6,8){fr.,7-5} <012120> (12) 4-26: (0,3,5,8){min7,maj6} <012111> (24) 4-27: (0,2,5,8){hd7} [0,3,6,8]{dom7} <004002> (3) 4-28: (0,3,6,9){dd7}



<CCCCC6> (1) (0,1,2,3,4,5,6,7,8,9,A,B){chromatic}


<AAAAA5>(12) (0,1,2,3,4,5,6,7,8,9,A)


<988884> (12) (0,1,2,3,4,5,6,7,8,9) <898884> (12) (0,1,2,3,4,5,6,7,8,A) <889884> (12) (0,1,2,3,4,5,6,7,9,A) <888984> (12) (0,1,2,3,4,5,6,8,9,A) <888894> (12) (0,1,2,3,4,5,7,8,9,A) <888885> (6) (0,1,2,3,4,6,7,8,9,A)


<876663> (12) 9-1: (0,1,2,3,4,5,6,7,8) <777663> (24) 9-2: (0,1,2,3,4,5,6,7,9) [0,2,3,4,5,6,7,8,9] <767763> (24) 9-3: (0,1,2,3,4,5,6,8,9) [0,1,3,4,5,6,7,8,9] <766773> (24) 9-4: (0,1,2,3,4,5,7,8,9) [0,1,2,4,5,6,7,8,9] <766674> (24) 9-5: (0,1,2,3,4,6,7,8,9) [0,1,2,3,5,6,7,8,9] <686763> (12) 9-6: (0,1,2,3,4,5,6,8,A) <677673> (24) 9-7: (0,1,2,3,4,5,7,8,A) [0,1,3,4,5,6,7,8,A] <676764> (24) 9-8: (0,1,2,3,4,6,7,8,A) [0,1,2,4,5,6,7,8,A] <676683> (12) 9-9: (0,1,2,3,5,6,7,8,A) <668664> (12) 9-10: (0,1,2,3,4,6,7,9,A) <667773> (24) 9-11: (0,1,2,3,5,6,7,9,A) [0,1,2,4,5,6,7,9,A] <666963> (4) 9-12: (0,1,2,4,5,6,8,9,A)


<765442> (12) 8-1: (0,1,2,3,4,5,6,7) <665542> (24) 8-2: (0,1,2,3,4,5,6,8) [0,2,3,4,5,6,7,8] <656542> (12) 8-3: (0,1,2,3,4,5,6,9) <655552> (24) 8-4: (0,1,2,3,4,5,7,8) [0,1,3,4,5,6,7,8] <654553> (24) 8-5: (0,1,2,3,4,6,7,8) [0,1,2,4,5,6,7,8] <654463> (12) 8-6: (0,1,2,3,5,6,7,8) <645652> (12) 8-7: (0,1,2,3,4,5,8,9) <644563> (12) 8-8: (0,1,2,3,4,7,8,9) <644464> (6) 8-9: (0,1,2,3,6,7,8,9) <566452> (12) 8-10: (0,2,3,4,5,6,7,9) <565552> (24) 8-11: (0,1,2,3,4,5,7,9) [0,2,4,5,6,7,8,9] <556543> (24) 8-12: (0,1,3,4,5,6,7,9) [0,2,3,4,5,6,8,9] <556453> (24) 8-13: (0,1,2,3,4,6,7,9) [0,2,3,5,6,7,8,9] <555562> (24) 8-14: (0,1,2,4,5,6,7,9) [0,2,3,4,5,7,8,9] <555553> (48) 8-Z15: (0,1,2,3,4,6,8,9) [0,1,3,5,6,7,8,9] 8-Z29: (0,1,2,3,5,6,7,9) [0,2,3,4,6,7,8,9] <554563> (24) 8-16: (0,1,2,3,5,7,8,9) [0,1,2,4,6,7,8,9] <546652> (12) 8-17: (0,1,3,4,5,6,8,9) <546553> (24) 8-18: (0,1,2,3,5,6,8,9) [0,1,3,4,6,7,8,9] <545752> (24) 8-19: (0,1,2,4,5,6,8,9) [0,1,3,4,5,7,8,9] <545662> (12) 8-20: (0,1,2,4,5,7,8,9) <474643> (12) 8-21: (0,1,2,3,4,6,8,A) <465562> (24) 8-22: (0,1,2,3,5,6,8,A) [0,1,3,4,5,6,8,A] <465472> (12) 8-23: (0,1,2,3,5,7,8,A) <464743> (12) 8-24: (0,1,2,4,5,6,8,A) <464644> (6) 8-25: (0,1,2,4,6,7,8,A) <456562> (12) 8-26: (0,1,2,4,5,7,9,A) <456553> (24) 8-27: (0,1,2,4,5,7,8,A) [0,1,3,4,6,7,8,A] <448444> (3) 8-28: (0,1,3,4,6,7,9,A){octatonic}



Sets of 5 pitch classes, 10 intvls (35 vectors, 66 qualities, 792 total) <432100> (12) 5-1: (0,1,2,3,4) <332110> (24) 5-2: (0,1,2,3,5) [0,2,3,4,5] <322210> (24) 5-3: (0,1,2,4,5) [0,1,3,4,5] <322111> (24) 5-4: (0,1,2,3,6) [0,3,4,5,6] <321121> (24) 5-5: (0,1,2,3,7) [0,4,5,6,7] <311221> (24) 5-6: (0,1,2,5,6) [0,1,4,5,6] <310132> (24) 5-7: (0,1,2,6,7) [0,1,5,6,7] <232201> (12) 5-8: (0,2,3,4,6) <231211> (24) 5-9: (0,1,2,4,6) [0,2,4,5,6] <223111> (24) 5-10: (0,1,3,4,6) [0,2,3,5,6] <222220> (24) 5-11: (0,2,3,4,7) [0,3,4,5,7] <222121> (36) 5-Z12: (0,1,3,5,6) 5-Z36: (0,1,2,4,7) [0,3,5,6,7] <221311> (24) 5-13: (0,1,2,4,8) [0,2,3,4,8] <221131> (24) 5-14: (0,1,2,5,7) [0,2,5,6,7] <220222> (12) 5-15: (0,1,2,6,8) <213211> (24) 5-16: (0,1,3,4,7) [0,3,4,6,7] <212320> (24) 5-Z17: (0,1,3,4,8) 5-Z37: (0,3,4,5,8) <212221> (48) 5-Z18: (0,1,4,5,7) [0,2,3,6,7] 5-Z38: (0,1,2,5,8) [0,3,6,7,8] <212122> (24) 5-19: (0,1,3,6,7) [0,1,4,6,7] <211231> (24) 5-20: (0,1,3,7,8) [0,1,5,7,8] <202420> (24) 5-21: (0,1,4,5,8) [0,3,4,7,8] <202321> (12) 5-22: (0,1,4,7,8) <132130> (24) 5-23: (0,2,3,5,7) [0,2,4,5,7] <131221> (24) 5-24: (0,1,3,5,7) [0,2,4,6,7] <123121> (24) 5-25: (0,2,3,5,8) [0,3,5,6,8] <122311> (24) 5-26: (0,2,4,5,8) [0,3,4,6,8] <122230> (24) 5-27: (0,1,3,5,8) [0,3,5,7,8]{min9} <122212> (24) 5-28: (0,2,3,6,8) [0,2,5,6,8] <122131> (24) 5-29: (0,1,3,6,8) [0,2,5,7,8] <121321> (24) 5-30: (0,1,4,6,8) [0,2,4,7,8] <114112> (24) 5-31: (0,1,3,6,9) [0,2,3,6,9]{7-9} <113221> (24) 5-32: (0,1,4,6,9) [0,2,5,6,9]{7+9} <040402> (12) 5-33: (0,2,4,6,8){9+5,9-5} <032221> (12) 5-34: (0,2,4,6,9){dom9} <032140> (12) 5-35: (0,2,4,7,9){pentatonic,Quar-5}


