Post-Tonal Music

Set Theory Pattern matching for contemporary music. Note that many musical/math set processes do not have corresponding counterparts!

Mathematical Set Theory A set: {7,4,8} Another set: {7,4,1} Curly brackets Typically unordered (order does not matter)

Comparing sets Symbols representing several ways in which sets can be more formally compared.  is an element of  is not an element of  is a subset of  is not a subset of  the empty set; a set with no elements  union (collection of sets)  intersection (overlap of sets)

Example Example of a set proof: A  (B  C) = (A  B)  (A  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, intersection, subsets)

Musical 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 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 sets [2,6,9] [6,9,2] [9,2,6]

Equivalency The same unordered set [2,6,9] [2,6,9] [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 Clockface

Setting 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.

Forms 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 groupings

Analysis

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

How many trichord sets are 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 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

Hmmmm Dyads (2 elements - an interval) and Dodecachords (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) 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 Inversion

Variance 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 Classes The complete set of Tn and TnI is called a Set Class as in [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 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]

Vectors 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-Relations Two different sets having the same vector. Z does not stand for anything Z-related sets considered close cousins Example:

Symmetrical relationships

Webern’s Concerto for 9 Instruments

Analysis All sets = [0,1,4]

Examples of composing with sets Improvise with one set Two sets as separate entities One set evolving into another set Polysets One set harmony,other set melody Three sets in counterpoint Related sets

Large-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 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.

Assignment 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.

Triplum

Continued