Pitch-class sets Chapter 2 February F ebruary 3rd, 2014

Pitch-Class Sets pc set: an unorder unordered ed collection of pitch-classes.

Used for identifying pitch motives (no rhythm, timbre, order, etc.) The motive from Nacht 4, 7, 3 would be a good example of a pc set. Compositionally speaking, using pc sets is one way to create unity in the piece while having great variety in other domains like timbre and rhythm. A pc set can be written in any order.

Normal Form Normal form: the most compressed version of a pc set.

To ﬁnd normal form (pg. 36): Eliminate all repeated notes. Compress all the pcs to ﬁt within a single octave. Rotate the set until the boundary interval is the smallest possible. If there is a tie, choose the set that is the most “closely packed” to the left. If there is still a tie, choose the form that begins with the smallest integer. Normal form is somewhat similar to root position in triads (but without hierarchical implications). Normal form is notated in square brackets, with commas between each element.

Normal Form Put this string of notes into normal form on the musical staff.

Transposition (Tn)

Transposition (Tn) Traditionally, transposition just means that the pitch level of something was changed. As long as the contour is the same, transposition is recognized; it doesn’t matter if the intervals were transposed exactly. With pc transposition, contour can be changed, since pitch class assumes octave equivalence.

Transposition (Tn) Transposition is very easy to calculate in integer notation: simply add the same number to all pcs (mod 12). The transposition operation is notated as Tn , where the subscript number indicates the amount of transposition in number of semitones (the number added to the original pc to yield the transposed pc). Transpose the following set of integers by T 5: 370389 805812

Transposition (Tn)

Transpose the following pc sets as indicated. T3 [8, 0, 3] T9 [1, 4, 7, 10]

Inversion (TnI)

Inversion (TnI) Like with transposition, inversion of pcs is both similar and different to the traditional concept of inversion. Pcs are inverted around 0, by convention. E.g., 3 is inverted to -3 (i.e., 9); 7 is inverted to -7 (5); 6 is inverted to -6 (6). The inversion operation is abbreviated as TnI because it is actually a compound operation which involves transposing and inverting. Sometimes you will see the abbreviation In but the meaning is the same.

Inversion (TnI) Like transposition, inversion preserves ordered pc intervals, but the direction is reversed. Sets that are mirror images of each other are inversions of each other.

Inversion (TnI) Solve T0I(3). 1. Invert 3. This gives 9. 2. Transpose 9 by 0 semitones. This gives 9. 3. T0I(3) = 9.

Inversion (TnI) Index numbers is another way to

conceptualize inversion that many people ﬁnd easier to understand. When two sets in normal form are related by inversion, you can add up the numbers of the sets and it will yield the same index number each time. This number is the same as ’n’ in TnI. To ﬁnd the index number for two elements within inversionally-related sets, simply add them together.

Inversion (TnI) To perform TnI on a set, subtract each element of the set from n. T4I on [e, 1, 2, 6] 4 4 4 4 -11 1 2 6 5 3 2 10 Normal form: [10, 2, 3, 5]

Inversion (I ) x y

There is still another way to conceptualize inversion. x y

I stands for the inversion that maps x onto y. x and y may represent any two pcs. There can be multiple names for the same kind of inversion.

Inversion Invert the following pc sets as indicated. T9I [9, 10, 0, 2] T6I [4, 7, t, 0] What is the index number of these pairs of sets? [5, 9, e] [7, 9, 1] [4, 5, 8, e] [t, 1, 4, 5] Invert [B, C, D, E, F, G] about I D D .

Set Classes (sc)

Set Classes (sc) Each of these pc sets are members of the same set class

because they are related by transposition and/or inversion.

Set Classes (sc) A set class (sc) is a group of pc sets that are related by transposition and/or inversion. Normally, a set class will have 24 different pc sets as members, but some have fewer. Why 24? Why sometimes fewer? What is the fewest number of members a set class could have?

Prime Form

Prime Form Prime form is one instantiation

of a set class that is representative of the set class. Prime form is the form that begins on 0 and is most closely packed to the left. Two begin on 0… But [0, 1, 4] is more closely packed to the left. (014) is the prime form of this set class.

Prime Form

Prime Form You can also use the clock face to help envision and ﬁnd prime form. Allan Forte created a table of all the set classes in their prime forms, and gave each prime form a number. This is in Appendix 1. Notice how few trichords there are!