An Introduction to Pc Set Analysis

An Introduction to Pc Set Analysis

A Brief Introduction to Pitch-Class Set Analysis

Introduction

I. Gathering Sets

1. Basic axioms: pitch classes 2. Intervals and interval classes 3. Analysis and segmentation 4. Pitch-class sets and normal form 5. Pitch-class set classes and prime form 6. Set classes and interval-class content 7. Sizes of sets 8. Set-class tables; Forte's set-class names 9. Quick review: levels of abstraction

II. Interpreting Sets

10. The uses of pc-set analysis 11. Invariance relations and set symmetry 12. Inclusion relations: subsets and supersets 13. Z-relations 14. Complement relations 15. Similarity relations, complexes, and genera 16. Referential pcs, centricity, and tonality 17. Ordered sets; serial music

Pc-Set Calculator, by David Walters Table of pc-set classes Other sources

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Introduction

Introduction

When we analyze tonal music, the basic principle of tonality -- that a single pitch class (pc) is the gravitational centre for a work's pitch structure -- generates most of the terms of reference for our analysis. For instance, in analyses of "common-practice" tonal music (much of the music written between c.1650 and c.1900), such concepts as scale steps, chord functions, and tonicization help to describe a system -- an evolving system, to be sure -- of pitch relations.

Some twentieth-century music, however, is not tonal. Not only does such music avoid the conventions of common-practice tonality; it avoids projecting any clear sense of a central pitch class. Analyzing this music is challenging. There appears to be no generalized system of non-tonal pitch structure; instead, each piece seems to create its own contextual structure. And it may be unclear just what the bases of such a contextual structure might be.

An approach that has gained favour with musicians intrigued by non-tonal music is pitch-class set analysis. Like other analytical methods, pc set analysis has a few essential features:

● basic axioms, assumptions upon which the analytical method is founded, ● abstract concepts used for classifying and interpreting musical events, ● common analytical operations with which to probe the music, and ● conventions for describing the relationships and patterns the analysis discovers.

Just what use we make of pc set analysis depends on our interests and skills. The music of Schoenberg, Berg, Webern, Stravinsky, Bartók, Debussy, Scriabin, and their contemporaries has formed the core repertory to which pc set analysis has been applied, but it has also been used to examine later and earlier music, including tonal music. Indeed, studying pc set analysis doesn't just give us a new toolbox for probing many pieces of music; it can give us a new way of thinking about pitch design in general.

Pc set analysis builds upon the observation that, in the absence of tonality's centralizing force, pitch structure often seems to be grounded in the intervallic relationships among pitches. At times these relationships also throw certain pitches or pitch classes into prominence. Some intervallic relationships may be obvious to the listener. Many others may not be, and pc set analysis has proved useful at uncovering and categorizing these relationships.

The present guide is in two parts.

I. Gathering Sets This part introduces the primary axioms and concepts of pc set analysis, including the notions of pitch class, interval class, pc set, pc set class and interval vector.

II. Interpreting Sets This part defines the kinds of relationships and patterns among sets and set classes for which analysts usually look. Uncovering these relationships and patterns is the aim of pc set analysis. Note: At present (September 2001), Part II is still under construction.

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Introduction

This guide includes some exercises, some of which you are guided through, with answers, to give you practice in dealing with analytical concepts and operations. In addition, the guide includes a pc set-class table and an annotated bibliography of basic printed and on-line resources in pc set analysis.

Acknowledgements

This guide in its present form has principally been developed over two summers, with the financial support of two grants and with the much appreciated assistance of two of my students. A grant from Mount Allison University's Innovative Teaching Fund supported the work of Jordan Fleming In the summer of 1999, and one from Mount Allison's Purdy Crawford Centre for Teaching allowed me to work with David Walters in the summer of 2001. I extend my thanks to the University and especially to Jordan and David. Their computer (and especially JavaScript) knowledge and their rapid assimilation of pc set analysis and of my aims made working with both of them a pleasure and an education. In addition to his many valuable insights throughout this guide, David is wholly responsible for the excellent JavaScript set calculator that accompanies it. I am solely responsible, of course, for any mistakes remaining in this guide.


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1. Basic axioms: pitch classes

Systems of musical analysis are always built upon certain axioms. An axiom is a "given": a statement accepted as true -- in this case for the purpose of developing an analytical method. (Of course, we can always try not accepting an axiom, to see what different analytical insights might result.)

Our introduction to pitch-class set analysis can begin with two underlying axioms.

Axiom 1: Octave equivalence

As in tonal music, the octave seems to play a special role in non-tonal music. Pitches distant from each other by one or more octaves can be treated as equivalent.

We must be clear about what the word "equivalent" means. It does not mean "identical"; pitches an octave apart are different pitches, and a composer would never use one when he means the other. Rather, "equivalent" means "conceptually equal to each other" for the purpose of our analysis.

The axiom of octave equivalence underlies the concept of pitch class (pc): all pitches that are separated by one or more octaves are given the same name and are considered to belong to the same class. This is a familiar concept among musicians. When we speak, for instance, about a piece of music in D major, it's not any particular pitch D that we think of as the tonal centre, but the pitch class D. Notice that pitch class is not itself a primary musical experience -- we hear pitches, not pitch classes -- but a concept abstracted from that experience.

Axiom 2: Enharmonic equivalence

In tonal music, pc A-sharp is not equivalent to pc B-flat; pitches belonging to these two classes appear in different tonal contexts and with different structural meanings. In non-tonal music, however, such differences no longer seem to matter. Here, enharmonically paired pcs usually seem equivalent, and we shall assume such equivalence from now on. Enharmonic equivalence reduces the number of pcs from at least twenty-one (actually more, counting double-sharps and double-flats) to just twelve.

To reflect enharmonic equivalence, we shall give the pcs new names, ones that carry no tonal biases. Using integer notation, we shall simply number the pcs from 0 to 11. You may find that the new names take a bit of getting used to, but they will simplify some of the operations we shall later use in analysis.

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In Example 1-1 this integer notation is used to label the pitch classes represented in a passage from a song by Webern.

Example 1-1. Anton Webern, "Dies ist ein Lied für dich allein," op. 3, no. 1, mm. 4-6. Pitch classes

Exercise 1-1. The integer names of pitch classes

Key concepts on this page:

● axiom ● equivalence ● octave equivalence ● pitch class ● enharmonic equivalence ● integer notation

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2. Intervals and interval classes

Like the traditional pitch-class names, the traditional names of intervals are also biased by notions of tonal context; for instance, in tonal music, a minor 7th (say, C up to B-flat) and an augmented 6th (C up to A-sharp) are different intervals. Since such contexts are absent in non-tonal music, however, enharmonically paired intervals are also now equivalent: both a minor 7th and an augmented 6th are simply counted as an interval of 10 semitones. In all of our discussions of intervals, we shall use this system of semitone counts, again written with integers.

When we speak of intervals, we often use a few subtle levels of abstraction. In fact, we can think of four such levels, two when we measure the distance between pitches and two when we measure the distance between pitch classes.

Pitch intervals

Closest to our aural experience is the concept of ordered pitch intervals: we hear intervals of a particular size and direction between successive pitches. One pitch, for instance, might be 9 semitones above the previous one, or 15 semitones below. In Example 2-1, the ascending intervals between the pitches in Webern's melody are named with a "plus" (+) sign, descending intervals with a "minus" (-) sign.

Example 2-1. Webern, op. 3, no. 1, mm. 4-6. Pitch intervals

At times we may be concerned only with a pitch interval's size, not its direction. Then we speak of unordered pitch intervals. Removing the + and - signs from the interval names in Example 2-1 leaves only information about their size in semitones. Though this is a slight abstraction from our aural experience of melodic intervals, unordered pitch intervals are indeed a part of that experience when pitches are sounded harmonically. Then we hear intervals neither up nor down, just "between."

Pitch-class intervals

More important to our analysis will be two ways of thinking about the distance between pitch classes. We must understand a crucial difference between pitch classes and pitches: one pc isn't really "higher" or "lower" than another. Rather, the twelve pcs


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form a closed modulo 12 system, like the numbers on a clock face. We can think of measuring intervals between pcs like measuring clock time (see Example 2-2 below). What, for instance, is the interval between pc 4 and pc 11? There are two possibilities: "ascending" -- going clockwise -- it's 7 semitones; "descending" -- counterclockwise -- it's 5 semitones. Taking these two directions into account, we can measure pitch-class intervals.

Example 2-2. The pitch-class clock face

In measuring pc intervals we use modulo 12 arithmetic. The "ascending" clockwise interval from pc 4 to pc 11 is 7 semitones: 4 + 7 = 11.

The "descending" counterclockwise interval is 5 semitones: 4 - 5 = 11.

