VIII. 20th- and 21st-Century Techniques
Set Class and Prime Form
Brian Moseley and Megan Lavengood
Key Takeaways
- A set class is a group of pitch-class sets related by transposition or inversion.
- Set classes are named by their prime form: the version of the set that is transposed to zero and is most compact to the left (compared with its inversion).
- You can find prime form mathematically or by using the clock face.
- All possible set classes are summarized in the set class table, and are available on Wikipedia and many other websites.
The simplest way to define set class is “a group of pitch-class sets related by transposition or inversion.” This may initially seem confusing, but it’s just another kind of class. As you have learned in other chapters, “class” is another name for “group.” Recall the other kinds of classes you have already learned about.
- Pitch vs. pitch class: A pitch occurs at a specific octave, and often we conceive of it with a specific spelling. A pitch class is a group of pitches that is related by transposition or inversion.
- Interval vs. interval class: An interval has a specific distance in semitones, while an interval class is a group of intervals that are inversions of each other or related by octaves.
Introduction
Pitch-class set vs. set class (pitch-class set class) is the topic of this chapter. The reason the definition of “set class” may seem more confusing is that it involves two kinds of groups: classes and sets.
- A class is a group that is related in some way.
- A set is a group that is not necessarily related in any specific way.
As an analogy, consider biology and the way different living things are categorized. Plants in the same class are are all biologically related in a specific way: Angiospermae is a class of plants that produce flowers. But we can group together plants for other reasons: the group of plants in someone’s front yard, for example. That would be a set of plants, but not a class of plants.
So a pitch-class set is a group of pitches that the analyst decides to put together for some reason. The pitch-class set class—a term that is very unwieldy, so theorists have agreed to shorten it to the last two words, set class—is the group of groups of pitches that are all related by transposition or inversion.
Why transposition and inversion?
One way of analyzing a lot of post-tonal music is by studying the transpositional and inversional relationships between pitch-class sets. Take the short example below: two passages from Béla Bartók’s “Subject and Reflection” ( ). Comparing across the two passages, the two sets that comprise the right hand, [10, 0, 2, 3, 5] and [3, 5, 7, 8, 10], are related by T5. The two left-hand sets are also related in the same way..Now looking within each passage, the right and left hands are related to each other by inversion. In the first passage, they are related by I8; in the second, by I6.
To quickly explain why these snippets of notes all sound the same, we can say they are all members of the same set class.
Major and minor triads may be a helpful and more familiar example. Major and minor triads all sound “the same” compared to tone clusters, quartal harmonies, or even augmented or diminished triads. The reason for that is that all major and minor triads are transpositionally or inversionally related to one another.
- Triads of the same quality are transpositionally related.
- T2 of a C major triad [0, 4, 7] is a D major triad [2, 6, 9]
- T2 of an A minor triad [9, 0, 4] is a B minor triad [11, 2, 6].
- Triads of opposite quality (major vs. minor) are inversionally related.
- I0 of a C major triad [0, 4, 7] is an F minor triad [5, 8, 0].
- I2 of a C major triad [0, 4, 7] is a G minor triad [7, 10, 2].
Prime Form
Just as pitch class sets are named by their normal form, set classes are named by their prime form: the version of the set that is transposed to zero and is most compact to the left (compared with its inversion).
Set classes are named by their prime form, just as.
Note that prime form is just a label for a set class. It does not have any special status—it’s not significant if a composer uses [0, 1, 4] as a pitch-class set just because it shares the same integers as the prime form (014).
Mathematical process
Here is the process to put a pitch-class set in prime form, with an example using the motive from
.| Step | Example |
|---|---|
| 1. Put the pitch class set in normal order. | [10, 0, 2, 3, 5] |
| 2. Transpose the set so that the first pitch class is 0. | T2 = [0, 2, 4, 5, 7] |
| 3. Invert the results from Step 2 and put the result in normal order. | I0 = [5, 7, 8, 10, 0] |
| 4. Transpose the set from Step 3 so that the first pitch class is 0. | T7 = [0, 2, 3, 5, 7] |
| 5. Compare the sets in Step 2 and Step 4. Whichever set is most compact to the left is the prime form. Write the prime form in parentheses with no commas. |
(02357) is the prime form ( | ).
