Introduction to Post-Functional Music Analysis:
Set Theory, The Matrix, and the Twelve-Tone Method

Robert T. Kelley
2002


The essential part of this essay to read in order to understand the purpose of the Set Analyzer is The Basics of Atonal Theory. The essential parts of this essay to read in order to understand the purpose of the Matrix Generator are The Twelve-Tone Method, and The Matrix. Combinatoriality is discussed under Advanced Techniques. The essential part of this essay to read in order to understand the purpose of the Contour Analyzer is Contour Analysis. However, I recommend reading the entire document, especially if you aren't familiar with twentieth-century music.

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The Basics of Twelve-Tone Atonality

Two Equivalence Assumptions (Pitch Class Equivalence):

  1. Octave equivalence
  2. Enharmonic equivalence

Octave equivalence has been an assumption in tonal music, and it does not require extra attention here except to mention that the terminology used in "atonal" theory is much more specific. The term "pitch" refers to a specific tone with a specific frequency. The term "pitch class" (abbreviated "pc") employs the notion of octave equivalence and refers to any member of the set of pitches of the same "name" (i.e. C-sharp).

When looking at the non-diatonic relationships among the twelve chromatic pitches, it can be useful to look at the chromatic scale as a mathematical group (the Z-12 additive group). For reasons of convenience, integers are assigned to the chromatic scale members. One reason for this is that it simplifies the naming of pitch classes which have historically been named according to their tonal function. A-sharp was seen as a theoretically different entity from B-flat. Music that does not use this fundamentally tonal distinction between chromatic scale members that are enharmonically the same is easier to analyze without the confusion of this system. This integer system of note naming employs the notion of enharmonic equivalence (i.e. B, C-flat, and A double-sharp are all pc 11), and is organized thus:

PC#Names
0C,B#,Dbb
1C#,Db,B##
2D,C##,Ebb
3D#,Eb,Fbb
4E,Fb,D##
5F,E#,Gbb
6F#,Gb,E##
7G,F##,Abb
8G#,Ab
9A,G##,Bbb
10*A#,Bb,Cbb
11*B,Cb,A##

*The numbers 10 and 11 are often represented with letters so that they are not confused with a "1" followed by a "0", or two "1"s in a row. The letters traditionally used for 10 and 11 in base-12 arithmetic are "A" and "B", but some theorists object to these because they are already symbols in common use for designating musical pitch classes (pc 9 and pc 11, or, in the German system, pc 9 and pc 10). Some theorists have used "T" for 10 and "E" for 11, but, once again, "E" is already a note name. "T" and "L" are used in one recent dissertation; and, in my own writing, I have often used "X" and "Y" ("X" is the Roman numeral for 10), although this precludes the use of "X" and "Y" as variable names for algebraic constructions.

The numbering is arbitrary in that 0 could just as easily be assigned to A-flat as it is to C. However, the use of integers 0 - 11 is not arbitrary, as it makes the mod-12 arithmetic that is characteristic of the mod-12 group possible and applicable to the musical scale.

Pitch-Class Set Equivalence

Collections of pcs (harmonic or melodic) can be analyzed according to "Set Theory" as proposed by Allen Forte. Forte assumed two additional types of equivalence related to collections:

  1. Transpositional equivalence
  2. Inversional equivalence

Transpositional equivalence has also been in place in tonal theory. Any sonority (collection of pcs, or "pc set") can be transposed to another pitch level and retain its character (though the function may change in tonal music). In other words, an F-major triad and an A-major triad are highly related in sound, and thus equivalent at some level. The operation of addition mod-12 on the pitch classes in the set corresponds to transposition of the chord or melody represented by the set.

Inversional equivalence is the most problematic of all of the equivalence assumptions made so far. If the collection is flipped upside down, it retains the same interval content as before, however the chord also can sound somewhat different. For example, if a major triad is inverted, it becomes a minor triad. There is equivalence because they both contain one perfect fifth (fourth), one major third (minor sixth), and one minor third (major sixth). However, in most musical contexts they are distinguished because of their difference in sound rather than their equivalence. So in the case of inversional equivalence, the equivalence will be assumed for the purposes of identifying pc sets by their equivalence classes, but it may often be more useful to distinguish between a sonority and its inversion.