In modulo 12 arithmetic any addition or subtraction that goes past 12 or below 0 takes into account that 12 = 0 on the clock face.

In Example 2-3 the intervals between the adjacent pcs represented by Webern's pitches are measured both "ascending" and "descending." Notice that the pairs of pc intervals are complementary: their integers always add up to 12.

Example 2-3. Webern, op. 3, no. 1, mm. 4-6. Pc intervals

We also use modulo 12 arithmetic to calculate intervals when we transpose pcs and groups of pcs. Though, as we've just seen, each calculation could be done in two directions, by convention pc transposition levels are calculated only in the "ascending" direction. In Example 2-4, the first five-note gesture of the Webern melody we've been considering is presented along with two other gestures, one that occurs earlier in the same song and one that occurs later.

Example 2-4. from Webern op. 3, no. 1. Pc transposition


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The three gestures are clearly related: they are transpositions of each other. In terms of pc intervals, gesture a (pcs 1 5 4 7 2) may be transposed by 2 semitones (T2) to yield gesture b (pcs 3 7 6 9 4). In turn, gesture b may be transposed by 9 semitones (T9) to produce gesture c (pcs 0 4 3 6 1). We could easily work out the other transposition relationships among these sets, taking care to calculate clockwise.

Exercise 2-1. Modulo 12 arithmetic and pc intervals

Interval classes

In tonal theory, the axiom of octave equivalence underlies another familiar notion, that of interval "inversion". The perfect 4th and perfect 5th are said to be "inverses" of each other, as are the minor 3rd and major 6th, the major 2nd and minor 7th, and so on.

This is also true of their atonal counterparts: the complementary pc intervals of 7 and 5 semitones, of 3 and 9 semitones, of 2 and 10 semitones, etc. We shall use this relationship to propose that the pc intervals of each pair are equivalent, that they each belong to a single interval class (ic). In the following table, we can see how pc interval equivalence produces just seven interval classes, each named, by convention, for the smaller of its two pc intervals. (Ic 0 will not affect our analysis, so we'll be ignoring it).

pc intervals: 0 / 12 1 / 11 2 / 10 3 / 9 4 / 8 5 / 7 6

interval classes: 0 1 2 3 4 5 6


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The concept of interval classes is an abstraction that will prove useful when we begin to consider the properties of pc sets.

Summary

The table below summarizes the four levels at which we consider intervals. The example illustrates how the progressively more abstract concepts we apply have the effect of collapsing the distinct pitch interval types we hear into a small number of classes. Conversely, we can say that the few interval classes are each represented in actual music by several types of intervals. Ic 5, for instance, encompasses the eight ordered pitch intervals listed below, along with many more.

Example 2-5. Table of interval levels

Exercise 2-2. Pitch intervals and interval classes


● pitch interval (ordered and unordered) ● pitch-class interval ● modulo 12 arithmetic ● interval class




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3. Analysis and segmentation

"Analysis" means "taking apart." Nearly all pieces of music provide such complex experiences that, if we are to study them, we must dissect them. One immediate strategy in this regard is to focus our attention separately on the elements that make up music. In pc set analysis, for instance, we investigate music�s pitch organization, leaving as secondary issues of rhythm, texture, timbre, and the other elements with which pitches are bound up in our musical experience.

Another necessary strategy is segmentation: we partition the music into analyzable fragments. Two familiar types of fragments are chords, units whose pitches sound together; and motives, units whose pitch, contour, and/or rhythmic designs are used repeatedly. In pc set analysis, the principal unit is the pitch-class set. A pc set is a (usually) unordered collection of (usually) between three and nine pitch classes. (The term "set" is a mathematical one. It betrays the fact that the axioms and operations of this analysis can be viewed as -- mostly rudimentary -- applications of mathematical set theory.)

Pc sets can be abstracted from the pitches in melodic, in harmonic, and in mixed musical segments. Deciding upon these segments -- segmentation -- is the first step when when we analyze a piece of music. Analysts usually base their decisions on "common-sense" contextual clues gained from careful score reading and listening. Sometimes, however, the intricacies of atonal music can make segmentation difficult. It is a process in which the skill, experience, and aims of the analyst all influence decisions.

In Example 3-1 the first six measures of Webern�s op 3., no. 1 song have been given sample segmentations. Example 3-1. Webern, song op. 3, no. 1, mm. 1-6


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This example illustrates several common segmentation strategies, and it suggests a few ways of indicating segments.

● Each instrument's material, considered separately. Here melodic segments in the voice are indicated by brackets. Segments in the piano�s more harmonic material are mostly enclosed by boxes. Naturally, segments can also encompass more than one instrument; the thick brackets of segments j, k, and l, for instance, are meant to embrace material above


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them for both piano and voice. (In many other places here the voice adds nothing new to the piano's pitch class material.)

● The work�s motivic gestures. This song happens to feature much varied imitation of short melodic gestures, and all appearances of these gestures might be considered segments. The material in vocal segments b, c, m, and n, for instance, recurs in several other segments.

● Gestures bounded by rests or grouped by beams or slurs. These criteria suggest segmenting the four overall phrases of the vocal part as well as several gestures in the piano�s material.

● In vocal music, gestures spanning textual units. Here the first two lines of text ("Dies ist ein Lied / für dich allein") suggest breaking segment a into segments b and c.

● Vertical sonorities � chords � with or without adjacent "non-harmonic" pitches. In common-practice harmonic analysis, there are secure criteria for omitting non-harmonic tones from harmonic segments. The analyst may or may not wish to apply -- and defend! -- these or similar criteria in pc set analysis.

● Inclusive segments. In measure 4, for instance, the separate segments formed by materials in the piano�s two staves are also combined into larger segments; and in measure 1, segment e adds a bass note to segment f.

● Overlapping segments. In measure 2, segments g, h and i overlap in the piano part. In measures 5-6, the final vocal phrase has been subjected to a similar technique, called imbrication. Its first four notes suggest themselves as a segment, n, since they present a retrograde version of the pcs in segment b. The remainder of this phrase, however, is then systematically parsed in overalapping groups of four notes: 2-5, 3-6, and 4-7.

● Non-contiguous segments. Nearly all the segments presented here feature adjacent notes, notes that we might expect to be related to each other somehow. Grouping non-adjacent pitches into segments is a less secure enterprise, and the analyst should have some defensible criterion for relating pitches distant from each other. One possibility is exemplified by segment d, which comprises the opening notes of the four vocal phrases; the notes are stemmed and linked by a beam. The discovery that these notes mirror the pitches of segment b, suggests their status as a viable segment.

The segmentations presented above are certainly not exhaustive, and it's likely that not all of them will yield analytically useful results. They illustrate, however, that you should be both open to trying alternatives in your segmentation and able to defend the choices you make. One ideal for segmentation is that -- since you will be looking for pitch-class content in these segments -- the criteria for segmentation should be non-pitch based. In practice, however, the analyst usually tempers this ideal as patterns and relationships among pc sets emerge during the analysis.


● segmentation● pitch-class set




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4. Pitch-class sets and normal form

Pitch-class sets

Suppose we are analyzing Webern's "Dies ist ein Lied für dich allein," and we come upon the two segments presented in Example 4-1.

Example 4-1. from Webern op. 3, no. 1. a: m. 2, piano; b: mm. 5-6 voice

These two five-note segments share only one notated pitch, and they don't project any clear tonal design. Might they nonetheless be related in pitch structure?

If we apply the axioms of octave and enharmonic equivalence, we see that segments a and b are indeed related in a basic way: they comprise the same pitch classes: 2, 1, 3, 7, and 6. We can say, then, that these segments represent the same pitch-class set, and we can invent a set name for both segments that expresses their equivalent status. It could be any name, as long as it's the same one for both segments. We'll use a name that simply lists the set's pc content: "[1,2,3,6,7]." Labeling any future segments that have the same pc content "[1,2,3,6,7]" would express our perception that all such segments are equivalent according to our two basic axioms.

(By the way, we're used to segment equivalence in tonal analysis. Applying just the octave equivalence axiom, we routinely judge any segment that embodies pitch classes C, E, and G to be a "C-major triad." This includes the four different segments in Example 4-2, as well as countless others. The label "C-major triad" is a pc set name, isn't it?)

Example 4-2. Some C-major triads

Naming sets: normal form


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One of the abstractions of pc sets is that they are unordered, that is, the pcs may be embodied in any order in the musical segments. Segments in which pcs 1, 2, 3, 6, and 7 appear in the order 2, 1, 3, 7, 6 or 6, 3, 1, 2, 7 or 3, 1, 6, 1, 2, 3, 7, 2, or indeed simultaneously are all classified as belonging to the same set. Since we usually name a set by citing its pc content, it will help to have a standard order for listing the pcs; then all examples of the set will have the same name.