Using the clock face
The video in
explains the differences between normal form and prime form and reviews how to find each by using the clock face.
The Set Class Table
There are a finite number of set classes and prime forms. The table below arranges the set classes by the cardinality of the set classes. You’ll notice another number-based name for each set class in the format X–X; this is the Forte number of the set. Set class tables also pair sets together by their complements (the set which, together with the original set, will complete the twelve-tone collection). Other features of the set class table, such as the interval class vector, are discussed in later chapters.
The table has two vertical halves. The smaller set classes are in the left half, and the larger in the right. Together, the two set classes are complementary (together, they create the chromatic set).
| Prime Form |
Forte Number |
Interval Class Vector |
Prime Form |
Forte Number |
Interval Class Vector |
|---|---|---|---|---|---|
| (012) | 3–1 | <210000> | (012345678) | 9–1 | <876663> |
| (013) | 3–2 | <111000> | (012345679) | 9–2 | <777663> |
| (014) | 3–3 | <101100> | (012345689) | 9–3 | <767763> |
| (015) | 3–4 | <100110> | (012345789) | 9–4 | <766773> |
| (016) | 3–5 | <100011> | (012346789) | 9–5 | <766674> |
| (024) | 3–6 | <020100> | (01234568T) | 9–6 | <686763> |
| (025) | 3–7 | <011010> | (01234578T) | 9–7 | <677673> |
| (026) | 3–8 | <010101> | (01234678T) | 9–8 | <676764> |
| (027) | 3–9 | <010020> | (01235678T) | 9–9 | <676683> |
| (036) | 3–10 | <002001> | (01234679T) | 9–10 | <668664> |
| (037) | 3–11 | <001110> | (01235679T) | 9–11 | <667773> |
| (048) | 3–12 | <000300> | (01245689T) | 9–12 | <666963> |
| (0123) | 4–1 | <321000> | (01234567) | 8–1 | <765442> |
| (0124) | 4–2 | <221100> | (01234568) | 8–2 | <665542> |
| (0125) | 4–4 | <211110> | (01234578) | 8–4 | <655552> |
| (0126) | 4–5 | <210111> | (01234678) | 8–5 | <654553> |
| (0127) | 4–6 | <210021> | (01235678) | 8–6 | <654463> |
| (0134) | 4–3 | <212100> | (01234569) | 8–3 | <656542> |
| (0135) | 4–11 | <121110> | (01234579) | 8–11 | <565552> |
| (0136) | 4–13 | <112011> | (01234679) | 8–13 | <556453> |
| (0137) | 4–Z29 | <111111> | (01235679) | 8–Z29 | <555553> |
| (0145) | 4–7 | <201210> | (01234589) | 8–7 | <645652> |
| (0146) | 4–Z15 | <111111> | (01234689) | 8–Z15 | <555553> |
| (0147) | 4–18 | <102111> | (01235689) | 8–18 | <546553> |
| (0148) | 4–19 | <101310> | (01245689) | 8–19 | <545752> |
| (0156) | 4–8 | <200121> | (01234789) | 8–8 | <644563> |
| (0157) | 4–16 | <110121> | (01235789) | 8–16 | <554563> |
| (0158) | 4–20 | <101220> | (01245789) | 8–20 | <545662> |
| (0167) | 4–9 | <200022> | (01236789) | 8–9 | <644464> |
| (0235) | 4–10 | <122010> | (02345679) | 8–10 | <566452> |
| (0236) | 4–12 | <112101> | (01345679) | 8–12 | <556543> |
| (0237) | 4–14 | <111120> | (01245679) | 8–14 | <555562> |
| (0246) | 4–21 | <030201> | (0123468T) | 8–21 | <474643> |
| (0247) | 4–22 | <021120> | (0123568T) | 8–22 | <465562> |
| (0248) | 4–24 | <020301> | (0124568T) | 8–24 | <464743> |
| (0257) | 4–23 | <021030> | (0123578T) | 8–23 | <465472> |
| (0258) | 4–27 | <012111> | (0124578T) | 8–27 | <456553> |
| (0268) | 4–25 | <020202> | (0124678T) | 8–25 | <464644> |