Prime Form, an Expression of Sets' Equivalence

According to these rules of equivalent intervallic content, Forte devised a list of all possible collections of chromatic pitches. The number of possible combinations without equivalence classes is astronomical. However, Forte devised a "lowest ordering" that is considered the standard form for all members of an equivalence class (set class). All members of any equivalence class can be reduced to this prime form by a single method. This results in an ascending version of the collection's members within the span of an octave that is transposed to 0 (the first pc in the collection is 0) and is rotated and inverted to be "left-packed". The entire list of Forte sets can be found online at Larry Solomon, The Table of Set Classes. One can determine the set class of any collection of pitch classes using the Set Analyzer Applet. The technique for determining the equivalence class of a pc set (without the use of the Set Analyzer) is to follow the technique given below, and then look up the result in the table. (If you cannot find the prime form in Solomon's table, it is because Solomon uses Forte's original method of finding prime form, which differs slightly from the more algorithmically consistent method given here. Please use this method if looking up the prime form on Solomon's table.)

Procecure for finding Normal Form (from Straus, see Bibliography):

  1. Excluding doublings, write the pitch classes ascending within an octave. There will be as many different ways of doing this as there are pitch classes in the set, since an ordering can begin on any of the pitch classes in the set.

  2. Choose the ordering that has the smallest interval from first to last (from the lowest to highest).

  3. If there is a tie under Rule 2, choose the ordering that is most packed to the left. To determine which is most packed to the left, compare the intervals between the first and second-to-last notes. If there is still a tie, compare the intervals between the first and third-to-last notes, and so on.

  4. If the application of Rule 3 still results in a tie, then choose the ordering beginning with the pitch class represented by the smallest integer. For example, (A, C#, F), (C#, F, A), and (F, A, C#) are in a three-way tie according to Rule 3. So we select [C#, F, A] as the normal form since its first pitch class is 1, which is lower than 5 or 9.

Procedure for finding Prime Form (from Straus):

  1. Put the set into normal form (see above).

  2. Transpose the set so that the first element is 0.

  3. Invert the set and repeat steps 1 and 2.

  4. Compare the results of step 2 and step 3; whichever is more packed to the left is the prime form. (Use the procedure for determining "left-packedness" given above in Rules 3 and 4 of finding Normal Form.)

Some important terms and concepts in atonal theory that are not explicitly mentioned above are briefly defined in the Post-Functional Theory Terminology glossary. The analysis of music based on parsing melodies and chords into these small collections of pitch classes is useful in non-functional music where no serial process is involved, and also it can be useful in analyzing the construction of a particular tone row and the atonal results of its use in a complex work. The next section of this brief tutorial describes the history, function, and structure of the tone row.


The Twelve-Tone Method

One of the most dominating ideas in twentieth century music has been the development of techniques for systematic ordering of musical events. The first significant advance in this trend toward serialism in music was the twelve-tone atonal composition system devised by the German experimental composer Arnold Schönberg in the late 1910s. In order to promote consistency and order in atonal composition Schoenberg adopted specific precepts for his "method of composing with twelve notes only related to each other":

  1. The row, or series, must contain all twelve pitch-classes of the chromatic scale in a specific and predetermined order with no repetitions of any one pitch-class.

  2. The permissible row forms include a row's original or prime form, inversion, retrograde, and retrograde-inversion and the twelve transpositions of each. The total number of row forms, or permutations, is forty-eight and can be represented concisely in a form of chart called a matrix.

  3. Consistently atonal treatment of the row requires that no notes be doubled at the octave, tonal melodic or harmonic elements (intervals) are to be avoided, and no note should be sustained to the point where it becomes a focal pitch.

  4. In order to maintain uniformity of musical material one must make exclusive use of one series per composition.

To Schoenberg and his students, the purpose of twelve-tone technique was primarily avoidance of tonality through the systematic creation of a "democracy of tones." Experimental music without tonality prior to the development of the twelve-tone process seldom succeeded in offering enough musical cohesion to allow for movements of considerable length. Schoenberg's method solved this problem by offering the opportunity for the creation of musically significant and orderly structures that offer a piece both unity and variety. Finally, Schoenberg's twelve-tone procedure could even help the composer to avoid the traditional notions of 'theme' and 'development' that were inextricably linked with tonal composition, while offering twelve-tone music's own unique brand of both.


The Matrix

The matrix is a concise representation of all 48 possible row permutations in a 12x12 grid. The four possible row forms are given by the four directions in which one can read the row notes off of the matrix for a given row. It is important to remember that a tone row is an ordered set of pcs, while a pitch class set is unordered.

Prime Form

First, the prime form of the row (not to be confused with prime form of a pc set) is read from the matrix from left to right. As there are 12 possible transpositions of the row, there are twelve left-to-right rows (as opposed to columns) in the matrix. The transpositions of the prime form of the row are labelled with the letter P and the pc number of the first pc in the row form (P0, P3, etc.).

Inversion

Inversion is the operation of turning the melody contour upside down, as if across a horizontal line of symmetry. Whereas the prime form of a series is always given in left-to-right rows on the series' matrix, the inversion row form is the top-to-bottom reading of a column of the matrix. The inversion forms are called by the letter I followed by the number of the first pitch class (I3, I8, etc.).