This standard order is called normal form. By convention, the normal form is the one that lists the pcs in ascending order and in the intervallically most compact form. Here are the steps for finding a set's normal form.

1. Examining your segment, list its pc content, eliminating all pc repetitions.

2. Arrange the pc integers in "ascending" (clockwise) orders. Remember: these integers form a modulo 12 number group. Not only can you "ascend" from 1 to 2, 2 to 3, 3 to 6, and 6 to 7; you can also "ascend" from 7 to 1 (7 + 6 = 1). There are always as many possible "ascending" orders as the set has pcs. For pcs 1, 2, 3, 6, and 7, for example, the five orders are

1 2 3 6 7 2 3 6 7 1 3 6 7 1 2 6 7 1 2 3 7 1 2 3 6

3. Now choose the most compact of the "ascending" orders: the one whose interval span between the first and last pcs is smallest. Do this by subtracting the first integer from the last.

"ascending" orders

overall span

1 2 3 6 7 7 - 1 = 6 semitones

2 3 6 7 1 1 - 2 = 11 semitones

3 6 7 1 2 2 - 3 = 11 semitones

6 7 1 2 3 3 - 6 = 9 semitones

7 1 2 3 6 6 - 7 = 11 semitones

Here the most compact "ascending"ordering is 1 2 3 6 7. This is the set's normal form, and the conventional name of the set will be [1,2,3,6,7]. The square brackets, commas, and lack of spaces between digits are also a convention in naming unordered sets (though some analysts use other conventions).

4. The above steps are often enough to find a set's normal form. With some sets, however two or more orders tie for overall compactness. The rule then is to choose the set that is most compact towards the left. Measure the intervals from first to second-last note. Still tied? Measure from first to third-last note. Keep going until one set wins in compactness. For example, name the set made up of pcs 1, 4, 7, 8, and 10.

"ascending" orders

overall span

span 1st to 2nd-last pc

span 1st to 3rd-last pc

1 4 7 8 10 10 - 1 = 9 8 - 1 = 7

4 7 8 10 1 1 - 4 = 9 10 - 4 = 6 8 - 4 = 4

7 8 10 1 4 4 - 7 = 9 1 - 7 = 6 10 - 7 = 3


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8 10 1 4 7 7 - 8 = 11

10 1 4 7 8 8 - 10 = 10

As you can see, three of the five ascending orders are equally compact overall, spanning 9 semitones. Two of these still tie for compactness when we compare first to second-last pcs. Only when we compare first to third-last pcs does order 7 8 10 1 4 emerge as this set's normal form, the one most compact towards the left. We shall call this set [7,8,10,1,4].

5. With sets of great intervallic regularity, no amount of interval measuring will break the tie. Then choose the ordering that begins with the lowest number. For example, name the set comprising pcs 2, 4, 8, and 10

"ascending" orders

overall span

span 1st to 2nd-last pc

span 1st to 3rd-last pc

2 4 8 10 10 - 2 = 8 8 - 2 = 6 4 - 2 = 2

4 8 10 2 2 - 4= 10

8 10 2 4 4 - 8 = 8 2 - 8 = 6 10 - 8 = 2

10 2 4 8 8 - 10= 10

Since interval measurement here doesn't produce a single most compact order, we simply choose [2,4,8,10] rather than [8,10,2,4] as the set's name.

Exercise 4-1. Naming sets using normal form


● pitch-class set● normal form● steps for determining normal form



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5. Pitch-class set classes and prime form

When our analysis uncovered musical segments a and b, cited again in Example 5-1 below, we found that these segments are related in a basic way: their pitch-class content is identical. We say they represent the same pc set, which we've named [1,2,3,6,7]. Suppose now that, in continuing to analyse Webern's song, we find two more segments, c and d, which we'll consider in turn.

Example 5-1. from Webern op. 3, no. 1

Axiom 3: Transpositional equivalence of sets

Now, even without pc set analysis, it's clear that segment c is quite similar to b: it repeats the same melodic gesture, just transposed. In pc set terms, however, c is considered to be related to both a and b: its pc set, named [2,3,4,7,8], is a transpositional equivalent of set [1,2,3,6,7]. If we were to take this new set and transpose each of its pcs by 11 semitones (T11), its pc content would be

2 3 4 7 8


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+ 11 11 11 11 11 = 1 2 3 6 7 set [1,2,3,6,7]

Theorists say that set [2,3,4,7,8] would "map onto" set [1,2,3,6,7] by operation T11. (And, of course, [1,2,3,6,7] maps onto [2,3,4,7,8] by operation T1.) Accepting an axiom of transpositional equivalence, we are going to to say that sets [2,3,4,7,8] and [1,2,3,6,7] belong to the same "type of set"; or, put another way, they are members of the same pc set class.

A moment's reflection will confirm that there are twelve pc sets like the two we've just met. Here they all are, listed in "ascending" order beginning on 0:

[0,1,2,5,6] [3,4,5,8,9] [6,7,8,11,0] [9,10,11,2,3] [1,2,3,6,7] [4,5,6,9,10] [7,8,9,0,1] [10,11,0,3,4] [2,3,4,7,8] [5,6,7,10,11] [8,9,10,1,2] [11,0,1,4,5]

They could all be mapped onto each other by transposition, so they are all sets of the same type -- all members of the same pc set class. We're going to need a label for this class, again to help us express our sense of set relatedness if we come upon more sets of this same class. Adopting one common convention, we'll call it class "(01256)". So, to sum up our analysis so far, segments a, b, and c in Example 5-1 embody two different sets, [1,2,3,6,7] and [2,3,4,7,8], but these two sets belong to the same set class, (01256).

(Again, by the way, we're used to transpositional set equivalence and pc set classes in tonal music. A C-major triad (containing pcs C, E, G) and an F#-major triad (F#, A#, C#) are transpositionally equivalent. They, and all the other major triads, are members of a pc set class whose class name is "major triad".)

Exercise 5-1. Set transposition and transpositional equivalence of sets

Axiom 4: Inversional equivalence of sets

Finally, what about segment d in Example 5-1? Its pc content makes it an example of yet another set, [10,11,2,3,4], but one that cannot be mapped onto [1,2,3,6,7] or [2,3,4,7,8] by transposition (Notice that it's not in the list of 12 transpositionally equivalent sets above). It could be made to map, however, if we inverted it -- turned it upside down -- intervallically before transposing it (an operation called TnI).

Let's see what would happen if we inverted this set. The common convention is to invert sets using 0 as an "axis of inversion" -- the central point around which the inversion takes place. To invert a set around 0, simply subtract each of its pcs from 0 (= 12). In such an inversion, pc 1 always maps to 11 (and vice versa), pc 10 to 2, pc 3 to 9, pc 4 to 8, pc 5 to 7 and both pcs 6 and 0 to themselves. So, set [10,11,2,3,4] inverts as follows:

pc: 10 11 2 3 4




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inverts to pc: 2 1 10 9 8 normal form: [8,9,10,1,2]

Here's a graphical representation of the same inversion operation, in which the pcs are "flipped" around the 0 axis:

By doing this inversion, we have produced a new set, [8,9,10,1,2], and it happens that this is a set we could transpose to map onto the others (it is in the list above). Acording to a final axiom, Webern's set [10,11,2,3,4] is inversionally equivalent to sets [1,2,3,6,7] and [2,3,4,7,8]. Because of this, we shall classify it also as belonging to the same type of set -- as a member of set class (01256).

Of course, there are twelve sets that are transpositionally equivalent to [10,11,2,3,4] also. Here they are.

[0,1,4,5,6] [3,4,7,8,9] [6,7,10,11,0] [9,10,1,2,3] [1,2,5,6,7] [4,5,8,9,10] [7,8,11,0,1] [10,11,2,3,4] [2,3,6,7,8] [5,6,9,10,11] [8,9,0,1,2] [11,0,3,4,5]

These are all inversionally equivalent to all the 12 sets of our first list, so we now have a total of 24 sets that, through axioms of transpositional and inversional set equivalence, are considered to be members of the same set class: (01256).

Exercise 5-2. Set inversion and inversional equivalence of sets

Briefly to recap: what is a pc set class? You can think of it as family of pc sets whose members -- according to axiom or convention -- are all related to each other by transposition (Tn) or by inversion (TnI). Some authors refer to pc set classes as "Tn / TnI types." If two sets can be mapped onto each other through one of these two operations they are classified as equivalent: they are held to belong to the same class. If not, they belong to different classes. Normally, a set class will have 24 member sets, as we've found class (01256) to have.

