| Twelve-Tone Composition |
| If you can call up a mental image of the piano keyboard, or if you can find a concrete one nearby, do so. Upon examining the image, you will find that the keyboard has a repeating unit: a certain organization of black and white keys. In fact, in each repeating unit of keys there are 5 black keys and 7 white keys. Each unit is a particular octave. Note that each repeating unit is a collection of all of the pitch values commonly employed in the Western harmonic system. This is called the chromatic scale, and I have given the names of the notes below: c c# d d# e f f# g g# a a# b There are, as we may have expected from the amalgamation of 5 black and 7 white, 12 pitches. The letters with a "#" after them are heard by striking the black keys, the remainder by striking the white. It will be useful later on to know that the distance between two adjacent pitches (eg. f# and g) is a "semitone." A tone is the distance between two pitches seperated by one pitch (eg. d and e). Once we have grasped these facts, it is only a small step to understanding the basic premise of the twelve-tone system. This is that each of the 12 pitches of the chromatic scale must be heard before any of them can be heard again. For example, if I selected a key on a piano and struck it, I would have to strike all other keys in that 12 pitch unit (in any octave) before I could strike that first key once more (again in any octave). The only stumbling block to understanding this premise is it's strangeness. More, and more difficult, stumbling blocks are still to come. The order in which the pitches are played, called the row, is extremely important. It will be repeated in exactly its original form and will be subject to a variety of transformations. The original ordering of the twelve tones is referred to as the prime (P) form of the row. This can be employed in a composition simply by repeating it over and over. It is important to note that the particular octave from which an instance of a pitch is drawn is relevant to the sounds generated in a performance of the music, but not to the pitch order of a resultant row. Each pitch is represented by a number: "c" is zero and "b" is 11. So the row (g d# b a# a c e c# f f# d) would be written as (7 3 11 10 8 9 0 4 1 5 6 2); this is our prime row. The prime row can be altered by transposition. In other words, all pitches of the row can be increased or decreased by any number of semitones, as long as every pitch is changed by the same amount. A statement of our prime row is notated P(7), meaning that it begins with g, the pitch numbered 7. P(5) is the row created by decreasing each note in P(7) by 2 semitones: (5 1 9 8 6 7 10 2 11 3 4 0). (Any number less than 0 must be increased by 12 to get a valid integer. Similarly, any number greater than 11 must be reduced by 12. Only the integers 1 to 11 count.) The prime row can also be altered by reversing its order. This is done by simply ordering the notes of the row from last to first. This transformed version is called the retrograde (R). The retrograde of P(7) is called R(7) and is (2 6 5 1 4 0 9 8 10 11 3 7). Transposition and Retrograde transformations can be combined. For example, R(11) would be the prime row transposed upwards by 4 semitones and played backwards. The most difficult to understand transformation is the inversion (I). Described simply, it is the upside-down version of the prime row. If all notes in a prime row were taken from the same octave, a jump from a lower note to a higher note would become, in the inverted form, a jump from a higher note to a lower one. I(7) is the inverted row starting on g. To obtain the inverse of the row, we leave the first note as it is and subtract each of the others from 14. If the result of this subtraction is greater than 11, we subtract 12 from that number. The row's transposition number is assigned based on its first pitch. So I(7) is (7 11 3 4 6 5 2 10 1 9 8 0). We have seen that transposition can be combined with both retrograde and inversion. Retrograde and inversion can also be combined with eachother, producing retrograde inversion (RI). RI(11) is the prime row raised by 4 semitones, inverted, and played backwards. All of the above transformations can be executed in a single chart called the matrix. A program created by Dan Cavanagh that will create a matrix can be accessed via this hyperlink. In all, the prime row can be altered in 47 ways, so that a composition restricted to a single row may actually use up to 48 different rows. However, certain rows (e.g. palindromic rows) produce matrices in which several of the transformed rows are identical. The row (7 9 11 1 3 5 6 4 2 0 10 8) can be transformed into only 23 unique rows. In this row, "5 6" is the axis of symmetry in the row. Notice that each jump on the left half of the row is identical to the corresponding jump on the right. This feature of palindromic rows was employed by Berg in extremely clever and complex ways. The basic tenet of the use of rows is that every note in a piece of music must be derived from the prime row. In Berg's 12-tone music, the basic tenets of serialism are often changed. Berg employed matrices and all of the transformations but often mixed twelve-tone music with non-twelve-tone music. He would also use more than one prime row within a single composition. Often Berg would divide the 12 tones of a row into two 6-note collections, or hexachords. Any two rows having the same hexachordal content would be the same row. This is referred to as an unordered row. In composing the opera Lulu, Berg assigned a seperate row to each of the principal characters of the opera. Thus purely abstract note progressions have dramatic meaning. This is of course very similar to Wagner's leitmotive, the musical phrase that represented a character, object, or state of being. Berg used true leitmotive in both of his operas. The row transformations found in Berg's twelve-tone music themselves have dramatic significance. |
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