Retrograde

The retrograde, or backward form of the row, is the third possible manipulation that can be used in serial composition. Since it is simply the reverse of the prime form, it is most conveniently found in a matrix by reading the desired transposition of the prime form backwards (from right to left). Because there is a one-to-one correspondence between the prime forms and the retrograde forms that are their backwards-renditions, the retrograde forms are labelled with an R and the number of the pc that ends the row form (R3, etc.). This is so that R0 is always the retrograde of P0 and not some other row form. Although a composer's main concern with using retrograde material is the question of whether the listener will recognize that two melodies are from the same source when one is presented backwards, this concern seems not to have bothered composers who work with series.

Retrograde-Inversion

The final operation that can be performed on a tone row is retrograde inversion. As this upside down and backwards presentation of a tone row is simply the inversion form in reverse, it is found on the matrix by reading the desired transposition of the inversion form backwards (from bottom to top). Once again the retrograde-inversion transpositions will be labelled with RI plus the pc number that ends the row form (RI3, etc.). Thus RI0 is always the reverse of I0.

To see how a matrix looks and functions, please try the Matrix Generator applet. If you're at a loss for what row to enter, try the row from Webern's Symphonie, Op. 21 (0, 3, 2, 1, 5, 4, T, E, 7, 8, 9, 6). To see a complete list of rows used in the works of Schoenberg, Webern, and Berg, see jlondon@carleton.edu, Row Forms in the Serial Works of Schoenberg, Berg, and Webern. Next, some of the more advanced concepts with relation to serial music will be briefly summarized.


Advanced Twelve-tone Techniques

The twelve-tone technique quickly gave rise to several further developments of the system that help to promote consistency in atonal music.

Combinatoriality

The first of these advanced techniques of twelve-tone composition, hexachordal combinatoriality, was first used by Schoenberg, although the specific terminology and the full extent of its possible uses were only adopted later by serial composer Milton Babbitt. A combinatorial series is constructed in such a way (see the Atonal Theory Glossary for an explanation) that when it is simultaneously stated with another of its permutations, each half (hexachord, six notes) of the prime form combines with the corresponding half of the other row form to create an aggregate, which is a new twelve-tone series that is not a form of the piece's primary series. To see an example of this, enter the row from Schoenberg's Op. 33A Piano Piece (0, 7, 2, 1, E, 8, 3, 5, 9, T, 4, 6) into the Matrix Generator, and see how the first hexachord of P0 and the first hexachord of I5 are literal complements (as are the second hexachords) in that they combine to form the entire chromatic collection. (The Set Analyzer, when the first six notes of P0 and I5 are entered, will analyze them as literal complements.)

Row Inbreeding (Invariance)

Another development upon twelve-tone technique that was developed primarily by Anton Webern is a sort of "row inbreeding" or the use of a series with fewer than 48 possible permutations. This is accomplished by using intervallic similarity within the row with the intention of making two or more of the row forms identical. The resulting "inbred" series is called a "derived row". To see this in action, try entering the chromatic scale (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, T, E) into the Matrix Generator, and see how this reduces the number of unique row forms. An ancillary side-effect of this technique is that invariant rows are also combinatorial.

Series with Fewer than Twelve Tones

Many who tried twelve-tone composition also experimented with applying the technique to sets of fewer than twelve tones. When Russian composer Igor Stravinsky began to compose using the twelve-tone technique in the early 1950s, some of his pieces successfully displayed serialism with fewer than twelve pitch-classes. The easiest analytic technique with this type of row is to "complete the aggregate" (i.e. fill out the rest of the chromatic scale when generating a matrix). This technique allows all 12 transpositions of the row forms to be seen, even if analytically extraneous pitch classes appear in the matrix.

Tonal Implications within the Row

Another manipulation of the twelve-tone method is found in many composers' return to tonal materials and structures within the twelve-tone system. This seems to represent a regression in technique rather than an advance because it undermines Schoenberg's most basic precept that he had developed his method in order to create orderly atonal structures that could replace tonality entirely (without reference to the tonal system). The most notable early examples of this trend in twelve-tone composition can be found in many of Schoenberg's own late compositions and in much of his student Alban Berg's twelve-tone music. Many other more recent composers have experimented with serial technique within an extended (not always harmonically functional) tonal language.

