More on Twentieth Century Pitch Theory

I. Visualizing pc set content; Mod-12 arithmetic and the clock face

It is a good idea to keep a clock face in mind when manipulating pc sets: 

Zero or 12 is at the top. (Whichever pc we select as zero is at 12:00.)

Ascending motion is motion clockwise. (Going "up a half step" is the same as moving from 12:00 to 1:00.)

Descending motion is motion counterclockwise.(Going "down a half step" is the same as moving backwards from 12:00 to 11:00.)

                       Like our method of telling time, pc set manipulation is a matter of modulo 12 arithmetic.It is 7:00.We are on a train that will arrive at its destination in six hours.In order to figure out when we will arrive, we "wrap around" the clock face and figure out that the train will arrive not at thirteen o'clock (there is no such hour, at least as we tell time in the U.S.), but rather at 1:00, since we are adding 7+6 modulo 12, i.e. in a system that moves to 12 and then "resets".Such a system is called mod-12 for short.(By the way, the hands of a real clock move in only one direction  -- clockwise.We can move either clockwise or counterclockwise on our mod-12 pc clock, but generally we will prefer clockwise motion to counterclockwise motion, all other things being equal.)

By convention, C-natural is pc 0, as are any of its other possible spellings (B-sharp, D-double flat, etc.); pc 1 is C-sharp or D-flat, and so on up to B-natural, pc 11.Often we will use T for pc 10 (B-flat or A-sharp) and E for pc 11 for the sake or notational clarity.

II. Ordered and unordered sets; normal form

                       Obviously, in the actual music we are studying, the order in which the notes occur matters a great deal!But in analyzing the pitch class sets that make up much of the music of the twentieth century, we can often ignore the order in which the notes that make up each set occur, and focus on the content or membership of the sets.Ways to discuss and compare pc set content include finding a set's interval vector (already discussed) and the pc set class to which it belongs.

                       In order to facilitate comparison between pc sets and operations on pc sets, we often place a given set in its normal form.This involves ordering the pcs from lowest to highest, and selecting the first pc of the ordering that creates the smallest number of semitones between the first and last pcs.If more than one ordering provides such a smallest outside interval, the ordering that has the smallest intervals on the left is the one that is chosen.The normal form of the set C D E F G A B (pc integers 0 2 4 5 7 9 E) is actually B C D E F G A (E 0 2 4 5 7 9) -- why?

III. Operations on pc sets

                       There are two basic operations which can alter pc set membership: transposition and inversion.In addition, retrograding is an operation that alters pc set order; we will discuss it later in relation to serial music. 

                       Transposition means moving each member of the set up or down the same number of semitones. Consider the trichord (0, 3, 7), i.e. a minor triad on C.Transposition of the triad "up a minor third" means (0+3, 3+3, 7+3) = (3, 6, 10) ? yielding an E flat minor triad.Transposition "down a fifth" means (0-7, 3-7, 7-7) = (5, 8, 0) -- an F minor triad. To use the clock face idea, transposing pc 11 (B-natural) up a perfect fifth (i.e. clockwise seven semitones), for instance, leads to pc 6 (F sharp), since adding 7 to 11 lands you on 6 on the clock face.

                       For simplicity, we express all pc set transposition operations in terms of clockwise motio.We use the letter T with a subscript to describe transposition."T1" means "transposition up one semitone," while "T2" means "transposition up two semitones," and so on.We use only positive (clockwise) values for the T subscript, and thus express transposition in terms of the integers 0 through 11 (or 0 through E).This means that instead of saying T-3 for "transposition down a minor third," we say T9-- "transposition up a major sixth."These operations are equivalent in pitch class space, which ignores register (all C-naturals are the same C-natural and all A-naturals are the same A-naturals, so C up to A is the same as C down to A).

                       Inversion -- We can think of every normal-form pc set as a series of clockwise moves around the clock face from its first pitch to its last pitch.We can consider the set C E G, or 0 4 7, as a set that is created by starting at C, then moving around the clock face twice, once +4 (clockwise four semitones) to E, then once again +3 (clockwise three more semitones, for a total of seven) to G; 12+4 followed by 12+7.We invert this set form by making a series of counterclockwise moves of the same size.Our 0,4, 7 set becomes (12-4, 12-7) = (0, 8, 5) -- an F minortriad (note that major and minor triads are members of the same set class).More general and powerful ways to talk about inversion exist, but for our present purposes using the clock face in this way provides a sufficient understanding of the concept.Commit to memory the following list of mod-12 complements as an aid to understanding pc set inversion: (0, 0), (1, 11), (2, 10), (3, 9), (4, 8), (5, 7), (6,6).Note that pc 0 and pc 6 are their own complements.

IV. Set and set class

A pc set consists of any number of distinct pitch classes.  If two pc sets can be related by transposition or inversion, they are said to be members of the same set class.  In other words, if set #1 can be transformed into set #2 by inverting it, transposing it, or both, sets #1 and #2 both belong to the same set class.A few things about set classes:

(1)  Two sets can be members of the same set class only if they each have the same number of pitches: a given trichord may be a subset of a given tetrachord, for instance, but it cannot be a member of the same set class as the tetrachord.

(2)  Since we can transpose either the original or the inverted form of a given set so that it begins on any one of the twelve pitch classes, there can be a maximum of twenty-four distinct set forms in each set class.Set classes with a high degree of intervallic symmetry, e.g. the whole tone or octatonic scales, actually have fewer distinct set class forms.(There are only two distinct forms of the whole tone scale, and three distinct form of the octatonic; can you discover why?)

Prime form --For ease of comparison and reference, we select one of the 24 potential forms of each set class as a referential norm.We call this referential norm the prime form of the set class.The prime form is the normal form that (1) starts on C natural and (2) has the smallest possible intervals packed to the left.