Naming set classes: prime form

As with naming sets, we need a conventional naming scheme for set classes, so that all sets we assign to the same class -- as we've done with [1,2,3,6,7], [2,3,4,7,8], and [10,11,2,3,4] for instance -- will be class-labeled identically. Since classes are defined by axioms of transpositional and inversional set equivalence, the conventional class name is one that reduces the (usually 24) normal form names of all the sets in the class to one supra-normal form called prime form. We cite the prime form of a set by putting it in a normal form, most compact towards the left, that begins on pc 0.

Here are the steps for finding the prime form of a set:




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1. Make sure your set is in normal form.2. Now invert this set, and place the result also in normal form.3. Now, transpose both normal forms to begin on 0.4. Finally, compare the two 0-transposed sets. Which is more compact towards the left? That one is taken as the prime form

-- and the name of the class to which your original set belongs.

The tricky part can come in step 2. Normally if you invert a set, subtracting each of its integers from 0, the result will appear in "reverse-normal" order. You simply have to re-reverse this order to place your inversion in its own normal form. You can then proceed to steps 3 and 4. For example, here's the procedure for finding the prime form of set [2,3,4,7,8]:

However, the intervallic makeup of some classes of sets means that the normal form of a set and of its inversion are not always simply reverses of each other! For example, look what happens if we try to find the prime form of set [8,10,11,1,2,5]:

It turns out that the normal form of a set with pcs 4, 2, 1, 11, 10, and 7 is [10,11,1,2,4,7] -- something you should be able to verify for yourself by now. In this case, if you were to carry out Step 2 carelessly, merely reversing the digits of your inversion, the normal form -- and then the prime form -- you end up with would be incorrect! Since you can never tell when you'll face a set of this sort, you must always take care when normalizing sets.

Finally, some sets are inversionally symmetrical. As the example below shows, making your final choice of prime form is easy here: the original set, and its inversion produce the same result when transposed to 0. Let's try set [4,7,8,11].


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Exercise 5-3. Classifying sets using prime form


● transpositional equivalence● pc set class● inversional equivalence● prime form● steps in finding prime form






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6. Set classes and interval-class content

What relates all the pc sets in a set class is that they are transpositional and/or inversional equivalents of each other. These, we must remember, are the two axiomatic properties that define a set class. There is also, however, another attribute that all sets in the same class share: they all have the same interval-class content. A set's interval class content is the complete inventory of the interval classes that the set contains.

Let's consider sets [1,2,3,6,7] and [10,11,2,3,4] from class (01256) again and take inventory of their interval classes. Each pc in a set forms an interval with every other pc, so a set with five pcs will have ten such intervals. In the tables below, we subtract the first integer from the second to measure the interval. We then list the classes to which these intervals belong.

[1, 2, 3, 6, 7] ic 1 2 11 3 21 6 51 7 6

2 3 12 6 42 7 5

3 6 33 7 4

6 7 1

[10, 11, 2, 3, 4] ic 10 11 110 2 410 3 510 4 6

11 2 311 3 411 4 5

2 3 12 4 2

3 4 1

We see that both of these sets have the same interval-class profile: each set features three instances of ic 1, two instances of ics 4 and 5, and one instance of ics 2, 3, and 6. These same interval classes are also found among the pcs of the other twenty-two sets in class (01256).

This shared ic profile is not quite as abstract as it might seem. A set's ic content tends to give the set a particular sound quality no matter how the set is disposed in the music. It gives all the sets in a class a similar quality. (Again, we know this from our experience with tonal music. What is it that makes all major triads sound much alike? Their interval-class content, which features single intervals of a m3/M6, M3/m6, and P4/P5 (ics 3, 4, and 5) and lacks any m2/M7, M2/m7, and tritone intervals (ics 1, 2, and 6). Diminished triads have a different ic profile and sound markedly different.)

Interval-class vectors

All sets in the same class have the same ic content (and hence the same sound quality). Sets in different classes usually have different ic contents (and different sound qualities). To compare ic profiles easily, we need a standard way of writing them, and for this purpose we use the interval-class vector. The ic vector is a simple array of the interval classes from 1 to 6, with a listing of how often each class is represented. For example, we've just seen that, in all sets of class (01256), there are three instances of ic 1, two instances of ics 4 and 5, and one instance of ics 2, 3, and 6. When we array these in an ic vector we get

ics: 1 2 3 4 5 6


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number of occurrences: 3 1 1 2 2 1

We commonly say, then, that set class (01256) has an interval-class vector of 311221. The digits represent occurrances of the ics from 1 to 6. The vector for the major triad mentioned above, which features only single intervals of ics 3, 4, and 5, and none of ics 1, 2, and 6, is 001110.

Exercise 6-1. Interval-class content and ic vectors

The Z-relation

Now here's one of the curious facts about the world of pc set classes. What generally distinguishes sets of a particular class is a unique interval-class profile. In several cases, however, sets which cannot be mapped onto each other by transposition or inversion -- that is, sets of different classes -- nonetheless display the same profile! The sets of class (0123479) and those of class (0123568), for instance, all share a particular ic profile, 444342. Despite the fact that sets of these classes cannot be mapped onto each other, they do share the same sound quality. (By the way, you should already have found a pair of sets in Exercise 6-1 that share the same ic vector--and a curious vector at that!)

In the conventions of pc set analysis, these set classes are considered distinct (not equivalent), but closely related. For lack of a better label, the relation is called the Z-relation; sets classes (0123479) and (0123568) are said to be "Z-related". Remarkably, what seems to be a pretty abstract level of relatedness sometimes yields surprising concrete embodiments in atonal music.


● interval-class content● interval-class vector● Z-relation

Page last modified 8 August 2001 / GRT





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7. Sizes of sets

How many notes we group into a segment is a decision we make in analyzing music. The number of different pcs in a set is another matter. The maximum number is, of course, twelve -- and there is obviously only one set containing all twelve pcs. Since some composers liked to avoid hinting at tonality by using all the pcs democratically, the aggregate, as the 12-pc set is called, can play a central role in atonal pitch structure. (It's also the set used in 12-tone music, where the order in which the pcs appear becomes important.)

Sets containing just 2 different pcs and those containing 10 or 11 also can be significant in pc set analysis, but most attention is usually given to sets with from 3 to 9 pcs. This is partly because there are enough different set classes with these numbers of pcs that classifying the sets we find becomes analytically worthwhile.

The number of pcs a set contains is called its cardinality. Below are listed the usual names for the sets of various cardinalities, along with the number of different set classes there are for each cardinality.

cardinality name number of distinct set classes

2 dyad 6 (the interval classes from 1 to 6)

3 trichord 12

4 tetrachord 29 (including one Z-related pair)

5 pentachord 38 (including three Z-related pairs)

6 hexachord 50 (including fifteen Z-related pairs)

7 septachord 38 (including three Z-related pairs)

8 octachord 29 (including one Z-related pair)

9 nonachord 12

10 decachord 6

11 undecachord 1

12 the aggregate 1

In all there are 208 distinct set classes of cardinalities 3 to 9.

You've probably noticed that the list above has a pattern. The number of trichord and nonachord classes, the number of tetrachord and octachord classes, and the number of pentachord and septachord classes is the same. This is not an accident but is a natural property of the 12-pc universe.


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● the aggregate set● cardinality● names for set cardinalities

Page last modified 8 August 2001 / GRT



buttons_top.gif

8. Set-class tables; Forte's set-class names

There is a limited number of set classes: 208 classes of sets with between 3 and 9 pcs. While gathering set data, analysts often keep a table of the classes handy. Such a table is included with this guide. Each entry in this table has up to five columns of information:

1. Forte's set-class names. Though prime forms are often used as class names, these alternative names have also become quite popular among theorists since they were first proposed in Allen Forte's The Structure of Atonal Music (1973), one of the pioneering texts of pc set analysis. In Forte's list, each set class has a hyphenated name (for example, 4-27). The first number is the cardinality of the set class (so trichords all begin with 3-, tetrachords with 4-, and so on). The second number is simply a listing number: the first class in Forte's list of tetrachords (0123) is "4-1," the next (0124) is "4-2", and so on. (The order in which Forte assigned his names is a bit different from the one used in our table. That's why some Forte numbers appear out of order in this table.) A convenient name for class (01256), then, is "class 5-6".

2. Prime forms. Within each of the cardinality groups, the table lists classes in ascending order of pc content [Note]. In some of the larger classes, letter "T" stands for pc 10.

3. Interval-class vector.One of the shared characteristics of all sets in a class is an identical interval-class content. The ic vector is a way of listing this content. The six number positions in the vector stand for ics 1 to 6. The numbers filling those positions show how many times that ic is represented in any set in the class.

