Post-Tonal MusicPost-Tonal Music

Set TheorySet Theory

Pattern matching for contemporary music.

Note that many musical/math set processes do not have corresponding counterparts!

Mathematical Set TheoryMathematical Set Theory

A set:

{7,4,8}

Another set:

{7,4,1} Curly brackets Typically unordered (order does not

matter)

Comparing setsComparing sets

Symbols representing several ways in which sets can be more formally compared.

is an element of is not an element ofis a subset ofis not a subset of the empty set; a set with no elements union (collection of sets)intersection (overlap of sets)

ExampleExample

Example of a set proof:A C) C)

The union of A and the non-union of B and C equals the union of the non-union of A and B and A and C.

A+B+C does not equal A+B+A+C

Venn diagram (union, Venn diagram (union, intersection, subsets)intersection, subsets)

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Musical set theoryMusical set theory

Milton Babbitt and then Allen Forte Set: [9,3,5] Brackets Ordered or unordered Modulo 12 (pitch classes) Ordered version of above:[9,3,5] Normal (unordered/most compact) version of

above [3,5,9] Prime version (unordered/invertible) of above

[0,2,6]

Modulo12: set theory for musicModulo12: set theory for music

[47,55,66]=[11,7,6] ordered (in the music) and based on 60=middleC, 61 C#, etc.

[6,7,11] unordered (yes, incremental, but that’s for convenience) with 0=C, 1=C#, etc.

Music setsMusic sets

[2,6,9] [6,9,2] [9,2,6]

EquivalencyEquivalency

The same unordered set

[2,6,9] [2,6,9] [2,6,9]

Most compact version of [2,6,9]Most compact version of [2,6,9]

[2,6,9] = 7[6,9,2] = 8[9,2,6] = 10

If identical outer distance, then the inner voices count in compactness toward the pc that you begin counting from.

The ClockfaceThe Clockface

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Setting to 0 for comparisonSetting to 0 for comparison

[2,6,9] and [4,8,E] both equal [0,4,7] when transposed beginning on 0.

The [1,6,10] pitch-class set resolves to 9 distance but the smaller range of [6,10,1] is 7 distance and thus becomes the normal form.

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Forms of setsForms of sets

[6,9,2] ordered form[2,6,9] normal form (most

compact)[0,3,7] prime form (includes

inver, as in clockwise and

counterclockwise)

Prime form analysis of 4 Prime form analysis of 4 groupingsgroupings

AnalysisAnalysis

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Prime formsPrime forms

All sets of previous example = [0,1,3,6,8,9]

How many trichord sets are How many trichord sets are there?there?Cardinal 3 (3 elements) - 12– [0,1,2] [0,2,5]– [0,1,3] [0,2,6]– [0,1,4] [0,2,7]– [0,1,5] [0,3,6]– [0,1,6] [0,3.7]– [0,2,4] [0,4,8]Note: Major triad not present

Set types by number of elementsSet types by number of elements

Trichords (3 - 12) Tetrachords (4 - 29) Pentachords (5 - 38) Hexachords (6- - 50) Septachords (7 - 38) Octachords (8 - 29) Nonachords (9 - 12) Forte’s list 208

HmmmmHmmmm

Dyads (2 elements - an interval) andDodecachords (12 elements) not

generally counted since they either contain too few or too many to be useful

T(n)andT(n)I relationships T(n)andT(n)I relationships

T(n) refers to another set class whose pitches are all transposed up by n semitones from the original. For example, if your original set class is [1,2,7], then T(3) would be [4,5,10]. T(n)I just means that first you invert your original set, and then perform the transposition (the n here is referred to as the index number). So to get T(3)I of [1,2,7] you would first invert [1,2,7] to get [11,10,5], and then transpose [11,10,5] up by 3 to get [2,1,8].

Transposition and InversionTransposition and Inversion

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Variance and InvarianceVariance and Invariance 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 operations. Composers often 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.

Set ClassesSet Classes

The complete set of Tn and TnI is called a Set Class as in

[2,5,6] [6,7,T] [3,6,7] [7,8,E] [4,7,8] [8,9,0] [5,8,9] [9,T,1] [6,9,T] [T,E,2] [7,T,E] [E,0,3] [8.E,0] [0,1,4] [9,0,1] [1,2,5] [T,1,2] [2,3,6] [E,2,3] [3,4,7] [0,3,4] [4,5,8] [1,4,5] [5,6,9]

Compiling variations of the same Compiling variations of the same set:set: [DB D E ] [ D EB F] [EB E F#] [E F G] [F F# AB] [F# G A] [G AB BB] [AB A B] [A BB C] [BB B DB] [B C D] [C DB EB] [DB EB C] [D E DB] [EB F D][E F# EB] [F G E] [F# AB F] [G A F#] [AB BB G] [A B AB] [BB C A] [B DB BB] [C D B] [DB BB B] [D B C] [EB C DB] [E DB D] [F D EB] [F# EB E][G E F] [AB F F#] [A F# G] [BB G AB] [B AB A] [C A BB] [DB EB E] [D E F] [EB F F#] [E F# G] [F G AB] [F# AB A] [G A BB] [AB BB B] [A B C][BB C DB] [B DB D] [C D EB] [DB BB C] [D B DB] [EB C D] [E DB EB] [F D E] [F# EB F] [G E F#] [AB F G] [A F# AB] [BB G A] [B AB BB] [C A B][DB D B] [D EB C] [EB E DB] [E F D] [F F# EB] [F# G E] [G AB F] [AB A F#] [A BB G] [BB B AB] [B C A] [C DB BB]

VectorsVectors

Intervals that are inverted onto one another are in the same "interval class.” (Intervals 1 and 11 are interval class 1; 2 and 10 are interval class 2; 3 and 9 are interval class 3, and so on.)There are 6 unique interval classes, ranging from 1 to 6. Note that intervals are not the same as pitches! For example, the interval between pitches 2 and 9 is 7, which belongs to interval class 5. The interval class vector is a 6-member tally of the number of occurrences of each interval class found in a set.To obtain the tally, you find the interval between every possible pairing of notes in a set and increment the tally of that interval class. For example, consider the set [2,3,9]. There is one occurrence of interval class 1 (between the 2 and the 3), one occurrence of interval class 6 (between the 3 and the 9) and one occurrence of interval class 5 (between the 2 and the 9). Therefore the interval class vector for set [2,3,9] is 100011.

Z-RelationsZ-Relations

Two different sets having the same vector.Z does not stand for anythingZ-related sets considered close cousinsExample:

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Symmetrical relationshipsSymmetrical relationships

= major third

= minor second

= major second

= minor third

Webern’s Concerto for 9 Instruments

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AnalysisAnalysis

All sets = [0,1,4]

Examples of composing with setsExamples of composing with sets

Improvise with one setTwo sets as separate entitiesOne set evolving into another setPolysetsOne set harmony,other set melodyThree sets in counterpointRelated sets

Large-scale setsLarge-scale sets

Sets can be used structurally as well as contiguously as long as the set has some musical clarity (highest notes, same instrument, lowest notes, same dynamic, same articulation, etc.)

Referential SetsReferential Sets

Larger sets from which all of the sets of the composition are subsets. This kind of usage is especially effective if the referential set itself is used (preferably near the end of the composition in which it is used). Triplum uses the chromatic scale as a referential set.

AssignmentAssignment

Create a work or part of a work using two different related sets such that most (if not all) of the music could be analyzed accordingly. Choose a different instrumentation than your first assignment and one which you think you can get live performers to play.

TriplumTriplum1. 2. 3.

4.

5.

ContinuedContinued6. 7.

8. 9.