Sets of 7 pitch classes, 21 intvls (35 vectors, 66 qualities, 792 total) <654321> (12) 7-1: (0,1,2,3,4,5,6) <554331> (24) 7-2: (0,1,2,3,4,5,7) [0,2,3,4,5,6,7] <544431> (24) 7-3: (0,1,2,3,4,5,8) [0,3,4,5,6,7,8] <544332> (24) 7-4: (0,1,2,3,4,6,7) [0,1,3,4,5,6,7] <543342> (24) 7-5: (0,1,2,3,5,6,7) [0,1,2,4,5,6,7] <533442> (24) 7-6: (0,1,2,3,4,7,8) [0,1,4,5,6,7,8] <532353> (24) 7-7: (0,1,2,3,6,7,8) [0,1,2,5,6,7,8] <454422> (12) 7-8: (0,2,3,4,5,6,8) <453432> (24) 7-9: (0,1,2,3,4,6,8) [0,2,4,5,6,7,8] <445332> (24) 7-10: (0,1,2,3,4,6,9) [0,2,3,4,5,6,9] <444441> (24) 7-11: (0,1,3,4,5,6,8) [0,2,3,4,5,7,8] <444342> (36) 7-Z12: (0,1,2,3,4,7,9) 7-Z36: (0,1,2,3,5,6,8) [0,2,3,5,6,7,8] <443532> (24) 7-13: (0,1,2,4,5,6,8) [0,2,3,4,6,7,8] <443352> (24) 7-14: (0,1,2,3,5,7,8) [0,1,3,5,6,7,8] <442443> (12) 7-15: (0,1,2,4,6,7,8) <435432> (24) 7-16: (0,1,2,3,5,6,9) [0,1,3,4,5,6,9] <434541> (24) 7-Z17: (0,1,2,4,5,6,9) 7-Z37: (0,1,3,4,5,7,8) <434442> (48) 7-Z18: (0,1,4,5,6,7,9) [0,2,3,4,5,8,9] 7-Z38: (0,1,2,4,5,7,8) [0,1,3,4,6,7,8] <434343> (24) 7-19: (0,1,2,3,6,7,9) [0,1,2,3,6,8,9] <433452> (24) 7-20: (0,1,2,4,7,8,9) [0,1,2,5,7,8,9] <424641> (24) 7-21: (0,1,2,4,5,8,9) [0,1,3,4,5,8,9] <424542> (12) 7-22: (0,1,2,5,6,8,9){hungar-min} <354351> (24) 7-23: (0,2,3,4,5,7,9) [0,2,4,5,6,7,9] <353442> (24) 7-24: (0,1,2,3,5,7,9) [0,2,4,6,7,8,9] <345342> (24) 7-25: (0,2,3,4,6,7,9) [0,2,3,5,6,7,9] <344532> (24) 7-26: (0,1,3,4,5,7,9) [0,2,4,5,6,8,9] <344451> (24) 7-27: (0,1,2,4,5,7,9) [0,2,4,5,7,8,9] <344433> (24) 7-28: (0,1,3,5,6,7,9) [0,2,3,4,6,8,9] <344352> (24) 7-29: (0,1,2,4,6,7,9) [0,2,3,5,7,8,9] <343542> (24) 7-30: (0,1,2,4,6,8,9) [0,1,3,5,7,8,9] <336333> (24) 7-31: (0,1,3,4,6,7,9) [0,2,3,5,6,8,9] <335442> (24) 7-32: (0,1,3,4,6,8,9){harm-min} [0,1,3,5,6,8,9] <262623> (12) 7-33: (0,1,2,4,6,8,A) <254442> (12) 7-34: (0,1,3,4,6,8,A) <254361> (12) 7-35: (0,1,3,5,6,8,A){diatonic}


<543210> (12) 6-1: (0,1,2,3,4,5) <443211> (24) 6-2: (0,1,2,3,4,6) [0,2,3,4,5,6] <433221> (48) 6-Z3: (0,1,2,3,5,6) [0,1,3,4,5,6] 6-Z36: (0,1,2,3,4,7) [0,3,4,5,6,7] <432321> (24) 6-Z4: (0,1,2,4,5,6) 6-Z37: (0,1,2,3,4,8) <422232> (24) 6-5: (0,1,2,3,6,7) [0,1,4,5,6,7] <421242> (24) 6-Z6: (0,1,2,5,6,7) 6-Z38: (0,1,2,3,7,8) <420243> (6) 6-7: (0,1,2,6,7,8) <343230> (12) 6-8: (0,2,3,4,5,7) <342231> (24) 6-9: (0,1,2,3,5,7) [0,2,4,5,6,7] <333321> (48) 6-Z10: (0,1,3,4,5,7) [0,2,3,4,6,7] 6-Z39: (0,2,3,4,5,8) [0,3,4,5,6,8] <333231> (48) 6-Z11: (0,1,2,4,5,7) [0,2,3,5,6,7] 6-Z40: (0,1,2,3,5,8) [0,3,5,6,7,8] <332232> (48) 6-Z12: (0,1,2,4,6,7) [0,1,3,5,6,7] 6-Z41: (0,1,2,3,6,8) [0,2,5,6,7,8] <324222> (24) 6-Z13: (0,1,3,4,6,7) 6-Z42: (0,1,2,3,6,9) <323430> (24) 6-14: (0,1,3,4,5,8) [0,3,4,5,7,8] <323421> (24) 6-15: (0,1,2,4,5,8) [0,3,4,6,7,8] <322431> (24) 6-16: (0,1,4,5,6,8) [0,2,3,4,7,8] <322332> (48) 6-Z17: (0,1,2,4,7,8) [0,1,4,6,7,8] 6-Z43: (0,1,2,5,6,8) [0,2,3,6,7,8]

<322242> (24) 6-18: (0,1,2,5,7,8) [0,1,3,6,7,8] <313431> (48) 6-Z19: (0,1,3,4,7,8) [0,1,4,5,7,8] 6-Z44: (0,1,2,5,6,9) [0,1,4,5,6,9] <303630> (4) 6-20: (0,1,4,5,8,9) <242412> (24) 6-21: (0,2,3,4,6,8) [0,2,4,5,6,8] <241422> (24) 6-22: (0,1,2,4,6,8) [0,2,4,6,7,8] <234222> (24) 6-Z23: (0,2,3,5,6,8) 6-Z45: (0,2,3,4,6,9) <233331> (48) 6-Z24: (0,1,3,4,6,8) [0,2,4,5,7,8] 6-Z46: (0,1,2,4,6,9) [0,2,4,5,6,9] <233241> (48) 6-Z25: (0,1,3,5,6,8) [0,2,3,5,7,8] 6-Z47: (0,1,2,4,7,9) [0,2,3,4,7,9] <232341> (24) 6-Z26: (0,1,3,5,7,8) 6-Z48: (0,1,2,5,7,9) <225222> (24) 6-27: (0,1,3,4,6,9) [0,2,3,5,6,9] <224322> (24) 6-Z28: (0,1,3,5,6,9) 6-Z49: (0,1,3,4,7,9) <224232> (24) 6-Z29: (0,1,3,6,8,9) 6-Z50: (0,1,4,6,7,9) <224223> (12) 6-30: (0,1,3,6,7,9) [0,2,3,6,8,9] <223431> (24) 6-31: (0,1,3,5,8,9) [0,1,4,6,8,9] <143250> (12) 6-32: (0,2,4,5,7,9){min11} <143241> (24) 6-33: (0,2,3,5,7,9) [0,2,4,6,7,9]{dom11} <142422> (24) 6-34: (0,1,3,5,7,9) [0,2,4,6,8,9] <060603> (2) 6-35: (0,2,4,6,8,A){wholetone}

Totals: Total unique interval vectors: 200 Total prime forms: 208 (according to the Forte designations, does not include 0, 1, 2, 10, 11, 12 element PC sets) Total unique chord qualities: 351 (the prime forms plus inversions of all PC sets shown above) Total pitch collections: 4095 (includes all transpositions and inversions of all PC sets shown above)

PITCH-CLASS SETS & RELATIONS

A. Pitches & Intervals

1.Numbering pitch-classes

-The numbers 0 to 11 refer to the twelve different pitch-classes in ascending semitones.

2. Fixed- & Movable-Zero-Two methods of assigning the number 0 to a pitch-class:

a) Movable-Zero notation- assigning 0 to whatever pitch-class is a convenient focal point.

b) Fixed-Zero notation- using 0 for the pitch-class C.

3. Calculating harmonic Intervals

-Harmonic Intervals are calculated by subtracting the lower pitch from that of the upper pitch. (note: if the number of the upper pitch is smaller than the lower, add 12 to the upper before subtracting )-Compound intervals (larger than an 8va) are reduced by subtracting 12 or multiples of 12 until the number is between 0 and 11.