4. Invariance. We normally expect twenty-four distinct sets in any set class: twelve that are equivalent under transposition (Tn) and twelve more that are equivalent to the first twelve under inversion-plus-transposition (TnI). With 81 of the 208 classes, however, some of these twenty-four sets duplicate each other: they have the same (invariant) pc content. This column lists the number of invariant sets under Tn and under TnI. The former number is always at least 1, since T0 of a set is naturally invariant. Those classes with entries higher than 1 / 0 display some degree of intervallic symmetry; the greater the symmetry, the greater the amount of invariance among sets.

5. "Z-mates" While most set classes have unique interval-class vectors, some pairs of classes happen to share their ic profiles. By convention, these pairs are called "Z-related" classes, and their Forte names include the "Z". So hexachords 6-Z6 and 6-Z38 are distinct classes, but both share ic vector 421242. It's useful to have the Z-mate of such classes listed in the table, because one sometimes finds concrete embodiments of this seemingly abstract relation in atonal music.

EXERCISE 8-1. The Pc set-class table (not yet available)


http://www.mta.ca/faculty/arts-letters/music/pc-set_project/pc-set_new/pages/pc-table/packed.html

http://www.mta.ca/faculty/arts-letters/music/pc-set_project/pc-set_new/pages/pc-set_new/exercises/exer6-1.html

buttons_top.gif


● The pc set-class table● Forte's set-class names

Page last modified 25 September 2004 / GRT



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9. Quick review: levels of abstraction

Our aim in using pitch-class set analysis is to explore the pitch structure of (usually) atonal music. Music analysis always invokes abstract concepts that help us to classify and make sense of our concrete musical experience. As you have discovered, pc set analysis deals in a few levels of abstraction, some of which we have been routinely using on tonal music -- perhaps without considering them consciously. We'll quickly review these levels.

1. The pitch material of music consists of pitches. These pitches are separated in pitch-space by pitch intervals: ordered (directed) ones between melodic pitches, unordered ones between harmonic pitches.

2. We classify the pitches we hear into pitch classes (pcs) using axioms of octave and enharmonic equivalence. There are 12 pitch classes.

3. We then consider groups or sets of pitch classes, abstracted from the musical segments we analyze, for example, sets [1,2,3,6,7], [2,3,4,7,8], and [10,11,2,3,4]. We usually use the normal form names for these sets. We are mostly interested in sets of between three and nine pcs.

4. Using axioms of transpositional and inversional set equivalence, we can classify sets into pc set classes. We usually name set classes by citing either their prime forms, for example, (01256), or their Forte set-class names, for example, 5-6.

5. With the growing abstraction of pitch-related concepts, concepts of interval also grow abstract. We measure the distance between pitch classes -- pitch-class intervals -- using modulo 12 arithmetic. As with tonal pc intervals, interval pairs which add together to make an octave are considered "inverses" of each other. We classify such pairs as equivalent, grouping them into interval classes (ics). Pc set classes are usually characterized by unique ic contents (conveniently written as ic vectors), though some pairs of classes ("Z-related" classes) happen to share their ic profile.

Each of these levels of abstraction and of classification tells us something about the relatedness of concrete pitches and groups of pitches to each other.

We can say that the process of analyzing an atonal work is a three-stage one, though in real analysis we likely interweave these stages:

1. segmenting the music 2. gathering and classifying the pc sets that we find in the segments, and 3. interpretating the set data we've gathered.

The last, interpretive, stage is, of course, the real point of any analysis. The first stage lays the crucial (if at times problematical) groundwork for the analysis. The second stage is the one involving the basic mechanics of pc set analysis, mechanics we have been practising. Happily, many of these sometimes tedious mechanics can be computerized, leaving us more time to spend on interpretation. This guide is accompanied, for instance, by an excellent Web-based pc-set calculator, written by David Walters.


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(See the list of Other Sources for some other on-line set calculators.) As with arithmetic and the hand-held calculator, however, we must learn the mechanics and their rationale before we can sense how the data we gather might be interpreted.




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10. The uses of pc-set analysis.

What use you make of the set and set-class data you gather depends on your interests, analytical skill, and imagination. The interests of musicians using pc set analysis vary from the concrete to the abstract. On the concrete end, you can analyze a single work, exploring the structural and expressive aspects of the work's pitch organization. Broadening out, you can attempt to characterize the pitch-structural "language" found in a repertoire, perhaps the works of a single composer like Bartók or of a cultural milieu like the Second Viennese School. At the most abstract, you can explore the inherent properties of set classes, gaining insights into the 12-pc universe on which almost all of our western music is built. Such insights can be of interest to composers as well as to those who are just intellectually intrigued by the world of tones.

A basic concern for most music analysis is that the analyzed segments of the music be shown to relate to each other. It's usually felt that most pieces demonstrate coherence among their parts. Unity may also be a hallmark of many works -- a sense that their coherence is governed by one central force or principle. (To be sure, twentieth-century music does offer examples of works whose apparent aims are multiplicity rather than unity and fractured rather than coherent experience.)

A search for coherence clearly animates the basic operations of pc set analysis: the grouping of pitches into just twelve pitch classes, the categorizing of pc sets into a limited number of set classes. The same search also drives most of the terms by which pc set data are analyzed. Most are concern with relations among pcs, pc sets, and pc set classes. Again, which relations an analyst chooses to focus on is up to the analyst. There is, however, a number of standard relations whose investigation forms a common thread in many pc set analyses. Pc set theory has developed mathematical operations to handle some of these relations. In the present guide we shall touch, at least briefly, on the main types of set and set-class relations.


http://www.mta.ca/faculty/arts-letters/music/pc-set_project/pc-set_new/pages/page10/page10.html [3/15/2008 8:00:59 PM]


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11. Invariance relations and set symmetry.

When sets are transposed (Tn) or inverted and transposed (TnI), their pc content may

● Completely change. Transposing set [3,5,6,9] up by 5 semitones (T5) yields set [8,10,11,2]. This new set shares none of its pcs with [3,5,6,9]; it is wholly variant from [3,5,6,9]. Likewise T1I of [3,5,6,9] yields set [4,7,8,10], also completely variant from the original set.

● Partly change. T3 of [3,5,6,9] yields set [6,8,9,0], preserving pcs 6 and 9. T10I of [3,5,6,9] yields set [1,4,5,7], this time preserving just pc 5. Both of these new sets are partly invariant from the original set (though they vary completely from each other).

● Remain completely the same. Both T0 (of course) and T6 of set [2,3,8,9] yield [2,3,8,9] again. And both T5I and T11I of [2,3,8,9] return those same four pcs. Set [2,3,8,9] happens to remain wholly invariant under these four operations.

Composers often appear to make use of variance and invariance properties among sets of the same class. For instance, partial pc invariance among sets can be a marker that certain pcs (the invariant ones) are being stressed or made salient or that these pcs are acting as links among different sets. Conversely, a composer can avoid unwanted stress on pcs by making sure that pc content changes among different sets.

(You've long ago experienced the concrete effects of controlled pc invariance. Sets of class 7-35 (013568T)--the diatonic scale class--preserve 6 of their 7 pcs under T5 and T7, that is, when transposed by a perfect 4th or perfect 5th. That's why in tonal music the subdominant and dominant keys are so closely related to the tonic key.)

SymmetryComplete invariance among sets is a marker of a set class's inherent symmetry: the more aspects of symmetry a set's intervallic arrangement possesses, the more times it will display complete invariance when transposed or inverted. We can illustrate this fact by looking at some sets on the 12-pc clockface. In Example 9-1 are displayed (in red) sets representing classes 3-9 (027) and two tetrachords, 4-9 (0167) and 4-28 (0369).

Example 9-1.

a. set 3-9 [0,2,7] b. set 4-9 [0,1,6,7]) c. set 4-28 [0,3,6,9]


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a. Notice the interval pattern of trichord [0,2,7] as we travel fully around the clockface. If we begin and end at pc 7, we can

arrange this pattern symmetrically: 5 - 2 - 5. Now, this pattern does not allow set [0,2,7] to be transposed (rotated about the clock) and still comprise pcs 0, 2, and 7. So it's only invariant under the static case of transposition T0. The red line, however, reveals that this set does has an axis of inversional symmetry; notice that the line splits the clockface into two mirror images. The set can be inverted ("flipped") about this axis and retain the same pcs. Sure enough, T2I of [0,2,7] is [0,2,7]. Since any set in class 3-9 will be invariant with an inverted version of itself, there are only twelve, instead of twenty-four, distinct sets in this class.

b. Sets of class 4-9 show more symmetry. First of all, its repeated interval pattern around the clockface, 1 - 5 - 1 - 5, means that it can be transposed invariantly. We can see that if set [0,1,6,7] is rotated half-way around the clockface (that is, operation T6), it will remain invariant. Furthermore, its two axes of inversional symmetry indicate that at two inverted levels, (T1I and T7I), the set will again be [0,1,6,7]. With four invariant versions of this pretty symmetrical set, there are but six, rather than twenty-four, distinct sets in class 4-9 (0167).

c. Tetrachord 4-28 (familiar as the diminished-seventh chord) is more symmetrical still. It has an extremely regular interval pattern around the clockface: 3 - 3 - 3 - 3. Four different rotations of [0,3,6,9]--T0, T3, T6, and T9--will return the same pcs. And the four inversional-axis lines tell us that four inverted forms--T0I, T3I, T6I, and T9I--are also invariant. So we have eight invariant versions and only three distinct sets in class 4-28, a fact that you probably already knew about the diminished-seventh chord.

