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Week 1
Operations
Contrapuntal Operations
Imagine a melody moving upward a sixth and then downward a second. Call this motive PRIME. Notice that PRIME has properties of interval size and direction: two sizes (LARGER & SMALLER) and two directions (ASCENDING & DESCENDING). In the case of PRIME the first interval is the larger (and it ascends) while the second interval is the smaller (and it descends).
Three operations can be applied to PRIME that will change either the interval order or direction or both. The first of these, called RETROGRADE is a backward reading of PRIME. RETROGRADE can be produced by flipping the music horizontally. To assist you in visualizing this relationship, I've prepared a handout which you can print and then flip. After you've printed the handout, use your browser's BACK button to return here.
The second operation, called INVERSION, involves reading intervals of PRIME in the same order but moving them in the opposite direction (or the same pitches can be produced by moving the INVERSION of the original interval in the SAME direction). If you've printed the handout, the INVERSION can be produced by flipping the page vertically. As this type of inversion operates upon a single melodic line, I like to call it "melodic inversion" to distinguish it from "contrapuntal inversion" which we shall study later.
The third operation, called the RETROGRADE-INVERSION, involves reading PRIME backward AND upside-down. Flip this page horizontally (retrograde) then vertically (invert) to produce the RETROGRADE-INVERSION. You will notice that the same effect can be created by rotating the page 180 degrees.
So let's review. The order and direction of intervals in any motive can be transformed by contrapuntal operations. The effect of each operator can be illustrated by flipping the motive horizontally (RETROGRADE), vertically (INVERSION), or by rotation (RETROGRADE-INVERSION). Use your handout now to experiment with all the various "flippings" and rotations to see how the PRIME form of the motive can be transformed. You might need to hold the paper up to the light. After you have done this for awhile you should be able to predict, without moving the page, the effect each operation will have upon interval order and direction.
When you have anticipated the effect an operation would produce, use the smart table to check your answer. Click on the music to cycle through each operator. Then click the interval size (or direction) to toggle through various possibilities until you arrive at the order and direction you would predict to result. If your prediction is correct, the statement should read as TRUE. Use the table to check your answers to the questions on the handout.
Webern "Concerto for Nine Instruments"
In Assignment No. 2 you will be required to identify contrapuntal operations in the first ten measures of Webern's "Concerto" (Op. 24). The remaining paragraphs will prepare you to do this. The Concerto employs the following tone row. The row can be parsed into four trichords bearing a fundamental relationship to each other. Each row has two intervals (M3 & m2) and two directions (up & down). Sound familiar? If you think of the first trichord as PRIME, what contrapuntal operators would generate the remaining contours? [Answer]
As noted earlier, there are two ways to create the melodic inversion. Original intervals could moved in the OPPOSITE direction, or INVERSIONS of the original interval could be moved in the SAME direction. Consider, for example, the first interval of Webern's row: G ascending a M3 to B. The inversion could be produced in two ways: (1) G descending a M3 to Eb, or (2) G ascending a m6 to Eb. Either way, the same pitch is produced.
Observe that inversions of trichord (a) relocate the smaller interval (which had been at the top) to the bottom of the texture. Which trichords in the row place the m2 at the bottom? [Answer] Of the two inverted trichords, one is the INVERSION (without retrogradation) and the other the RETROGRADE-INVERSION. You can distinguish between these two by noting the interval order. In trichord (a), which we called PRIME, the order was M3-m2. So, any trichord reversing that order would be a RETROGRADE. Which trichords are retrogrades? [Answer] Notice that trichord (b) met the criteria for both retrograde and inversion.
One final note on contrapuntal operators. You've probably noticed that retrogradation is dependent upon time but that inversion is not. Let's illustrate with inversion. Squish trichord (a) PRIME together and play all of its pitches simultaneously, that is, as a chord. Now do the same for trichord (d) its INVERSION. Notice that the structure of these two chords is quite different. PRIME puts the m2 at the top of the chord, while the INVERSION puts the m2 at the bottom. Remember this: the distinction between PRIME and INVERSION does not require pitches to be placed one after the other. In other words, there is no time element to inversion.
By contrast, any type of RETROGRADE requires pitches to be played over time. Return to PRIME as a chord and compare it with trichord (c), its RETROGRADE, as a chord. The quality of these chords is identical! In other words, PRIME and RETROGRADE cannot be distinguished from each other by chord structure but only by pitches sounding in time. The time element need separate only one pitch from the other two, as the following illustrates.
SUMMARY: the distinction between PRIME and INVERSION can be made in the absence of time--that is, even if all of the pitches in the motive were to sound simultaneously as a chord. By contrast, the distinction between PRIME and RETROGRADE can be made only when pitches sound one after the other. If all the pitches in the motive were to sound simultaneously, one could not distinguish between PRIME and its RETROGRADE.
If you have understood the concepts of this page, you should be prepared to do Assignment No. 2. I suggest, however that you proceed to the next paragraph of Week 1 which contains some additional reading and listening.