4. Inverting Intervals-To invert an interval, subtract its number from 12. (its complement)

5. Interval-Classes -An interval-class contains an interval, its complement, and all compounds:

a) 1,11 - semitones & major seventhsb) 2,10 - whole tones & minor seventhsc) 3,9 - minor thirds & major sixthsd) 4,8 - major thirds & minor sixthse) 5,7 - perfect fourths & fifthsf) 6 - tritones

B. Pitch-Class Sets

1. the Pitch-class set

-A group of pitch-classes, each different from the others.. ex: [0,2,4,7,9]

2. Transposition

-A pitch-class set is transposed by adding the interval of transposition to each pitch number,

subtracting from 12 any pitch numbers over 11.

Ex: [0, 2, 4, 7, 9]+4 +4 +4 +4 +4

=[4, 6, 8, 11, 1]

3. Inversion

-Replacing every interval in a set with its complement.

Ex: [0,2,6] 12 12 12

-0, -2, -6

= [0, 10, 6]

4. Naming Pitch-class sets-The form of a pitch-class set in normal order with the first integer being 0 is the Prime form. -Three main steps for figuring out this lowest ordering:

a) Notate all the pitches in ascending order within an octave. (starting on any pitch class)b) Find the largest interval between consecutive pitches.

Reorder the pitches, beginning with the upper pitch of the largest interval. Number from 0. (If the last interval is larger than the first interval, this is the lowest ordering).c) If the last interval in step2 is the same size or smaller than the first interval, rewrite the pitches from right to left, write the complement of each number, and transpose to begin on 0. Compare this new result to the result(s) in step 2 to find the lowest ordering.

Ex: [0,2,5,6] 6,5, 2, 0 (reversed order)6,7,10,0 (the complement of each)

[0,1,4,6] (transposed to begin on 0)

5. Pitch-class sets in Nontonal Music-Pitch-class sets in nontonal music are analogous to scales in tonal music . Like tonal scales, nontonal pitch class sets provide the notes out of which melodies and harmonies arise.

C. Interval Content

1. the Interval Vector-The interval content of a pitch-class set is the total of all its intervals.

a) Take the first number (other than 0) and subtract each of the other numbers of the set working your way down to one digit.

Ex: [0,1,5,6,7] 0 1 5 6 7 4 5 6

1 2

1

b) Count the number of digits representing each interval class:Ex: # of instances: 3 1 0 1 3 2

Interval-class: 1 2 3 4 5 6

D. Related Pitch-Class Sets

1. Equivalence relations-Two pitch class-sets are said to be equivalent if and only if they are reducible to the same prime form by transposition or by inversion followed by transposition. (see above for details)

2. Inclusion relations-Subsets & Supersets- A subset is a pitch-class set that is part of a larger set or superset.

Ex: 4-19: [2,3,6,10] is a subset of 5-30: [2,3,6,8,10]

3. Complement relations-The set of 12 pitch-class integers comprises the universal set U, the set of all elements from which sets of cardinal number less than 12 are drawn.

a) the Literal complement- the remaining pitches not found in that set.

b) the transformed complement- the transposition and/or inversion of one or both

complement-related pc sets without losing the fundamental association between them.

4. Set Complexes -A group of sets related to a single set, called the nexus set, by the inclusion relation:

a) set complex K- Given a set A and its complementB, the set complex K consists of all the sets

that are in an inclusion relation (subset or superset) with either A or B. those sets are said to be in the set complex K about the pair A/B. (Ex: 6-Z44 is a member of the set complex K about 5-32/7-32)

b) set complex Kh- For a set to be a member of the set complex Kh about a pair A/B, it must be in

an inclusion relation to both A and B. (Ex: Set 6-27 is a subset of 7-32 and also contains 5-32)

5. Similarity relations-Pitch-class sets (whether transpositions or inversions of one set or different sets) can also be related to one another by the number of pitches or the number of intervals they have in common.

Relation Interpreted as:

Rp Maximum similarity with respect to pitch classR0 Minimum similarity with respect to interval class

R1 Maximum similarity “ “ “

(note: the relation maximum similarity with respect to both pitch class and interval class will be regarded as more significant than pitch or interval class similarity alone.)

Twelve-Tone Technique

Twelve-tone technique (also dodecaphony) is a method of musical composition devised by Arnold Schoenberg. Music using the technique is called twelve-tone music. Josef Matthias Hauer also developed a similar system using unordered hexachords, or tropes, at the exact same time and country but with no connection to Schoenberg. Other composers have created systematic use of the chromatic scale, but it is Schoenberg's method which is historically the most prevalent and considered easily the most important.

Technique

The basis of twelve-tone technique is the tone row, an ordered arrangement of the twelve notes of the chromatic scale (the twelve equal tempered pitch classes). The tone row chosen as the basis of the piece is called the prime series (P). Untransposed, it is notated as P0. Given the twelve pitch classes of the chromatic scale, there are 12! (12 factorial) unique tone rows.

When twelve-tone technique is strictly applied, a piece consists of statements of certain permitted transformations of the prime series. These statements may appear serially, or may overlap, giving rise to harmony.Appearances of P can be transformed from the original in three basic ways:

•transposition up or down, giving P?.•reversal in time, giving the retrograde (R)•reversal in pitch, giving the inversion (I): I(?) = 12 - P?.•The various transformations can be combined. The combination of the retrograde and inversion transformations is known as the retrograde inversion (RI).

P, R, I and RI can each be started on any of the twelve notes of the chromatic scale, meaning that 47 permutations of the initial tone row can be used, giving a maximum of 48 possible tone rows. However, not all prime series will yield so many variations because tranposed transformations may be identical to each other. This is known as invariance. A simple case is the ascending chromatic scale, the retrograde inversion of which is identical to the prime form, and the retrograde of which is identical to the inversion (thus, only 24 forms of this tone row are available).

When rigorously applied, the technique demands that one statement of the tone row must be heard in full (otherwise known as aggregate completion) before another can begin. Adjacent notes in the row can be sounded at the same time, and the notes can appear in any octave, but the order of the notes in the tone row must be maintained. Durations, dynamics and other aspects of music other than the pitch can be freely chosen by the composer, and there are also no rules about which tone rows should be used at which time (beyond them all being derived from the prime series, as already explained).

In practice, the "rules" of twelve-tone technique have been bent and broken many times, not least by Schoenberg himself. For instance, in some pieces two or more tone rows may be heard progressing at once, or there may be parts of a composition which are written freely, without recourse to the twelve-tone technique at all. Offshoots or variations may produce music in which:

• the full chromatic is used and constantly circulates, but permutational devices are ignored• permutational devices are used but not on the full chromatic

Derivation

In music using the twelve tone technique a derived row is a tone row whose entirety of twelve tones is constructed from a segment or portion of the whole, the generator. Anton Webern often used derived rows in his pieces.

Rows may derived from a sub-set of any number of pitch classes that is a divisor of 12 (trichords, tetrachords, and hexachords), the most common being the first three pitches or a trichord. This segment may then undergo transposed, inversion, retrograde, or any combination to produce the other parts of the row (in this case, the other three segments). The opposite is partitioning, the use of methods to create segments from sets, most often through registral difference

Invariance

Invariant formations are also the side effect of derived rows where a segment of a set remains similar or the same under transformation (inversion, retrograde, retrograde-inversion, multiplication). These invariants can take the form of pcs, subsets, and intervals (di) or interval classes (ic). Additionally, combinations of row forms may result in invariant sets of various sizes.

George Perle describes their use as "pivots" or non-tonal ways of emphasizing certain pitches. Invariant rows are also combinatorial.

Combinatoriality

In music using the twelve tone technique combinatoriality is a side-effect of derived rows where combining different segments or sets such that the pitch class content of the result fulfills certain criteria, usually the combination of hexachords which complete the full chromatic.

Hexachordal inversional combinatoriality refers to any two rows, one of which is an inversion and one is not. The first row's first half, or six notes, are the second's last six notes, but not necessarily in the same order. Thus the first half of each row is

the others complement, as with the second half, and, when combined, these rows still maintain a fully chromatic feeling and don't tend to reinforce certain pitches as tonal centers as would happen with freely combined rows. Babbitt also described the semi-combinatorial row and the all-combinatorial row, the latter being a row which is combinatorial with any of its derivations and their transpositions. Retrograde Hexachordal combinatoriality is considered trivial, since any set has retrograde hexachordal combinatoriality with itself. Combinatoriality may be used to create an aggregate or all twelve tones, though the term often refers simply to combinatorial rows stated together.

Semi-combinatorial sets are sets whose hexachords are capable of forming an aggregate with one of its basic transformations transposed.

