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Week 1: Contrapuntal Techniques

There is a venerable tradition in western classical music of generating new ideas from old. Or, put another way, western composers like to relate subsequent motives to earlier. This makes good musical sense in two ways. First, it contributes to the organic unity of a piece by relating the whole to its parts and the parts to the whole. Second, the job of the composer can be simplified if motives can be found that have the potential to enlarge themselves in various ways (or better yet, to accompany themselves outright).

The art of counterpoint represents, among other things, a method for "growing" new ideas out of old. Historically, counterpoint represented the core of musical study through the 18th century and therefore precedes the homophonic forms. From the procedural perspective, even in 19th century, during the flowering of the homophonic forms, composers employed counterpoint not only to stimulate the compositional muse, but also to connect and develop themes.


But before getting into the specifics of counterpoint, I'd like you better to acquaint yourself with this course by reading the Introduction to Online Resources. This reading should take less than an hour. The first homework assignment is embedded in these pages and must be completed by noon, Friday of the first week. So, click the link in the first sentence of this paragraph to go to the introductory pages of this course. Then keep clicking the "Next" links to thread your way through that material. If you do this properly, you'll end up at the next paragraph.

When you return to this paragraph you will have taken a detour, following the "Next" links, to read the course introduction pages. You may even have completed the first assignment. Anyway, it is time now to begin our actual study of contrapuntal procedures. Begin by reviewing chapter sixteen of Stein (pp. 121-126). These pages define, and give examples of, contrapuntal techniques: repetition, imitation and sequence; augmentation and diminution, double counterpoint, pedal point, change of mode, transposition, and stretto. You should be familiar with many of these techniques from your Harmony classes.

The most important section of the Stein reading has to do with the four contrapuntal operations of prime, retrograde, inversion, and retrograde-inversion. Because these are so important I have prepared a supplementary explanation of Contrapuntal Operations that I expect you to study with utmost care. This study will prepare you to analyze the first ten measures of Anton Webern's "Concerto for Nine Instruments." Every three-note segment of Webern's concerto is a contrapuntal derivative (transposed) of the pitches G, B, and B-flat. It shall be your task, in the second assignment of this unit, to identify contrapuntal operations used to generate eighteen trichord segments in those first ten measures. So, why don't you go to the Contrapuntal Operations page Next...

Listening: | Stop CD |

To impress upon you the amazing potential of contrapuntal operations to generate new ideas of old, I've selected three unusual pieces of puzzle music for your listening pleasure. The audio examples for these are found in the compact disk recording for Week 1. Please check that disk out now and insert it into the CD drive of your computer. So that you will not be charged for lost CD's be sure to eject the CD and sign it in when you are done.

These pieces are truly outstanding achievements of contrapuntal art, for they are actually two pieces in one. The first selection, Contrapunctus XII from Bach's Art of Fugue, contains two fugues which the composer called Contrapunctus XIIa and Contrapunctus XIIb. These fugues are mirrors of each other! Among other things, each part in Contrapunctus XIIb can be seen as the melodic inversion of a part in Contrapunctus XIIa. So that you can see this relationship I have transcribed XIIa on the top staff with its mirror on the staff directly beneath it. Click the page numbers between staves to toggle back and forth between the two. If you've inserted the CD for week 1 you should also hear them. If, after listening to them, the relationship between these mirror fugues is not clear, you might want to read What in the World is a Mirror Fugue.

The second selection is the Praeludium & Postludium from Paul Hindemith's Ludus Tonalis. I say "this selection IS" (rather than "these selections ARE") because these are really one and the same piece played right-side-up-and-forward (Prelude) and up-side-down-and-backward (Postlude)! In other words, these two pieces are retrograde-inversions of each other. To make this relationship plain, I have created a dynamic score that allows you to retrograde-invert the piece at any time by clicking the red "retrograde-invert" arrow. You will notice that the score does not actually turn upside-down and reverse. Instead, the clefs turn upside down and play the same lines and spaces backward. You might find yourself actually turning your head upside down to read the postlude! Notice that what had appeared in the treble clef of the Prelude, reinvents itself, backward, in the bass clef of the Postlude. These pieces are all the more amazing because Hindemith applies the same accidentals of the Prelude to the Postlude (albeit the accidentals are applied to different pitches in different clefs)!

In conclusion, I'd like you to listen to Fugue X of Hindemith's Ludus Tonalis. This fugue employs the same technique of melodic inversion as Bach's Contrapunctus XII. But, instead of creating two fugues out of one, Hindemith creates one fugue in which the second half is the melodic inversion of the first. Without access to the score for this fugue, I hope that you can hear this relationship by comparing the first and second halves of the fugue.
| Play the First Half | Play the Second Half | Stop CD | Homework:

  1. Assignment No. 1: Introduce yourself on the discussion page, get an email (Dana) account, and become acquainted with NAU's authentication process for submitting homework electronically.
  2. Assignment No. 2: Identify the Contrapuntal Operations of nineteen trichords in the first ten measures of Webern's "Concerto for Nine Instruments" (Op. 24).

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