Remember that the fourth column in our table of pc set classes cites information about the number of Tn levels (always at least T0) and TnI levels at which sets in each class remain invariant. Any number above "1, 0" in this column indicates some degree of intervallic symmetry.

It is clear that, as they grappled with composing outside of the systematic underpinnings of tonality, many early 20th-century composers gravitated towards symmetrical properties (and not just in pitch structure) to give coherence to their compositions. Hence the popularity with these composers of such highly symmetrical set classes as the whole-tone scale, class 6-35 (02468T), and the octatonic scale, class 8-28 (0134679T).

Determining invariance


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You may need at times to find out just how many pcs in a set will remain invariant under different levels of transposition or inversion. Here are some math shortcuts to help you. These shortcuts are based on the intervallic relations within and among sets. To save space, we'll just learn the shortcuts; later you can try figuring out their rationale (it's a great cure for insomnia).

Invariance under Tn

This one is simple: just use a set's interval-class vector, available in the table of pc set-classes. The number of pcs that will remain invariant when a set is transposed by n semitones is the same as the entry for n's interval class in the vector, except for ic 6, where it is double the entry. For example, let's try set [3,5,6,9] a set of class 4-12 (0236). The table tells us that the ic vector for this class is 112101.

● Notice that this set contains just one interval of ic 1. If, then, you transpose [3.5.6.9] by either T1 or T11, just one pc should remain invariant. Well, T1 of [3,5,6,9] is [4,6,7,10]. T11 of [3,5,6,9] is [2,4,5,8].

● The ic vector contains no entries for ic 5. T5 of [3,5,6,9] is the wholly variant [8,10,11,2]. T7 of [3,5,6,9] is [10,0,1,4].

● The ic vector entury again contains just one entry for ic 6. T6 of [3,5,6,9] is [9,11,0,3] as pcs 3 and 9 map onto each other.

So we can use the ic vector for any set to determine how it will behave when transposed.

Invariance under TnIIt's possible to construct vectors that can tell us about invariance under TnI levels, but they are a bother to use. Much easier is to construct an invariance matrix for the set you're examining. Below is such a matrix for set [3,5,6,9].

3 5 6 9

3 6 8 9 0

5 8 10 11 2

6 9 11 0 3

9 0 2 3 6

As in the example, we construct a matrix by

● writing out the set along the top and along the left side. ● adding all the pairs of numbers (in mod 12) to fill in the matrix.

The number of times any number n appears inside the matrix is the number of pcs that remain invariant under TnI. Moreover, following each matrix number back to the sides tell us which numbers map onto each other in the TnI operation.


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In our matrix, for example, number 10 appears just once in the matrix, as the sum of 5+5. T10I of [3,5,6,9] is set [1,4,5,7], retaining just pc 5. However, number 11 appears twice in the matrix, as the sum of 5+6 and of 6+5. T11I of [3,5,6,9] is [2,5,6,8], mapping pcs 5 and 6 onto each other.

3 5 6 9 3 5 6 9

3 6 8 9 0 3 6 8 9 0

5 8 10 11 2 5 8 10 11 2

6 9 11 0 3 6 9 11 0 3

9 0 2 3 6 9 0 2 3 6

Even constructing matrixes is a bit tedious, of course, so you can let the computer do it for you. Jay Tomlin's set calculator, listed on the Other Sources page of this guide, will construct invariance matrixes for inversion as well as for transposition.

Exercise 11-1. Invariance under Tn

Exercise 11-2. Invariance under TnI

Exercise 11-3. Set symmetry


● variance and invariance under Tn and TnI ● set symmetry ● axis of inversional symmetry ● determining invariance under Tn ● determining invariance under TnI



http://www.mta.ca/faculty/arts-letters/music/pc-set_project/pc-set_new/pages/bibliography/bibliography.html



http://www.mta.ca/faculty/arts-letters/music/pc-set_project/pc-set_new/exercises/11-2/11-2_a1.html

http://www.mta.ca/faculty/arts-letters/music/pc-set_project/pc-set_new/exercises/11-2/11-2_a1.html




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12. Inclusion relations: subsets and supersets.

Suppose in our analysis of a piece we have identified the sets given in Example 10-1 below. Can we perceive any relationships among the sets of segments a), b) and c)?

Example 12-1

The relationship of the overall set of segment a) to that of segment b) is a close and obvious one: b) is a subset of a), and a) is a superset of b). That is, set [10,1,2,4], is wholly included within the larger set [1,2,4,7,10]. We can also see that the upper voice of both segments features a shared subset, trichord [10,1,2]. Both [10,1,2,4] and [1,2,4,7,10] are supersets of [10,1,2]. This subset-superset relation is sometimes called the inclusion relation.

Now set c), [5,7,8,11], is clearly not a subset of set a); the two only share a single pitch class. It has, however, a more abstract class inclusion relation. That is, set class 4-12, of which [5,7,8,11] is a member, is a subset of class 5-31, of which [1,2,4,7,10] is a member. (Of course, [5,7,8,11] is also a class equivalent of [10,1,2,4]; both are members of class 4-12. The same can be said of their trichordal subsets, [7,8,11] and [10,1,2], both members of class 3-3.)

Both sets and set classes obviously have many possible subsets and supersets--for example, any pentachord has five tetrachordal and ten trichordal subsets--and most will probably not be musically significant in any given piece of music. At the set level, you should be sure that your segmentations make sense, that any subset-superset relations you identify have real, not contrived, musical importance.

Class inclusion relations might seem too abstract ever to have concrete musical importance. However, class subsets and supersets do share a familial resemblance based on interval-class content. We may be able to group some or many of the sets we find in a piece into families based on inclusion ties. For instance, we might find with more analysis that the piece from which the above segments are taken uses the octatonic scale as its basis for pitch structure. After all, classes 3-3, 4-12, and 5-31 are all subsets of the octatonic class 8-28. (In fact, the combined pc material of sets a), b) and c) is set 8-28 [1,2,4,5,7,8,10,11].)

At other times, shared class subsets may be used to associate two or more families of material. In some of Stravinsky's music, for example, octatonic (class 8-28) and diatonic (class 7-35) materials appear to be linked through shared subsets like the tonal triad (class 3-11), and dominant-seventh-type tetrachord (class 4-27). In fact, the prominence of subsets traditionally associated with diatonicism is one feature that made the octatonic scale attractive to several twentieth-century composers.


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● inclusion relations ● subsets and supersets ● class inclusion




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13. Z-relations.

We know (from page 6) that two sets are "Z-related" if--despite belonging to different classes--they share the same interval-class content. Since the "sound" of a set depends on this intervallic content, sets of Z-related classes will sound similar. Apart from this fact, the Z-relation, like other class-based relations, might seem abstract. Analysis, however, sometimes reveals the musical association of sets belonging to Z-related classes.

Exercise 13-1. Z-relations (Not yet available)


● Z-relations




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14. Complement relations.

The overall pc universe in which sets operate is the chromatic scale, the aggregate of twelve pcs. The complement of a set comprises whichever pcs in the aggregate that set excludes. The set complement of tetrachord 4-12 [3,5,6,9], for example, is octachord 8-12 [7,8,10,11,0,1,2,4]; together, these two sets make up the aggregate.

More abstractly, class 4-12 and class 8-12 are said to be complementary classes, because these classes include the above two complementary sets. If we take different sets in these same classes--for instance, 4-12 [0,3,4,6] and 8-12, [9,10,0,1,2,3,4,6]--such sets are considered to be "class complements" or "non-literal" complements. They no longer combine to make up the aggregate--in fact, in this example the tetrachord is also a subset of the octachord--but they do belong to the complementary classes 4-12 and 8-12.

By the way, these examples demonstrate a valuable aspect of Forte's set-class names: classes and their complements are given the same ordinal numbers in his list. Classes 4-12 and 8-12 are complements, for example, as are classes 3-10 and 9-10, and classes 5-Z38 and 7-Z38. Because there are twelve pcs in the aggregate, trichords will be complements of nonachords, tetrachords of octachords, and pentachords of heptachords.