All-combinatorial sets are sets whose hexachords are capable of forming an aggregate with any of its basic transformations transposed. There are six source sets, or basic hexachordally all-combinatorial sets, each hexachord of which may be reordered within itself:

• (A) 0 1 2 3 4 5 // 6 7 8 9 10 11• (B) 0 2 3 4 5 7 // 6 8 9 10 11 1• (C) 0 2 4 5 7 9 // 6 8 10 11 1 3• (D) 0 1 2 6 7 8 // 3 4 5 9 10 11• (E) 0 1 4 5 8 9 // 2 3 6 7 10 11• (F) 0 2 4 6 8 10 // 1 3 5 7 9 11

Partitioning

As strict linear ordering became less important in row composition, partitioning the row into sets composed of non-consecutive elements became increasingly important. These create what are known as secondary sets., i.e., sets formed from what may be non-consecutive pcs of the original row. Often these secondary sets are discrete equal units, but Schoenberg also began to use unequal units in his late compositions. By using features of invariance, these secondary sets can be organized and related to each other, overlapping and sharing subsets in a polyphonic texture.

These units may be composed so that their unordered contents are transpositions or inversions of one another, or have the same interval content (IV). This is called isomorphic partitioning.

Linear Set Presentation

Although Schoenberg used the row as a linear/melodic line in the earlier twelve-tone works his later use became more sophisticated. A strict linear statement is relatively rare in his mature compositions and is usually reserved for marking important structural or dramatic points in the composition, e.g., Moses and Aron. However, a linear statement is often given at the beginning in order to present the row in its clearest form, i.e., as an aid to perception.

Contrary to novice beliefs, pitches can be repeated in a row composition. A repetition is not considered different from holding a note, as in voice leading. Notes can be held and repeated but these repetitions should be at the same pitch level (rather than octave displaced). Further, a series subset may repeat, e.g., 1234 1234 567 456 78 78 9AB, again keeping the same pitch level. Although backtracking is permitted, it is the order that is maintained.

The series can be used to form chords where ordering is indeterminant. Thus, the pcs need not occur in any special arrangement (bottom to top, or vice versa). A horizontal presentation may be combined with a vertical one using various row segments.

Rhythm, dynamics, articulation, rests, texture, etc., are normally free in twelve-tone composition. But, to avoid confusion with a tonal conception and to maintain pc equality, there are compositional guidelines. In "total serialization" found in works by Milton Babbitt and Pierre Boulez, all elements of the composition are serialized, including dynamics and articulation.

Although the row was once believed to be rigid in its pc order, it is now used more flexibly. Often Schoenberg would partition his row into segments of three or four notes apiece, and although the order from one segment to another was maintained, the ordering within each segment was juggled. This principle was eventually extended to hexachordal segments. Thus, the linear rigidity of the row became much more flexible and began to function more as a governor of harmony.

Other

Some composers have used cyclic permutation, or rotation, where the row is taken in order but using a different starting note. (12345678 - 34567812)

Schoenberg also sought a way to create an analogue to the key modulations that tonal music contains. This was to provide for temporal unity and variation beyond the surface level. Thus, the first hexachord was treated as an analogue of tonic and was presented at the outset of a composition. Subsequent divergences from and return to this metaphorical "tonic" could then signal important formal divisions. Again, the hexachord becomes an important harmonic unit in this scheme. Although usually atonal, twelve tone music need not be - several pieces by Berg, for instance, have tonal elements.

Harmony in the twelve-tone music of Schoenberg is not arbitrary as is often claimed. In fact, one could argue that harmony was always the primary concern. The hexachord is used as the primary unit of harmony, but secondary sets are also very important. Non-adjacent elements were often combined to create local harmonies that repeat throughout a composition. Schoenberg used almost exclusively IH combinatoriality in his late works, which and regulates the possible harmonic combinations, effectively restricting them to one harmonic profile. Additionally, meter is often affected by the periodicity of this harmonic profile.

Set Theory Primer for MusicPart I. Nonlinear Set

Basic DefinitionsAbdo. see pitch numberdiv, or directed-interval vector, also interval string. The distance between successive (ordered) pcs cycling to an octave. A prime div isthe div of a prime form.Forte Prime. A generalized version of a set that includes its inversion.Index number. The transposition number, in semitones, above a reference pc. P5 would be a transposition up 5 semitones from P0.Interval. The distance between two pitches. In set theory intervals are measured by the number of semitones. Thus, CE is not a major third(M3) but 4 semitones, or simply 4. A minor sixth would be 8.Interval class (ic). The distance between two pitch classes, measured by the shortest distance. C to G may be the interval of 7, but itsinterval class is 5. Thus, the largest ic is the tritone (6).Interval String. see divModulo 12 (mod12). An arithmetic system nearly identical to that of a clock, where 13=1, 14=2 etc. However, in modulo 12 the number12=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 numbergreater than 11.Normal form or normal order. A cyclic permutation of a pc set arranged in ascending order as compactly as possible with respect to thefirst pc. Each pc is represented by a pitch number in the absolute-do system. The normal order of an F major chord would be 590.Pitch class (pc). All pitches with the same name plus their enharmonic equivalents; e.g. all C#s make up a single pitch class. But, Db andBx are also in the same class.Pitch number (pin). Each pc can be represented by a number from 0 to 11 in the twelve-tone system.

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 abdo (absolute-do) notation, where C is always zero (0). In the reldo (relative-do) notation, any pc may be set tozero, usually the first of an arbitrary rotation. Thus, FAC is represented as 590 in abdo, but as 047 in reldo.pcs or pitch-class set. A group of pitch classes.Protoprime. The prime form of a set without including its inversion.Reldo. see Pitch numberUnordered set, (or nonlinear set). A set whose temporal order is irrelevant, as in chords.

1. Identifying a Set Class, the PrimeThe first and most important way that a pc set is identified is by its protoprime, or simply prime. However, sometimes the

normal form, or normal order, is used. The normal order may be considered as a step on the way to the prime. So, we'll start by figuringit. As in the above glossary, the normal order is a cyclic permutation of a pc set arranged in ascending order as compactly as possible withrespect to the first pc. Each pc is represented by a pitch number in the absolute-do system. Thus, the normal order of an F major chordwould be 590. Here are the quickest steps for finding the normal form and then the prime:

1. Eliminate any duplicate pcs. For example, D# C G# F# A C G#, eliminate the duplicate G# and C. Thus, this pc set is initiallylabeled 30869, with the pitch duplications eliminated.

2. Place the numbers in ascending order, 03689.3. Figure the intervals between consecutive pairs of pitch numbers, cycling back to the initial pc. 3-0=3, 6-3=3, 8-6=2, 9-8=1, 0-

9=3. Thus, the intervals are 33213. This is called the directed-interval vector, or div.pins= 0 3 6 8 9div= 3 3 2 1 3

4. Find the index number by locating the largest interval number. In this case, the largest interval number is 3, but there are threeof them. When there are more than one of the largest number, choose the one with the smallest number following it (cyclically). This wouldbe the second 3 in the above example. The pin (pitch number) following this is the index number. In our example the index pin is 6. (If the"smallest" number occurs more than once following a tie, then the next number should be considered using the same criteria, etc.)

5. Arrange the pins ascending from the index number: 68903. This is the normal form. The normal form has very little use andcan be discarded.

6. The prime is figured from the normal form by setting the first pin to zero by transposition. This is done in our example bysubtracting 6. Subtract the same number from all the pins. [6-6=0, 8-6=2, 9-6=3, 0-6=6, 3-6=9]. The result is 02369. This is the prime,which may be simplified to 2369, omitting the superfluous leading zero.

A more elegant (simpler) way to get the prime is to use the div, or interval string. In our example this is 33213. The largestinterval followed by the smallest is the second 3. This points to the next interval as the starting interval for the prime. Therefore, cycling theintervals, we get 21333. The digits should always sum to 12 to complete an octave. If we build a set class from this we get 02369. To getthe Inversion (I), reverse the div: 33312. Then find the inversion's prime div by the same method: 12333. Building a set class from this weget: I=01369.

7. The Forte prime may or may not differ from the protoprime. To get the Forte Prime (after Allen Forte), compare the primewith its inversion. The one which is most compacted toward zero is the Forte Prime. In this case, compare P=02369 with I=01369. Thelatter is most compacted (smallest interval) near zero. So, the Forte Prime is 01369. Remember that Forte Primes do not discriminatebetween major and minor.

