Chapter 2. Pitch-Class Sets

l Pitch class setsà an unordered collection of pitch-classes. Register, rhythm, order are put away, and the basic pitch-class and interval-class identity of a musical idea remains. à basic structural unit that gives coherence.

l Schoenberg says, “The two-or-more-dimensional space in which musical ideas are presented is a unit.” No matter how it is presented (melodically, harmonically, or a combination of the two), a pitch-class set will retain its basic pitch-class and interval-class identity.

l A pitch-class set can be presented musically in a variety of ways. Conversely, many different musical figures can represent the same pitch-class set.

l Normal Formà the most compressed way of writing a pitch-class set. It makes it easy to see the essential attributes of a set and to compare it to other sets.

l Preference is for the one with the larger intervals toward the top of the stack, and whose notes thus cluster toward the bottom.

l In some ways, the normal form is similar to the root position of a triad. Both are simple, compressed ways of representing sonorities. However, the normal form has no particular stability or priority.

- 1. Excluding doublings, write the pitch classes as though they were a scale, ascending within an octave. There will be as many different ways of doing this as there are pitch classes in the set, since an ordering can begin on any of the pitch classes in the set.
- 2. Choose the ordering that has the smallest interval from first to last (from lowest to highest).
- 3. If there is a tie under Rule 2, choose the ordering that is most clustered away from the top. To do so, compare the intervals between the first and second-to-last notes. If there is still a tie, compare the intervals between the first and third-to-last notes, and so on.
- 4. If the application of Rule 3 still results in a tie, then choose the ordering beginning with the pitch class represented by the smallest integer.

l Transposition (Tn)à the transposition of a line of pitches. This operation preserves the ordered pitch intervals in the line. Because contour is such a basic musical feature, it is easy to recognize when two lines of pitches are related by transposition. T stands for transposition and n is the interval of transposition.

l A set is a collection with no specified order or contour. As a result, transposition of a set preserves neither order nor contour.

l The transpositionally equivalent sets contain the same unordered pitch-class interval; Transposition of a set of pitch classes changes many things, but it preserves interval-class content.

l We can represent these transpositonal relationships using a combination of *nodes* (circles that contain some musical element, such as a note or a set), and *arrows *(to show the operation that connects the nodes). Through these we can show the *network* and *mapping*.

l If a and b are corresponding elements in two sets related by Tn, then n equals either a-b or b-a. These two intervals of transposition (a-b + b-a) add up to 12. Ex) set1(7,8,10,11)-set2(10,11,1,2)=9, set2(10,11,1,2)-set1(7,8,10,11)=3. 9+3=12.

l Inversion (TnI)à refers to inverting a line of pitches, order is preserved and contour is reversed—each ascending pitch interval is replaced by a descending one, and vice versa. Inversion is a compound operation: it involves both inversion and transposition. Always invert first and then transpose.

l If you invert something (a note, a line, or a set) by some TnI and want to get back to where you started, just perform the same TnI again. However, if you transpose something at Tn, you will need to perform the complementary transposition, T12-n.

l Transposition and inversion of a line of pitch classes preserves the ordered pitch-class intervals, only now each interval is reversed in direction. They also have the same interval-class content.

l Generally when you invert a set in normal form, the result will be the normal form of the new set written backwards.

l Index number (sum)à offers a simpler way of inverting sets and of telling if two sets are inversionally related. When we compared transpositionally related sets, we subtracted corresponding elements in each set and called that difference the transposition number, n. However, when comparing inversionally related sets, we will add corresponding elements and call that sum an index number.

l When two sets are related by inversion and are written so that they are intervallic mirror images of each other (that will usually but not always, be true when they are written in normal form), the first note in one set will correspond to the last note in the other, the second to the second-to-last, and so on.

l TnI(a)=b, then n=a+b. In other words, inversionally related elements will sum to the index number.

l Ix/y à where x and y are pitch classes that invert onto each other. By specifying any one mapped pair, we are simultaneously specifying all of the others. All of these labels are equally valid—which one we choose depends on the specific musical context. x+y=n

l One advantage of Ix/y over TnI is that it does not emphasize inversion around C as 0, which may not have anything to do with the musical context. (IB/E=T3I) * The inversion of an inversion is always a transposition

l There are 12 possible inversions.

l Set class à a family of all sets that are related to all of the others by either transposition or inversion, and they all have same interval-class content.

l Prime Form à “most normal” of normal forms, that begins with 0 and is most packed to the left. It will be written in parentheses with no commas separating the elements. T=10 E=11 sc=set class

- 1. Put the set into normal form (1,5,6,7)
- 2. Transpose the set so that the first elements is 0. (0,4,5,6)
- 3. Invert the set and repeat steps 1, and 2. (1,5,6,7)à(11,7,6,5)à(5,6,7,11)à(0,1,2,6)
- 4. Compare the results of step 2 and step 3; whichever is more packed to the left is the prime form. (0,1,2,6)

l 12 trichordal set classes, 29 tetrachord classes, 38 pentachord classes, 50 hexachord classes

l Segmentation à It is a way of carving it up into meaningful musical groupings.

- 1. In a melodic line, consider all of the melodic segments. A rich interaction between phrase structure and set-class structure is a familiar feature of post-tonal music.
- 2. Harmonically, consider all of the simultaneities, that is, the notes sounding simultaneously at any particular point.
- 3. Notes can be associated by register. In a melody or a phrase, consider the highest (or lowest) notes, or the high points (or low points) of successive phrases.
- 4. Notes can be associated rhythmically in a variety of ways. Ex) successive downbeats, the notes heard at the beginning of a recurring rhythmic figure, the notes that are given the longest duration, etc.
- 5. Notes can be associated timbrally in a variety of ways

l The goal is to describe the richest possible network of musical relationships.

l Pitch à pitch-class

l Pitch intervals à ordered pitch-class intervals à interval classes

l Pitch-class set à Tn-type à Set class (Tn/TnI-type)