| (0347) | 4–17 | <102210> | (01345689) | 8–17 | <546652> |
| (0358) | 4–26 | <012120> | (0134578T) | 8–26 | <456562> |
| (0369) | 4–28 | <004002> | (0134679T) | 8–28 | <448444> |
| (01234) | 5–1 | <432100> | (0123456) | 7–1 | <654321> |
| (01235) | 5–2 | <332110> | (0123457) | 7–2 | <554331> |
| (01236) | 5–4 | <322111> | (0123467) | 7–4 | <544332> |
| (01237) | 5–5 | <321121> | (0123567) | 7–5 | <543342> |
| (01245) | 5–3 | <322210> | (0123458) | 7–3 | <544431> |
| (01246) | 5–9 | <231211> | (0123468) | 7–9 | <453432> |
| (01247) | 5–Z36 | <222121> | (0123568) | 7–Z36 | <444342> |
| (01248) | 5–13 | <221311> | (0124568) | 7–13 | <443532> |
| (01256) | 5–6 | <311221> | (0123478) | 7–6 | <533442> |
| (01257) | 5–14 | <221131> | (0123578) | 7–14 | <443352> |
| (01258) | 5–Z38 | <212221> | (0124578) | 7–Z38 | <434442> |
| (01267) | 5–7 | <310132> | (0123678) | 7–7 | <532353> |
| (01268) | 5–15 | <220222> | (0124678) | 7–15 | <442443> |
| (01346) | 5–10 | <223111> | (0123469) | 7–10 | <445332> |
| (01347) | 5–16 | <213211> | (0123569) | 7–16 | <435432> |
| (01348) | 5–Z17 | <212320> | (0124569) | 7–Z17 | <434541> |
| (01356) | 5–Z12 | <222121> | (0123479) | 7–Z12 | <444342> |
| (01357) | 5–24 | <131221> | (0123579) | 7–24 | <353442> |
| (01358) | 5–27 | <122230> | (0124579) | 7–27 | <344451> |
| (01367) | 5–19 | <212122> | (0123679) | 7–19 | <434343> |
| (01368) | 5–29 | <122131> | (0124679) | 7–29 | <344352> |
| (01369) | 5–31 | <114112> | (0134679) | 7–31 | <336333> |
| (01457) | 5–Z18 | <212221> | (0145679) | 7–Z18 | <434442> |
| (01458) | 5–21 | <202420> | (0124589) | 7–21 | <424641> |
| (01468) | 5–30 | <121321> | (0124689) | 7–30 | <343542> |
| (01469) | 5–32 | <113221> | (0134689) | 7–32 | <335442> |
| (01478) | 5–22 | <202321> | (0125689) | 7–22 | <424542> |
| (01568) | 5–20 | <211231> | (0125679) | 7–20 | <433452> |
| (02346) | 5–8 | <232201> | (0234568) | 7–8 | <454422> |
| (02347) | 5–11 | <222220> | (0134568) | 7–11 | <444441> |
| (02357) | 5–23 | <132130> | (0234579) | 7–23 | <354351> |
| (02358) | 5–25 | <123121> | (0234679) | 7–25 | <345342> |
| (02368) | 5–28 | <122212> | (0135679) | 7–28 | <344433> |
| (02458) | 5–26 | <122311> | (0134579) | 7–26 | <344532> |
| (02468) | 5–33 | <040402> | (012468T) | 7–33 | <262623> |
| (02469) | 5–34 | <032221> | (013468T) | 7–34 | <254442> |
| (02479) | 5–35 | <032140> | (013568T) | 7–35 | <254361> |
| (03458) | 5–Z37 | <212320> | (0134578) | 7–Z37 | <434541> |
| (012345) | 6–1 | <543210> | |||
| (012346) | 6–2 | <443211> | |||
| (012347) | 6–Z36 | <433221> | (012356) | 6–Z3 | <433221> |
| (012348) | 6–Z37 | <432321> | (012456) | 6–Z4 | <432321> |
| (012357) | 6–9 | <342231> | |||
| (012358) | 6–Z40 | <333231> | (012457) | 6–Z11 | <333231> |
| (012367) | 6–5 | <422232> | |||
| (012368) | 6–Z41 | <332232> | (012467) | 6–Z12 | <332232> |
| (012369) | 6–Z42 | <324222> | (013467) | 6–Z13 | <324222> |
| (012378) | 6–Z38 | <421242> | (012567) | 6–Z6 | <421242> |
| (012458) | 6–15 | <323421> | |||
| (012468) | 6–22 | <241422> | |||
| (012469) | 6–Z46 | <233331> | (013468) | 6–Z24 | <233331> |
| (012478) | 6–Z17 | <322332> | (012568) | 6–Z43 | <322332> |
| (012479) | 6–Z47 | <233241> | (013568) | 6–Z25 | <233241> |