C C# D D# E F F# G G# A Bb B0 1 2 3 4 5 6 7 8 9 10 110 1 2 3 4 5 6 7 8 9 A B

Ex 1.Consider the pitch set D#, Bb, F#, D, Ab, A#, G#.Assign pitch numbers, i.e., 3A628A8.Eliminate duplications, including enharmonics, and , i.e., 3A628.Put the numbers in order: 2368A.Compute the cyclic div by subtraction: 13224. (Remember that the last interval cycles the set back to the first pitch.)Find the largest interval (4), the index.Cycle the div startingwith the next interval, in this case the div is already in the required order: 13224.Starting with zero (0), generate the prime: P=01468 .

NOTICE: This is all that is required to identify the set class (The set class is the protoprime). The following are additional steps forcomputing the inversion and the Forte Prime.

To get the inversion, project the div intervals in the reverse order (42231).As before, find the largest interval: 42231.(The index here is 4)Cycle the div from the next interval: 22314Generate the Inversion from this: 02478.If desired, the Forte Prime can be computed by selecting the most compact form fromP and I, which in this case is 01468.

Primes from the KeyboardThe keyboard of a piano, organ, or other electronic keyboard, can be used to simplify the determination of the prime. For

example, a dominant seventh chord, e.g., a C7, can be imagined or played on a keyboard in four possible configurations or positions.

(1) (2) (3) (4)These are the cyclical rotations of this set, commonly known as root, first, second, and third inversions. From these, choose the

most compact form under the hand, i.e., number 2, which encompasses the smallest inverval of a minor sixth or 8 semitones. Using thisform, set the first pc to zero, i.e., using reldo, and figure the intervals in semitones above it; i.e., 0368. This is the prime. This method canbe used for any set. If two or more forms compete with the smallest span, choose the one with the smallest interval from the bass to the nextnote above.Interval String Notation

The most elegant way to represent a set class is with Interval String Notation (ISN), first documented by Ernst-Lecher Bacon inThe Monist, 27:1, October 1917, under the title "Our Musical Idiom". In this system a pc set is represented by a series of intervals (insemitones) that fills an octave. Thus, the minor 7th chord, 037, becomes 345 in ISN, the intervals between the pcs with one more tocomplete the octave. On close scrutiny it will be ascertained that a set class is really a directed-interval vector, or div, rather than a pitch-class set. A new catalog of sets may be constructed with this notation in which set identity is very elegant and economical, without the needfor set names; e.g., C7 chord, 0368 ( 4-27B), becomes 3324. The half-diminished seventh (0258 or 4-27) becomes simply 2334. In ISN, thedifference between major (435) and minor (345) chords are clearly nonequivalent, just as are the dominant seventh and half-diminishedseventh. The intervals in ISN should always add up to 12.

Why Does a (proto) Prime Differ from the Forte Prime?Allen Forte's "prime forms" are actually combined pairs of protoprimes (or simply, primes) that are not perceived or conceived as

such in our music. As a simple example, 047, the major chord, does not appear in Forte's table, but is subsumed into 037, the minor chord.(It is important to recognize that set theory calls these chords "inversions", which is not the same as the traditional concept of chordinversion as determined by the bass note.) Thus, it is impossible to distinguish these "inversions" in Forte's system, e.g., impossible todistinguish major chords from minor. This problem expands to all distinct pairs of prime inversions. The dominant-seventh (0368), asanother example, is subsumed into the half-diminished seventh (0258), making them indistinguishable. The same is true for more complexsets. The Table of Set Classes retains all the original Forte set-names, but identifies each distinct inversion as a "B" form, an identifyinglabel that is suffixed to the Forte name for each inversion. Thus, these additional primes are reinstated to their proper position in thepantheon of chords, distinct, yet related, to their inversions. In no way does this subtract from the information of set theory, nor does itchange Forte's foundational sets. Rather, it embraces them and expands upon them; i.e., more information is provided -- information that isomitted by subsumption of inversions into the same set class. It also has the additional benefit of simplifying the determination of the primeform by elimination of the steps that include the inversion, normal form, "best normal form", which are unnecessary and have little use.

It is maintained by some theorists that the reduction in the Forte primes is valid because of the "atonal" context for which settheory was designed; i.e., major and minor chords are the same in an "atonal" context. But, the division between tonality and atonality" isitself questionable, and objectively indefinable, just as is "atonal music". (See my web essay Tonality, Modality and Atonality. Schoenberghimself maintained that "atonality" is actually a misnomer and is indeterminate, . Even the concept "pantonality" can only be definedsubjectively. Forte himself uses set theory to analyze Stravinsky's Rite of Spring and other at least marginally tonal music, such as inScriabin's late music. Major and minor chords are found in the Rite, and they are rendered indistinguishable by Forteian analysis. I wouldcontend that these chords are not heard as identical sonorities in this, or any other context. They are simply not normally perceived asequivalent set classes. The division between tonal and and atonal music is very unclear. The problem is exacerbated in the "atonal" work ofSchoenberg, Ives, Satie, and others.

2. Identifying the Set Name and Set ClassThe set name is found in the Table of Pc Sets. The Prime, 02369, is found as set number 125 with the set name 5-31B. The Forte

Prime, 01369, is 5-31.*Definitions:

Set name. A tag used to identify a pc set. Allen Forte's set name consists of two numbers separated by a dash. An example is 4-27. The number before the dash indicates the cardinality of the set (the number of pcs it contains). The number after the dash, the ordinalnumber, indicates a catalog number determined by its alphanumeric order in the complete list of sets. Another way to identify a set is by itsprime form, which may also be its set name; 047, the major chord may be represented as simply 47 and minor as 37.

Set class. All the pc sets represented by a single prime form, including transpositions. In Forte's system the set class also includesthe inversion. Thus, 037 and 047 are in the same set class.*Note: another way to find the Forte Prime is to look up the Prime in a table of sets (try this link), and note its set-name (5-31B). Find thesame set-name without the B at the end (5-31). This is the Forte Prime, 01369.

3. Interval VectorInterval vector (iv). The total ic content of a pc set. This is normally represented by an array of six digits, where the first

indicates the number of semitones, the second the number of whole tones, the third the number of ic3, etc. The last digit indicates thenumber of tritones. The iv for a major chord is 001110.

Taking our example 02369, we can figure the interval vector using the following method:1. Subtract the first pin from the following pins: 23692. Subtract the second pin from the following pins: 1473. Subtract the third pin from the following pins: 364. Subtract the fourth pin from the last: 35. The intervals are: 2369147363. These need to be converted to ics; i.e., any number over 6 must be converted to a number less

than 7. This is done by subtracting any number over 6 from 12, thereby inverting the interval. There are two such numbers here: 9 and 7,which when subtracted from 12 become 3 and 5 respectively. These are their ic numbers. So, the original list becomes 2363145363.

6. Tally the number of each ic and place each number in the corresponding position of the iv array; i.e., there is one 1, one 2, four3s, one 4, one 5, and two 6s. Therefore, the iv is 114112.

Another way to find the iv is to look it up in the table. Notice that inversionally related sets have the same iv; thus, the FortePrime also has an iv of 114112.

4. Set RelationsComplement. All the pcs that are not in a given pc set.Subset Relation. Two sets are so related when one set is included within the other. The sets must be of differing cardinalities.Similarity Relation. Sets of the same cardinality may be related by their similarity. There are several different types, the most important ofwhich are described below.Directed Interval. The distance between two pins that are placed in an order. E.g., the di of E to C is 8, whereas the di from C to E is 4.Directed Interval Vector (div). the di between a series of pins; e.g., the div of ECGAC is 8723.

EquivalenceSince pc sets are not bound by the octave, two pc sets are equivalent if they map under rotation and/or transposition. EGC (470)

and CEG (047) are equivalent by rotation. GBDF (7B25) and FACEb (5903) are equivalent by transposition.The first operation involvesrotating the pitch numbers as in a circle. The second, transposition, is a matter of addition; add 2 to 5903 and the result is 7B25. Thus, pcsets are equivalent by these two operations. Sets FBGD (5B72) and CEGBb (047A) are equivalent after both operations, rotation andtransposition.

Inversion and Z-Related SetsInversion is achieved by projecting intervals of a set in the opposite direction. Mathematically, this is acheived by subtracting the

pins from 12 (the modulus). As an example, 047, the major chord is inverted by subtracting its pins from 12 to give: 085. When placed inprime form 085 becomes 037. Thus, 037, the minor chord is the inverse of 047, the major. In Forte's system inversionally related sets areequivalent and are made indistinguishable. Thus, 047 is subsumed into 037. In this system inversionally related sets are identified asdistinct but related by mutual inversion. Inversionally related sets always have the same interval vector.