What about hexachords, which contain half the aggregate? They form a special case. "Non-Z" hexachord classes are always complements of themselves. For example, the complement of the whole-tone hexachord 6-35 [0,2,4,6,8,10] is another whole-tone hexachord 6-35, [1,3,5,7,9,11]. Z-related hexachord classes, however, are complements of their Z-mates. For example, the complement of hexachord 6-Z6 [1,2,3,6,7,8] is hexachord 6-Z38 [9,10,11,0,4,5].

Complementary sets can be an important feature of 12-tone music and of other music where the full chromatic aggregate is repeatedly used. While certainly more abstract, class complement relations can also be of value. For one thing, class complements sometimes seem to pop up in formally important junctures in atonal music. For another, there is a family resemblance between sets of complementary classes, based on similarities of their interval-class profiles. For example, the ic vector set class 4-12 is 112101; that of class 8-12 is 556543. Notice that in the octachord, there are four more copies of each interval class, except just two more of ic 6. All complementary sets are related in this way:

The difference in ic vector entries is equal to the difference in the sets' cardinalities (and to half that difference in the case of ic 6).

So ic vector entries will differ by 6 (3 for ic 6) between a trichord and its nonachord complement. They will differ by 4 (2) between tetrachord and octachord complements. They will differ by 2 (1) between pentachord and heptachord complements. And they will differ not at all between complementary hexachords--which is why hexachords form the "special case" outlined above.


Page 14

It's also noteworthy that almost all sets larger than hexachords are capable of embedding their class complements as subsets. Composers may make use of such a combination of inclusion and complement relations. We may, for instance, find Stravinsky embedding sets of one of his favourite types, tetrachord 4-23 (0257) within passages based on the "diatonic octad" class 8-23 (0123578T).

Finally, you may find it useful, when calculating the prime forms of large sets, to deal with their complements instead. What, for instance, is the prime form of nonachord [7,8,9,10,0,1,2,3,4]? Well, the pcs it omits form set [5,6,11]. With a bit of experience, you can probably figure out in your head that the prime form of this trichord is (016) and that it belongs to class 3-5, so our nonachord must belong to class 9-5 (012346789).

Exercise 14-1. Complement relations


● complementary sets ● class complements ● ic profile relationships between complements






Page 15

15. Similarity relations, complexes, and genera.

We are used to the fact that in tonal music, motives and harmonies sometimes appear in variant guises, for instance, with one or two pitches or intervals changed. Generally we still regard these variants as variants, not as wholly unrelated musical segments. With its emphasis on pc and ic exactitude for classifying sets as equivalent, pc set analysis might be unable to address our perception of similar but non-equivalent sets. And, given the abstract nature of set classes, determining similarity among classes is not as intuitive as perceiving it among motives.

Some theorists have devised means to discuss the similarity of sets and classes according to shared pc and ic content. The conventions of naming pcs and ics with numbers has helped these theorists to formalize and quantify measurements of similarity in a mathematical way. In The Structure of Atonal Music, for instance, Allen Forte recognizes similarity relations of a few degrees between sets of equal size, based on their sharing all but one pc and on their having maximally or minimally similar ic contents (Forte 1973, 46-60). Other theorists have proposed their own criteria for determining similarity between sets, as well as ways to quantify degrees of similarity numerically.

The desire to express relatedness among sets in a work--and set classes in the abstract--has also led to theorizing--especially, again, by Allen Forte--about set complexes and set genera, super-classifications to which set classes can belong. In both, a single class or a small group of classes may be held to serve as a "nexus" or to have "generated" a whole web of classes, principally through inclusion relations (but also through complement and Z relations). Forte's theory of set complexes occupies the second part of The Structure of Atonal Music, pp. 93-177. His theory of set genera appeared in "Pitch-Class Set Genera and the Origin of Modern Harmonic Species," Journal of Music Theory 32 (1988): 187-271.




Page 16

16. Referential pcs, centricity and tonality.

Almost all the relations discussed above make no reference to the possibility that--even in the absence of common-practice tonal conventions--works may project a focus on certain pcs, privileging them above other members of the aggregate. It is clear, however, that a good deal of 20th-century music does indeed project what is sometimes called pc centricity; the music of Stravinsky, Bartók, and Debussy comes easily to mind here. This focus of certain pcs can be addressed with pc set analysis, for example, by noting pc invariance among different sets.

Analysts also learn to keep their eyes (ears!) open for certain referential set classes with "tonal" associations. For instance, the diatonic scale (class 7-35 (013568T)), the diatonic octachord (8-23 (0123578T)) and the octatonic scale (8-28 (0134679T)), as well their subsets, are all prominently represented in Stravinsky's music. They have associations with common-practice pitch structures. The ways in which these materials are used sometimes reproduce features of common-practice music, while also projecting differences. For instance, the dominant-seventh-type arpeggios heard at the beginning of Stravinsky's Symphony of Psalms may sound familiar, but analysis suggests that Stravinsky is using these sets of class 4-27 (0258) in an octatonic rather than diatonic context. Conversely, the opening of Petrushka is recognizably diatonic in context, though the set of class 4-23 (0257) Stravinsky uses here is not a common structure in traditional Western tonal music.

One of the most explored aspects of music early 20th-century music is whether or not music that projects pc salience or centricity is "tonal". Tonality has been defined in too many ways to answer this question definitively. Exploring the issue, however, has led to several attempts to combine pc set analysis with other analytical tools, especially with traditional harmonic analysis and with Schenkerian analysis.




Page 17

17. Ordered sets; serial music.

In analyzing atonal music, we often pay no attention to the sequence in which pcs are represented in the analyzed segments; we deal with unordered pc sets. At times, however, we may notice that the order in which pcs appear is itself a structural feature. For example, this order may be subject to patterned manipulation. If so, we would likely want to draw attention to any patterns we perceive. Although the word "set" is sometimes used for such ordered patterns, they are more often referred to as series (or rows); and music in which series are a prominent structural aspect is serial music.

Sets of any size can be manipulated serially, but the most significant kind of serial music has been twelve-tone music, in which the 12-pc aggregate set is so treated There is, of course, only one 12-pc set class, so structural designs cannot be created here on the basis of pc-content or ic-content alone. Only order manipulations can distinguish one use of the aggregate from another (and hence relate one to another). It was Arnold Schoenberg who, in the early 1920s, developed the basic operations used in 12-tone composition: the common order permutations of retrograde, inversion, and retrograde-inversion; and such secondary features as complement relations among serial subsets. Analysis of 12-tone music is beyond the scope of the present guide. In recent years, however, pc set analysis has proved quite useful in revealing structural aspects of serial music other than those dealt with by serial analysis alone.




PC Set Calculator

PC Set Calculator

0 1 2 3 4 5 6 7 8 9 10 11

Type your set here:

Forte name:

Subset:

Superset:

How to use this Calculator © 2001 David Walters

Normal form:

Prime form:

ic vector:

Z-Mate:

n values for Tn

invariance:

n values for TnI

invariance:

Subsets:

Supersets:

Matrix:

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PC Set Calculator

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Exercise

Table of pc set classes

3/9 4/8 5/7 6

Trichords Nonachords

Forte name

Prime form

Ic vector Invariance Tn, TnI

Z-mate

3-1 012 210000 1, 13-2 013 111000 1, 03-3 014 101100 1, 03-4 015 100110 1, 03-5 016 100011 1, 03-6 024 020100 1, 13-7 025 011010 1, 03-8 026 010101 1, 03-9 027 010020 1, 13-10 036 002001 1, 13-11 037 001110 1, 03-12 048 000300 3, 3

Forte name

Prime form


Z-mate

9-1 012345678 876663 1, 19-2 012345679 777663 1, 09-3 012345689 767763 1, 09-6 01234568T 686763 1, 19-4 012345789 766773 1, 09-7 01234578T 677673 1, 09-5 012346789 766674 1, 09-8 01234678T 676764 1, 09-10 01234679T 668664 1, 19-9 01235678T 676683 1, 19-11 01235679T 667773 1, 09-12 01245689T 666963 3, 3

3/9 4/8 5/7 6

Tetrachords Octachords

Forte name

Prime form


Z-mate

4-1 0123 321000 1, 14-2 0124 221100 1, 04-4 0125 211110 1, 04-5 0126 210111 1, 04-6 0127 210021 1, 14-3 0134 212100 1, 14-11 0135 121110 1, 04-13 0136 112011 1, 04-Z29 0137 111111 1, 0 4-Z154-7 0145 201210 1, 14-Z15 0146 111111 1, 0 4-Z29