| (012569) | 6–Z44 | <313431> | (013478) | 6–Z19 | <313431> |
| (012578) | 6–18 | <322242> | |||
| (012579) | 6–Z48 | <232341> | (013578) | 6–Z26 | <232341> |
| (012678) | 6–7 | <420243> | |||
| (013457) | 6–Z10 | <333321> | (023458) | 6–Z39 | <333321> |
| (013458) | 6–14 | <323430> | |||
| (013469) | 6–27 | <225222> | |||
| (013479) | 6–Z49 | <224322> | (013569) | 6–Z28 | <224322> |
| (013579) | 6–34 | <142422> | |||
| (013679) | 6–30 | <224223> | |||
| (023679) | 6–Z29 | <224232> | (014679) | 6–Z50 | <224232> |
| (014568) | 6–16 | <322431> | |||
| (014579) | 6–31 | <223431> | |||
| (014589) | 6–20 | <303630> | |||
| (023457) | 6–8 | <343230> | |||
| (023468) | 6–21 | <242412> | |||
| (023469) | 6–Z45 | <234222> | (023568) | 6–Z23 | <234222> |
| (023579) | 6–33 | <143241> | |||
| (024579) | 6–32 | <143250> | |||
| (02468T) | 6–35 | <060603> |
- Straus, Joseph N. 2016. Introduction to Post-Tonal Theory. 4th ed. Upper Saddle River, NJ: Prentice Hall.
- Blank clock faces (integer notation)
- Blank clock faces (letter names)
- Set Theory Quick Reference Sheet: summarizes the definitions of pitch vs. pitch class, intervals vs. interval classes, and sets vs. set classes.
- Set Class Composition prep worksheet (.pdf, .docx). Prepares students for the set class composition by asking them to find sets and transformations.
- Set Class Composition (.pdf, .docx). Builds on the prep worksheet. Asks students to compose and analyze a 24-bar ABA form piece for unaccompanied solo instrument using set classes.
A group of pitch-class sets related by transposition or inversion. Set classes are named by their prime forms; for example, (012) is a set class.
A group of pitch classes.
A name for a set class. The prime form is the version of the set class that is most compact to the left and transposed to begin on 0.
A group of pitches that are octave equivalent and enharmonically equivalent.
Unordered pitch-class intervals; that is, the smallest possible distance in semitones between two pitch classes. Thus, mi2 and ma7 are both IC 1; ma2, mi7, +6 are IC 2; mi3, ma6, +2 are IC 3, etc. The largest interval class is six semitones, because if order is disregarded, the tritone is the largest possible interval.
In set theory, a class is a group whose members are all equivalent in some sense—transposition, inversion, octave, enharmonic, etc.
In set theory, a group whose members are not necessarily related.
A chord composed entirely of seconds (major or minor), rather than thirds or any larger interval.
The act of moving pitch content by a certain interval.
The act of mirroring pitch content vertically, so that motion down becomes up and up becomes down. Inversion often preserves intervallic content.
The number of elements in a set or other grouping.
A nomenclature for set classes developed by Allen Forte; each set class has a unique Forte number. The first number refers to the cardinality of the set, and the second number is semi-arbitrary, but generally proceeds from the most compact to the most expanded set.
The set that, together with an original set, will make the complete twelve-tone collection. Complements are literal when referring to pitch class sets and abstract when referring to set classes.