Z related sets are sets that have the same interval vector. Additionally, when two sets have the same name except for the B endingon one, the sets are inversionally related. This makes it easy to establish these relations in the tables. Z-related sets are accompanied by anextension on the set name with two dots followed by an ordinal number. This identifies the ordinal number of another set having the sameinterval vector.

Mirror SetsA mirror set is actually not a relation between sets, but a relation that a set may have within itself. Such a set results in the equivalent setwhen inverted and is thereby called amirror set. All such sets are indicated in the table of sets with an asterisk after their set names. Thus,such a set has no distinct inverse, but instead, each is its own inverse.

Subset RelationWhen one set is included within another, they are said to be in the subset relation, also known as the inclusion relation. This

may be abbreviated S for the subset relation. A familiar example of this is the incomplete dominant-seventh chord. BGF is a subset ofGBDF. BDF, the diminished chord, is also a subset of the dominant seventh.

The subset relation is the only relation that two sets of differing cardinalities may have. However, Forte has also described specialset complexes that relate groups of such sets. The first is called the set complex K, where a set OR its complement are in the inclusionrelation (superset or subset) with all the other sets in its group. The second is called the set complex Kh, where a set AND its complementare in the inclusion relation with all the other sets in its group. The latter is more selective. The set to which all the others are so related iscalled a nexus set, which is used as a reference. A table of the set complexes Kh may be found in SAM, appendix 3.

Not all subset relations are equally significant. For instance, the statement that the major chord is a subset of the 12-note set,although true, is insignificant, because all sets are subsets of the 12-note set; i.e., the statement is not discriminating. The larger the supersetis, the less significant are its subsets. A 3-note subset is more significant if it is a subset of a 4-note set, than it is if it were a subset of an 9-note set. This observation leads to a method for establishing subset significance. It is suggested here that a subset whose cardinality is morethan half of the superset is more significant than one that is not. In this way one may make a distinction between significant-subsets(mapping>50%) by labelling them with an S, and a less-significant subset (mapping<=50%) by labelling it with a +.

How to Determine Subset RelationsTo determine if one set is contained in another place both in prime form. In some cases, the pins of the smaller set will be the

same as those of the larger set and can, thus, reveal the subset relation. But, most of the time this will not be the case.Figure the intervals between successive pins of both sets. These intervals should, as usual, be considered cyclic. If thediv of the

smaller set can be aligned with the larger or with successive sums of the larger, then the subset relation applies.Example 1. GBF, prime form 026, has a di content of 246 (cyclically). GBDF, prime form 0368, has a div of 3324 (Considered

cyclically the div is 3324332... etc.). 246 aligns with the latter starting with the 24. The 6 is a sum of the two 3s; i.e., 2433 = 246.Therefore, the two sets are in the subset relation.Example 2. Compare 023568 (div=212124) with 0134679A (div=12121212). The div of the first set aligns with the second when startingwith pin 7 [212124 = 21212(2+1+1)]. Therefore, they have the subset relation.

Complement RelationTwo sets are in the complement relation when one contains all the pcs that are excluded from the other. For example, a C major

scale excludes "black keys". The five black keys are its complement, and vice versa. Thus, a 7-note set has a 5-note complement, a 4-noteset has an 8-note complement, etc., where the complement cardinalities always add up to 12.

In the table of sets, sets with complementary cardinalities and the same ordinal number are complements. But, if one has a Bending, the complement does not. Exceptions to this are indicated with a< sign in the table of sets. This sign indicates that the complementhas the same name ending.

All hexachords have hexachord complements. Many are there own complements, except for the B ending. Those that are not theirown complements identify their complements by the ordinal number that follows two dots in the set name. These dots also identify Z-sets,those that have the same interval vector.Every set is in the subset relation to its complement except for the complementary couple 7-Z12/5-Z12.How to Determine if Two Sets are in the Complement Relation

The cardinality of the two sets must add up to 12. Write the pins of the complement of one of the sets and place it in prime form.If this matches the other set, the two sets are complements.

Similarity RelationsSimilarity relations are used to describe the relationships between sets of the same cardinality. These are based upon pc similarity

and ic similarity.Allen Forte describes four basic types of similarity relations, which he designates Rp, Ro, R1, and R2. Rp is determined by pc

similarity, and the other three are determined by ic similarity.Rp, maximum similarity of pc, exists when the two sets of cardinality C have at least one common subset of cardinality C-1,

which is the same as saying that there is one unmatched pc.Forte remarks that Rp by itself is "not especially significant" because it is too common, and leaves it at that. By Forte's Rp

criteria, for example, 014 would be just as similar to 047 (major chord) as is 037 (minor chord). And, 014 is just as similar to 037 (minorchord) as is 036 (diminished chord). This does not agree with the way these chords are commonly perceived.

For a revision of Rp that would agree more with our perceived notions of similarity, one could require that the unmatched pc pairbe within a semitone of a match. This makes the criteria for maximal similarity more selective, more distinguished, and corresponding moreclosely to our perceived notions of similarity. This type of similarity could be designated as simply P.

Ro, called "minimum similarity", exists when two set of cardinality C have no corresponding interval vector digits in common.R1 and R2 are known as maximum similarity of interval class. Such is the case when four out of their six iv digits are equivalent.

The remaining two dissimilar digits determine the difference between R1 and R2. If these are the same numbers but switched in position,then the relation is R1. If the two dissimiar digits are simply not equivalent, even if switched, the relation is R2.

When are Sets Maximally Similar?1. They must be the same cardinality C.2. They have four out of six iv digits corresponding and parallel (same position). (R1/R2)3. They have all but one pc correspondence. (Rp)Maximal similarity exists when two pc sets of the same cardinality can be mapped to one another, with the exception of one pc in

one of the sets. Additionally, the interval vectors of the two sets must have four out of six matches.These may be called the X relation (Forte's R1) when the unmatched digits are switched and the O relation (Forte's R2) when

they are not. (The reasons for changing Forte's symbols become apparent when a single letter is needed on a "Relations Triangle".Additionally, X and O are easier to remember, because they neatly describe their respective relations in the shape of the letters.)

The R RelationAnother type of maximal similarity, called simply the R relation, is more selective, and is based upon a perception model that is

statistically determined. By this criteria it is assumed that the perception of the similarity of small sets is easier than is the perception of thesimilarity of large ones; i.e., the similarity of, say, 9-note sets would be more difficult to perceive than their complements, 3-note sets.

Thus, a formula is constructed to simulate this difference in perception. To satisfy R the sets must meet the following criteria:1. Two pc sets of the same cardinality can be mapped to one another, with the exception of one pc in one of the sets, which must

be within a semitone of a match with the unmatched pc of the other set.2. There must be a minimum of interval correspondence T, where T equals the total number of ics corresponding in the two sets.

T must be equal or greater than SC/8, where S is the sum of the ics in the set cardinality (the sum of the digits in the iv), and C is thecardinality.

As an example, compare the following two sets, given in prime form with their interval vectors:013578 232341013579 142422C=6, the cardinality. Comparing the primes, only one pin pair is unmatched (8 and 9), and they are a semitone apart, which

satisfies condition 1. Then observing the ivs, the sum (S) of the ics in this cardinality is 15. (This is determined by adding the numbers ineither iv. All sets of cardinality C will have the same S.)

To determine T: The number of ics common in each iv position is equal to the smaller number (comparing each pair of digits inthe same position), and 132321 is the result. Adding these together gives T, which is 12 in this case.The number 8 is the cardinality chosen to represent a reasonable limit to the perception of R; i.e., sets having cardinalities greater than 7cannot have the R relation (although they may have other maximally similar relations).

Since SC/8=11.25 and T=12 is greater, the two sets of our example meet the criteria for the R relation.

PC InvarianceVarious operations are commonly performed on pc sets. These include rotations (CEG becomes EGC, GCE, ECG, etc.) and

registral displacement. These operations have no effect on pc content. That is, they remain invariant under these operations.Two operations that are commonly performed on pc sets that can alter pc content are (1) transposition and (2) inversion.

Transpositional PC InvariancePc sets often maintain some pc invariance after a transposition. A common example is the whole-tone scale, 02468A, or 6-35*.