Forte name

Prime form


Z-mate

8-1 01234567 765442 1, 18-2 01234568 665542 1, 08-3 01234569 656542 1, 18-4 01234578 655552 1, 08-11 01234579 565552 1, 08-7 01234589 645652 1, 18-5 01234678 654553 1, 08-13 01234679 556453 1, 08-Z15 01234689 555553 1, 0 8-Z298-21 0123468T 474643 1, 18-8 01234789 644563 1, 1

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Exercise

4-18 0147 102111 1, 04-19 0148 101310 1, 04-8 0156 200121 1, 14-16 0157 110121 1, 04-20 0158 101220 1, 14-9 0167 200022 2, 24-10 0235 122010 1, 14-12 0236 112101 1, 04-14 0237 111120 1, 04-21 0246 030201 1, 14-22 0247 021120 1, 04-24 0248 020301 1, 14-23 0257 021030 1, 14-27 0258 012111 1, 04-25 0268 020202 2, 24-17 0347 102210 1, 14-26 0358 012120 1, 14-28 0369 004002 4, 4

8-6 01235678 654463 1, 18-Z29 01235679 555553 1, 0 8-Z158-18 01235689 546553 1, 08-22 0123568T 465562 1, 08-16 01235789 554563 1, 08-23 0123578T 465472 1, 18-9 01236789 644464 2, 28-14 01245679 555562 1, 08-19 01245689 545752 1, 08-24 0124568T 464743 1, 18-20 01245789 545662 1, 18-27 0124578T 456553 1, 08-25 0124678T 464644 2, 28-12 01345679 556543 1, 08-17 01345689 546652 1, 1

8-26 0134578T* 456562 1, 1

8-28 0134679T 448444 4, 48-10 02345679 566452 1, 1

3\9 4\8 5\7 6

Pentachords Septachords

Forte name

Prime form


Z-mate

5-1 01234 432100 1, 15-2 01235 332110 1, 05-4 01236 322111 1, 05-5 01237 321121 1, 05-3 01245 322210 1, 05-9 01246 231211 1, 05-Z36 01247 222121 1, 0 5-Z125-13 01248 221311 1, 05-6 01256 311221 1, 05-14 01257 221131 1, 05-Z38 01258 212221 1, 0 5-Z185-7 01267 310132 1, 05-15 01268 220222 1, 15-10 01346 223111 1, 05-16 01347 213211 1, 0

Forte name

Prime form


Z-mate

7-1 0123456 654321 1, 17-2 0123457 554331 1, 07-3 0123458 544431 1, 07-4 0123467 544332 1, 07-9 0123468 453432 1, 07-10 0123469 445332 1, 07-6 0123478 533442 1, 07-Z12 0123479 444342 1, 1 7-Z367-5 0123567 543342 1, 07-Z36 0123568 444342 1, 0 7-Z127-16 0123569 435432 1, 07-14 0123578 443352 1, 07-24 0123579 353442 1, 0

7-Z18 0145679* 434442 1, 0 7-Z38


Exercise

5-Z17 01348 212320 1, 1 5-Z375-Z12 01356 222121 1, 1 5-Z365-24 01357 131221 1, 05-27 01358 122230 1, 05-19 01367 212122 1, 05-29 01368 122131 1, 05-31 01369 114112 1, 05-Z18 01457 212221 1, 0 5-Z385-21 01458 202420 1, 05-30 01468 121321 1, 05-32 01469 113221 1, 05-22 01478 202321 1, 1

5-20 01568* 211231 1, 0

5-8 02346 232201 1, 15-11 02347 222220 1, 05-23 02357 132130 1, 05-25 02358 123121 1, 05-28 02368 122212 1, 05-26 02458 122311 1, 05-33 02468 040402 1, 15-34 02469 032221 1, 15-35 02479 032140 1, 15-Z37 03458 212320 1, 1 5-Z17

7-7 0123678 532353 1, 07-19 0123679 434343 1, 07-13 0124568 443532 1, 07-Z17 0124569 434541 1, 1 7-Z377-Z38 0124578 434442 1, 0 7-Z187-27 0124579 344451 1, 07-21 0124589 434641 1, 07-15 0124678 442443 1, 17-29 0124679 344352 1, 07-30 0124689 343542 1, 07-33 012468T 262623 1, 1

7-20 0125679* 433452 1, 0

7-22 0125689 424542 1, 17-11 0134568 444441 1, 07-Z37 0134578 434541 1, 1 7-Z177-26 0134579 344532 1, 07-31 0134679 336333 1, 07-32 0134689 335442 1, 07-34 013468T 254442 1, 17-28 0135679 344433 1, 07-35 013568T 254361 1, 17-8 0234568 454422 1, 17-23 0234579 354351 1, 07-25 0234679 345342 1, 0

3/9 4/8 5 6

Hexachords

Forte name

Prime form


Z-mate

6-1 012345 543210 1, 1

6-2 012346 443211 1, 0

6-Z36 012347 433221 1, 0 6-Z3

6-Z37 012348 432321 1, 1 6-Z4

6-Z3 012356 433221 1, 0 6-Z36

6-9 012357 342231 1, 0

6-Z40 012358 333231 1, 0 6-Z11

Forte name

Prime form


Z-mate

6-Z10 013457 333321 1, 0 6-Z39

6-14 013458 323430 1, 0

6-Z13 013467 324222 1, 1 6-Z42

6-Z24 013468 233331 1, 0 6-Z46

6-27 013469 225222 1, 0

6-Z19 013478 313431 1, 0 6-Z44


Exercise

6-5 012367 422232 1, 0

6-Z41 012368 332232 1, 0 6-Z12

6-Z42 012369 324222 1, 1 6-Z13

6-Z38 012378 421242 1, 1 6-Z6

6-Z4 012456 432321 1, 1 6-Z37

6-Z11 012457 333231 1, 0 6-Z40

6-15 012458 323421 1, 0

6-Z12 012467 332232 1, 0 6-Z41

6-22 012468 241422 1, 0

6-Z46 012469 233331 1, 0 6-Z24

6-Z17 012478 322332 1, 0 6-Z43

6-Z47 012479 233241 1, 0 6-Z25

6-Z6 012567 421242 1, 1 6-Z38

6-Z43 012568 322332 1, 0 6-Z17

6-Z44 012569 313431 1, 0 6-Z19

6-18 012578 322242 1, 0

6-Z48 012579 232341 1, 1 6-Z26

6-7 012678 420243 2, 2

6-Z49 013479 224322 1, 1 6-Z28

6-Z25 013568 233241 1, 0 6-Z47

6-Z28 013569 224322 1, 1 6-Z49

6-Z26 013578 232341 1, 1 6-Z48

6-34 013579 142422 1, 0

6-30 013679 224223 2, 0

6-16 014568 322431 1, 0

6-31 014579* 223431 1, 0

6-20 014589 303630 3, 3

6-Z50 014679 224232 1, 1 6-Z29

6-8 023457 343230 1, 1

6-Z39 023458 333321 1, 0 6-Z10

6-21 023468 242412 1, 0

6-Z45 023469 234222 1, 1 6-Z23

6-Z23 023568 234222 1, 1 6-Z45

6-33 023579 143241 1, 0

6-Z29 023679* 224232 1, 1 6-Z50

6-32 024579 143250 1, 1


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Other Sources

Internet Sources

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Introduction to Set Theory, by Larry Solomon (Pima College, Tucson, AZ) includes a table of pc set classes. Note that Solomon's list of prime forms is based on transpositional but not inversional equivalence. He does, however, relate inversionally equivalent set classes in his application of Forte names. (His table lists, for instance, classes 4-27 (0258) and 4-27B (0368), a pair reduced to 4-27 (0258) in Forte's classification). Solomon's table also includes "descriptive names" for many set classes.

Set Helper, by James Halliday (Indiana University) a Java applet set calculator, which yields normal form, prime form, Forte name, and interval-class vectors of sets.

Print Sources

Forte, Allen. The Structure of Atonal Music. New Haven: Yale University Press, 1973. ML 3811 F66 An "Urtext" of pitch-class set analysis, by one of its chief developers. The book's focus is on analytical method.

Rahn, John. Basic Atonal Theory. New York: Longman, 1980. MT 40 R2 One of the basic sources for the study of pitch structure in atonal music. The explanations are rather mathematical, but Rahn also attempts to address the listening experience.

Straus, Joseph N. Introduction to Post-Tonal Theory. Englewood Cliffs, NJ: Prentice Hall, 1990. MT 40 S96 1990 An excellent introduction to both pitch-class set analysis and twelve-tone analysis. Straus takes a topic which often seems dauntingly mathematical and keeps its musical relevance clear. Includes analyses of twelve selected pieces.


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Documents

An Introduction to Pc Set Analysis