Integral Serialism: Carrying the Serial Process into All Other Parameters of the Music

Serialism, as defined as a musical trend encompassing more organizing principles than the twelve-tone system alone, came into full force around 1950 in association with the Darmstadt International Summer Course for New Music in Germany. The main idea developed by composers such as Olivier Messiaen, Karlheinz Stockhausen, and Pierre Boulez (and also by Milton Babbitt in America, though not associated with Darmstadt) is that Schoenberg's serial ordering process should be applied to all of the other musical perameters including rhythm, dynamics, timbre, the types of attacks and articulations used, etc. This so-called "Total Serialism" became the dominating force in music composition for much of the second half of the twentieth century. The process of serializing musical elements other than pitch involves the selection of twelve (or however many elements are in the series used in the piece) rhythmic values, dynamic levels, types of attacks, etc. The row orderings are then followed in adding all of these parameters to the music. The complexity and algorithmic precision of such composition techniques led composers to try allowing computers to carry out the compositional process once the series has been selected.


Contour Analysis

Post-tonal music that eschews tonal functionality certainly demands more of its audience because of the relative difficulty in perceiving harmonic relations without tonal function. Often listeners are forced (at least before intensively studying a work) to attend to more primitive meaning-creating structures that exist in music, based on perameters independent of harmony, diatonicism, intervallic size, and tonal content. These features of the music, including rhythm, motive, shape, and gesture, are often associated with the relative height of pitches regardless of pitch class or intervallic content. The techniques of contour analysis are designed to offer insights into the non-pitch-specific aspects of all music, both tonal and post-functional.

CSEGs and contour class Equivalence

Contour analysis thus ignores the exact notes and intervals of the musical material, instead attending to which notes are higher and lower. A contour segment (CSEG), is a numeric representation of the relative heights the notes in any melody or melodic fragment. It is an ordered collection, with as many elements as there are notes (n), each numbered from lowest to highest (beginning with 0) according to the height of the note within the segment. For instance, a motive that appears as (D4, C4, B4, F6) in one presentation, and as (F#5, F5, B5, D6) in a later presentation will be described using the CSEG <1,0,2,3>. This shows the motivic unity between these two melodic fragments, while ignoring their differences. (The two pc sets ((0136) and (0147)) have different (but similar) interval vectors.) Like twelve-tone rows, these ordered collections can be inverted, retrograded, and retrograde-inverted and still retain their motivic identity. For this reason, CSEGs can be organized into CSEG-classes, and a method for finding the "prime form" of contours has been determined as follows:

(n = the number of notes in the segment, or the cardinality of the CSEG)
  1. If necessary, translate the CSEG so that it consists of integers from 0 to (n - 1).

  2. If (n - 1) minus the last number in the CSEG is less than the first number in the CSEG, invert the CSEG.

  3. If the last number in the CSEG is less than the first number in the CSEG, retrograde the CSEG.

Contour Reduction Technique

The Contour Analyzer returns the prime form of the CSEG given. However, it also abstracts the general shape of longer melodic segments before determining the prime form. The process by which longer melodies can be reduced to simpler contours was introduced by Robert Morris. In a method loosely modelled upon Schenkerian analysis, the inner members of "melodically fluent" (Schenker's term, not Morris') lines between structurally important contour events can be removed from the analysis without detriment to the perception of contour. For example, a CSEG <2,0,1,3,5,6,4>, after removing the passing tones between 0 and 6, would reduce to <1,0,3,2>. Morris' criterion for deciding the structural importance of certain c-pitches is their status as "minima" and "maxima", or c-pitches at local low and high points in a melody. The first and last c-pitches are never reduced out, and the process is recursive, repruning the CSEG until it can be reduced no further (all c-pitches are maxima or minima).

A useful example of the application of contour analysis is given by Joseph Straus. He takes two excerpts from Ruth Crawford (Seeger)'s String Quartet, 1st mvt:

F#5G4Db5C6B5G#5C#5F#5E6F5G4Db5C6
<2013><2013><2013>

F#4G3B3A#4C5G#4C#4E4D#5D4A4F#4G3B3A#4
<2013><2013><2013>
<1320>
<0231><0231>

The CSEGs that Straus selects from the melody are all instances of CSEG class 4-4, <0,2,3,1>. Using Morris' technique on each melodic excerpt (using the Contour Analyzer) reveals another unifying feature: Both melodies exhibit the same general contour (CSEG class 4-7, <1032>). Further, within the process by which each is reduced (this can be seen in light grey in the Contour Analyzer) one can find a similar "middleground" contour exhibited by both melodies (<3,0,4,2,5,1,4> in the first case, and <3,0,5,2,6,1,4> in the second).

PC set comparison among CSEGs that are members of the same CSEG class or exhibit similarity relations (see Marvin/Laprade) may also provide useful insights into a piece's construction. Morris suggests that contour analysis may also be applicable to other perameters of the music other than relative pitch height, such as relative note duration and relative dynamic levels or accents. Contour analysis is thus a valuable analytical tool for extracting meaningful structures from both tonal and atonal pieces of music.


Bibliography

Atonal Analysis (and 12-Tone Analysis):

Contour Analysis:


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