When this set is transposed by 2 semitones all of its pcs are held invariant. This would, of course, have important compositionalconsequences. It would be important, then, to know when pcs are held invariant under the operation of transposition. We can determinethis by examining the iv of a pc set. 6-35* has an iv of 060603. The number that appears in each position indicates the number of pcs heldinvariant when transposed by its respective ic. Thus, the 6 that appears in ic position 2 reveals that when this set is transposed by 2semitones (t2), 6 pcs (all) are held invariant. Since this is an ic transposition it may be either up or down, i.e., t2 or t10. A zero in the firstposition reveals that when this set is transposed by 1 semitone no pcs are held constant (also true for t11). The same is true of t3 and t9.There is a 6 in the ic4 position, meaning that at t4 or t8, 6 pcs (all) are again held constant. The 3 in the ic6 position, however, may seempuzzling at first. But, recall that 6 is its own inversion; therefore, the number of invariant pcs is double the number in the ic6 position; i.e.,t6 also results in 6 invariant pcs).

As another example, consider set 6-7*, 012678, with iv=420243. The iv tells us that there will be 4 pcs invariant when t=1 (ort11), 2pcs invariant when t=2 (or t10), no pcs invariant when t=3 (or t9), 2 pcs invariant when t=4 (or t8), 4 pcs invariant when t=5 (or t7),and 6 pcs invariant when t=6.

Inversional PC InvarianceInversion can and often does lead to pc invariance. But, this operation needs to be considered in combination with transposition,

i.e., TnI, representing a transposition n of the inversion. The easiest way to determine this type of pc invariance is to form an addition tablewith the set represented horizontally and vertically. Let us take 2478 as an example.

Notice that the set is placed at the top, horizontally, and down the left side, vertically. The numbers within the table are the sumsat the intersections of the pins. By tallying these numbers we can ascertain the invariant pcs under TnI. For example, the two 9s reveal thatthere are 2 common pcs at T9I. Further, the table tells us which pcs are invariant under this T, namely those that create the intersection: 2and 7. The two 4s indicate 2 common pcs at T4I, and they are 2 and 8, etc.

2 4 7 8

2 4 6 9 A

4 6 8 B 0

7 9 B 2 3

8 A 0 3 4

Part 2. 12-Tone Composition & Serial or Ordered Sets

IntroductionThe construction of the set of twelve tones derives from the intention to postpone the repetition of every tone as long as possible.[Arnold Schoenberg, "Composition with Twelve Tones", Style and Idea, 246.

Schoenberg's purpose in his new compositional method of twelve tones was to supplant tonality with a new means of order, ofwhich he approved calling "pantonality", and disproved of the term "atonality". " Pantonality means an equality of tones, effectively ademocracy of tones. Thus, the basic principle behind composition with twelve tones is the even distribution of pcs with none dominatingthe others. Repeating some tones makes them sound more important, and hence contradicts the goal.

One way to guarantee the "postponement" of tone (pc) repetitions is to place all twelve in an order where none is repeated. Thisis called a twelve-tone row, or series. These pcs may be represented as letter names (C, D, F#, etc.) or by pitch numbers. The latter enablesmathematical operations on the set which facilitate various transformations and constructions. Thus, the number representation has definiteadvantages over the letter notations.Tone Row. A group of pcs (usually the 12 chromatic pcs) placed in a particular order to be so used in a composition.Pitch number (pin). Each pc can be represented by a number from 0 to 11 in the twelve-tone system.

How to Construct a 12-Tone MatrixA 12-tone matrix is a concise way of representing all 48 forms of a row on a grid (matrix). As an example, take the 12-tone row

G A# B F E C# C A G# D D# F#. This may be represented in pins as 7AB541098236 (hexadecimal).Since we will be using the relative-do (reldo) system here, set the first PC to zero by transposition (subtract 7): 034A9652178B.

This sets the pc G to zero, and the set that begins with it is Po. (Note that matrix construction would be different in the absolute-do [abdo]system.) To construct a reldo matrix first write the row horizontally. Then write its inversion vertically down the left side, starting with 0.Subtract the horizontal numbers from 12 for each successive pc of the inversion down the left side.

Now fill in the rest of the matrix by writing the rest of the transposed prime forms horizontally starting with each number thatappears in the first column. The second row will be P9, the third row P8, etc.

Table 1. The Row in Reldo

All 48 forms are represented on this matrix with inversions reading down from the top. Retrogrades are read from right to left,and RI forms are read from bottom to top. The first number becomes the index number of the transposition in reldo.

The following axioms apply to the Reldo system:Axiom 1. The Prime (Po) form starts with zero (reldo) which sets the index numbers for all other forms.*Axiom 2. Transpositions are determined by adding a transposition number to the pins (mod12).Axiom 3. The Inversion (I) pins are determined by subtracting the prime pins from 12.Axiom 4. The Retrograde (R) pins are the prime pins in reverse order.Axiom 5. The Retrograde Inversion (RI) pins are the reverse of the inversion pins.Axiom 6. The index or transposition number is equal to the first pin number of each set form.**It should be noted that the other system in use (abdo) calls the last pc of Po the zero transposition index of Ro.

An economical way to represent the 48 forms is with their divs (directed-interval vectors). For example, the row for Schoenberg'sSeptet, Op. 29 is 376A2B098451. The div for this set is 4B44919B818. This will be the div for all transpositions of the prime and is a moreelegant representation of a set. The inversion div is simply the pin complements of 12; i.e., 81883B314B4. RI is the reverse div of P:818B91944B4, and R is the reverse div of I: 4B413B38818 . A complete matrix of the divs can be represented:P: 4B44919B818I : 81883B314B4R: 4B413B38818RI:818B91944B4

0 3 4 A 9 6 5 2 1 7 8 B

9 0 1 7 6 3 2 B A 4 5 8

8 B 0 6 5 2 1 A 9 3 4 7

2 5 6 0 B 8 7 4 3 9 A 1

3 6 7 1 0 9 8 5 4 A B 2

6 9 A 4 3 0 B 8 7 1 2 5

7 A B 5 4 1 0 9 8 2 3 6

A 1 2 8 7 4 3 0 B 5 6 9

B 2 3 9 8 5 4 1 0 6 7 A

5 8 9 3 2 B A 7 6 0 1 4

4 7 8 2 1 A 9 6 5 B 0 3

1 4 5 B A 7 6 3 2 8 9 0

An even more elegant way to represent the set is by using thedcv (directed-class vector). The numbers over 6 are converted totheir complements as negatives. The negatives are here represented as smaller numbers on the matrix. They may be represented in longhand by placing a small minus sign before each negative number as in example 1 above.P: 41443131414I : 41443131414R: 41413134414RI:41413134414

With this last representation, it becomes easier to memorize the set, especially if it is partitioned into segments and visualized or"auralized": 4144, 3131, 414. Thus, one could start with any note and generate a transformationwith this dcv; e.g., Po = CED#GB,G#AF#F, C#DA#. Inversional forms simply switch positive and negative numbers:4144, 3131, 414. Thus, I2= DA#BGD#, F#FG#A, C#CE. Retrograde-Inversional forms (P/RI and I/R) reverse the order of`div.

Theorum 1. The div of a row inversion (I) is determined by subtracting the prime (P) div from 12.Theorum 2. The div of RI is the reverse of P.Theorum 3. The div of R is the reverse of I.Theorum 4. If the div numbers of P are complements of 12 about the central axis of symmetry, P=R and I=RI.Theorum 5. If the div numbers of P are equal about the central axis of symmetry, P=RI and I=R.

How to Select Organizational Properties of a Row: Derived RowsIf greatest unity is desired, a row may be constructed to have recurring subsets within the series, e.g., in four 3-note groups. Such

a constructed row would be used to explore the motivic and/or harmonic properties of the 3-note figures.Such a row is the one in Table 1. This row has very special properties: 034A9652178B. Its div is 316B9B9B613. Notice that the

numbers are symmetrical about the central B. Since RI has the reverse div of P and the div is symmetric, then P=RI. One can prove this bycomparing Po with the RI that starts with 0 (reading up)(RIo). The same relationship holds between I and R; so I=R. Comparing I4 with R4on the matrix, they are identical.

But there are even further principles organizing this row. If the first 3-note figure is taken as a little prime (p0), the secondtrichord is rA, the third trichord is i5 and the last is ri7! Thus, the row itself consists of an initial trichord with transformations derived fromit to create the rest of the row. Additionally, each hexachord is inversionally symmetric; the second is the inversion of the first. The row isalso combinatorial (Po+I5 and Po+IB). This is called a derived row and has profound implications for the composition that uses this row.Additionally, each trichord, if its unordered pc content is considered, consists of the Forte prime 014; i.e., they are identical in ic(unordered) content.

This row also has a most remarkable property if considered cyclic. All the trichords of all transformations maintain a constantordering. Therefore, effectively, P=I=R=RI, often considered to be an impossible condition. Also, each trichord is followed by its ownretrograde. Po, RI

This row demonstrates an extreme condition of maximum internal organization and therefore contains maximum repetition.Repetition eases comprehension. Anton Webern was fond of using derived rows in his compositions, which is a probable reason that hismusic is more accessible than that of Schoenberg's. (Another is that Webern's music is generally thinner in texture, hence more transparent,and shorter in length than Schoenberg's.)

Theorum 6: There is no transposition of P that is congruent with I (nor R with RI) unless the row is cyclic.Theorum 7: Interval classes are invariant after Inversion (TnI).

Unity vs Variety in Row Structure: All-interval Rows"Repetition is the means by which this unity, and comprehensibility, is achieved.... All formal construction is built upon it." [AntonWebern, The Path to the New Music, 17]"Composition with twelve tones has no other aim than comprehensibility. [Arnold Schonberg,Style and Idea, 103]

Derived rows are definitely an advantage if one is striving for the greatest amount of unity in a composition. Webern's chiefconcern was unity; thus, his rows are mostly of the derived type. For Schoenberg, however, variation was at least as important acompositional concern as unity. And, sometimes a composer prefers variation over repetition. In this case, a row might be chosen that has aminimum amount of internal repetition and maximum variation. Such rows are called 11-interval 12-tone rows, or "all-interval rows". Inthese rows all eleven intervals (di) are used without any repetition, assuring maximum variety.Such a row is found in Alban Berg'sLyric Suite: 5409728136AB. However, not all these rows have the same degree of variation, Forexample, Berg's row has a symmetrical dcv which makes P=R and I=RI. Here is a list of some all-interval rows expressed as div:

(1) B89A7652341 (Berg)(2) 4B295681A37(3) 529A16473B8(4) 453126AB978(5) 1432567A98B(6) 523146B897A(7) A195268B374(8) B29476583A1(9) 1A98567432B

CombinatorialityIf only one set is used the order of notes can be easily maintained without ambiguity. But, if two row forms are presented at once,

say in counterpoint, pcs will be repeated unless special procedures are followed. An aggregate is a collection that includes all twelve pcs.One of the aims of twelve tone composition is to complete an aggregate before repeating pcs, even when row forms are combined.

Hexachordal combinatoriality means that hexachords of the same order can be combined without pc duplication, or, in otherwords, when they complete an aggregate. Hexachordal combinations are the only ones that we will consider here, because they are the mostfrequently used. Other types are possible but are beyond the scope of this exposition.

All series are combinatorial with their retrograde at a transposition that corresponds with the last pin of Po (abdo). This isreferred to here as Ro combinatoriality. Since this is automatic, there is nothing special about it. Webern uses retrograde combinatorialityin his Piano Variations, Op. 27. At the beginning he uses Po+Ro where he aligns the first hexachord of each followed by their secondhexachords. This results in no pc duplication before all 12 pcs sound. The same result will hold when I is combined with RI at the propertranspositions. Since Ro combinatoriality is a property of all 12-tone rows, this form is trivial and will not be studied here. It is possible,however, for another transposition of R to be combinatorial, and this will be referred to here as Retrograde combinatoriality. This form isnot trivial and must be dealt with. R combinatoriality is possessed by any hexachord that has a 6 as one of its iv (interval vector) entries, ora 3 in the tritone position.

The prime form of a set may also be combinatorial with itself at some transposition. This may be called transpositional or Primecombinatoriality, or TH. And, a set may be combinatorial with its inversion. The last type, called hexachordal-inversional combinatoriality,or IH, has received considerable attention among 12-tone composers. A row may also have RI combinatoriality.

Composition of a row with TH is fairly easy. If for instance, one constructs the initial hexachord from some ordering of the firstsix notes of a chromatic scale, the second hexachord can be made from an ordering of the other six. When the whole row is transposed by atritone, i.e., P6 , it will be combinatorial with Po. Transpositional combinatoriality is possessed by any hexachord whose iv contains a zero.

Over half of all twelve-tone rows are IH combinatorial. The combinatorial properties can be determined from examination of thefirst unordered hexachord of the row. Combinatoriality does not depend upon how the notes are ordered but simply by their content.Thereby, these hexachords can be identified in the complete Table of (unordered) Pc Sets. On this list "comb" means combinatorial. Theforms of combinatoriality are listed following "comb". The transposition numbers apply only to the hexachord in prime form. The tableshows all forms of hexachordal combinatoriality except for the Ro type, which all hexachords possess.

Theorum 8. A necessary condition for IH is that the sum of pins of the same order number must be odd.

IH combinatorial row forms by themselves do not guarantee the avoidance of pitch class duplications. This also depends uponthe rhythmic alignment of the rows in a composition. Schoenberg avoided pc duplication in his late works by rhythmically aligning therespective IH hexachords. To extend this principle even further he created secondary sets that formed aggregates when two row forms wereused in succession.

InvariantsThrough the use of operations of transposition, I, R, and RI, certain properties of the original set may be preserved. These are

called invariants and take the form of pcs, subsets, and intervals (di) or interval classes (ic). Additionally, combinations of row forms mayresult in invariant sets of various sizes.

Theorum 9. Pc dyads are held invariant between inversionally related forms (TnI) if the transposition number is the sum of pins of thesame order. (e.g., Webern Op. 27/2)

Linear Set PresentationAlthough Schoenberg used the row as a linear/melodic line in the earlier twelve-tone works his later use became more

sophisticated. A strict linear statement is relatively rare in his mature compositions and is usually reserved for marking important structuralor dramatic points in the composition, e.g., Moses and Aron. However, a linear statement is often given at the beginning in order topresent the row in its clearest form, i.e., as an aid to perception.

Contrary to novice beliefs, pitches can be repeated in a row composition. A repetition is not considered different from holding anote, as in voice leading. Notes can be held and repeated but these repetitions should be at the same pitch level (rather than octavedisplaced). Further, a series subset may repeat, e.g., 1234 1234 567 456 78 78 9AB, again keeping the same pitch level. Althoughbacktracking is permitted, it is the order that is maintained.

The series can be used to form chords where ordering is indeterminant. Thus, the pcs need not occur in any special arrangement(bottom to top, or vice versa). A horizontal presentation may be combined with a vertical oneusing various row segments.

Rhythm, dynamics, articulation, rests, texture, etc., are normally free in twelve-tone composition. But, to avoid confusion with atonal conception and to maintain pc equality, there are compositional guidelines. In "total serialization" found in works by Milton Babbittand Pierre Boulez, all elements of the composition are serialized, including dynamics and articulation.

Although the row was once believed to be rigid in its pc order, it is now used more flexibly. Often Schoenberg would partitionhis row into segments of three or four notes apiece, and although the order from one segment to another was maintained, the orderingwithin each segment was juggled. This principle was eventually extended to hexachordal segments. Thus, the linear rigidity of the rowbecame much more flexible and began to function more as a governor of harmony.

PartitioningAs strict linear ordering became less important in row composition, partitioning the row into sets composed of non-consecutive

elements became increasingly important. These create what are known as secondary sets., i.e., sets formed from what may be non-consecutive pcs of the original row. Often these secondary sets are discrete equal units, but Schoenberg also began to use unequal units inhis late compositions. By using features of invariance, these secondary sets can be organized and related to each other, overlapping andsharing subsets in a polyphonic texture.

These units may be composed so that their unordered contents are transpositions or inversions of one another, or have the sameinterval content (IV). This is called isomorphic partitioning.

Twelve-Tone TonalitiesSchoenberg also sought a way to create an analogue to the key modulations that tonal music contains. This was to provide for

temporal unity and variation beyond the surface level. Thus, the first hexachord was treated as an analogue of tonic and was presented atthe outset of a composition. Subsequent divergences from and return to this metaphorical "tonic" could then signal important formaldivisions. Again, the hexachord becomes an important harmonic unit in this scheme.

Harmony and MeterHramony in the twelve-tone music of Schoenberg is not arbitrary as is often claimed. In fact, one could argue that harmony was

always the primary concern. The hexachord is used as the primary unit of harmony, but secondary sets are also very important. Non-adjacent elements were often combined to create local harmonies that repeat throughout a composition. Schoenberg used almostexclusively IH combinatoriality in his late works, which and regulates the possible harmonic combinations, effectively restricting them toone harmonic profile. Additionally, meter is often affected by the periodicity of this harmonic profile.

Atonality

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