Home
Topics by Week
  Week 3
      Clowning
      Mel Inv
    Contr Inv
      Evolutio

Contrapuntal Inversion

Contrapuntal Inversion happens when two melodies exchange registers--the high voice moving to the low and the low moving to the high. Unlike melodic inversion, which required only ONE melody to invert interval directions in relation to itself, contrapuntal inversion involves two or more melodies exchanging positions in relation to each other. Notice that when two melodies invert contrapuntally, they do not necessarily invert melodically. Bach had a special term for melodies that inverted both contrapuntally and melodically at the same time...he called it the Evolutio.

Most textbooks use "invertible counterpoint" or "double counterpoint" ("triple counterpoint" when three melodies are involved) in place of what I call "Contrapuntal Inversion." The reason I use "Contrapuntal Inversion" is to make a clear distinction between this type of flipping and melodic inversion. But you should be prepared to recognize that "contrapuntal inversion, double counterpoint," and "invertible counterpoint" refer to the same technique: multiple-object inversion--where two or more melodies exchange registers.

The subject of contrapuntal inversion is very complex and could occupy our attentions for a couple of weeks if we let it. In this course I do not require that you master the complexities, for example, of writing contrapuntal inversions. I do expect, however, that you know there are three types of contrapuntal inversion. You should also be able to recognize contrapuntal inversions and identify the interval of inversion. Before getting into specifics I want you to hear some contrapuntal inversions. Make sure that you have inserted the Inventionen & Sinfonien disk before playing the examples:

Examples at the 8va from the Inventions:

Whereas melodic inversions are less common in Bach's Two- and Three-Part Inventions, they liberally employ contrapuntal inversions--especially at the 8va. The reason: contrapuntal inversions at the 8va are easier to write--inverted dissonants stay dissonant and inverted consonants stay consonant.

Two-Part Invention No. 9 in F minor: Compare the high voice in mm. 1-3 (below) to the low voice in mm. 5-7...are they the same or different? Do they start on the same pitch or different? If the same pitch, how many octaves apart are they? Now compare the LOW voice in mm. 1-3 to the HIGH in mm. 5-7. Because analogous melodies start on the same pitches (albeit an 8va removed), this is an easy-to-recognize example of contrapuntal inversion at the octave. Do you see any evidence of single-object inversion (see melodic inversion) in this example?
| Play Invention No. 9 | Stop CD | (click staves to play excerpts)

Two-Part Invention No. 2 in C minor: Like the preceding example, this one illustrates contrapuntal inversion at the 8va. But, unlike the preceding, where the high voice transposed down an 8va and the low voice up two 8vas, this example moves the high voice down a 4th (plus an octave) and the low voice up a 5th. This renders the interval of contrapuntal inversion more difficult to recognize. Actually it is not that hard...just add the two intervals of transposition (4+5) and subtract one (9-1) and you have the interval of contrapuntal inversion (see How to Recognize Contrapuntal Inversions). Study this example carefully as you may be tested on your ability to identify the interval of contrapuntal inversion.
| Play Invention No. 2 | Stop CD | (click staves to play excerpts)

Two-Part Invention No. 6 in E Major: At what interval are mm. 5-8 the contrapuntal inversion of mm. 1-4? To what extent do you see this excerpt as illustrative of melodic inversion?
| Play Invention No. 6 | Stop CD | (click staves to play excerpts)

Examples at the 10th and 12th from the Art of Fugue:

Our examples of contrapuntal inversion at the 10th and 12th are taken from the Art of Fugue. Please eject the Inventions disk, check out the AOF disk (Die Kunst der Fuge), and insert it into the CD-ROM drive before proceeding.

Canon at the 10th: (From the Art of Fugue) Notice that the low voice in mm. 3-7 is transposed up an octave to become the high voice in mm. 44-46. Think of this octave as a unison (the number 1) By contrast, the high voice in mm. 3-7 is transposed down a 10th (the number 10). The interval of contrapuntal inversion is therefore a 10th (1+10=11, 11-1=10). If you don't understand what was in the parentheses read How to Recognize Contrapuntal Inversions. Notice that there is no parallel motion in this example. This is because when, inverted at the 10th, acceptable parallels become unacceptable. For example, parallel 3rds become parallel 8vas! Read How to Write Contrapuntally Invertible Melodies and see if you can predict what parallel 6ths would become when inverted at the 10th. Before the Mid-Term Exam you should study the detailed analysis of this canon, including the score. The Mid-Term Exam will have questions on contrapuntal inversion which may be involve this canon.
| Play Canon at the 10th | Stop CD | (click staves to play excerpts)

Canon at the 12th: (From the Art of Fugue) Parallel 3rds are desirable in contrapuntal inversion at the 12th. This is because they invert to 10ths (8va+3rd). Notice the frequency of parallel 3rds in this example. Read How to Write Contrapuntally Invertible Melodies once more and suggest why there are no parallel 6ths in this canon. As before, you should study the detailed analysis of this canon (and score) for the Mid-Term Exam.
| Play Canon at the 12th | Stop CD | (click staves to play excerpts)

Categories of Contrapuntal Inversion: In 18th-century style counterpoint there are three types of contrapuntal inversion. These categories are based upon the interval at which the two melodies are inverted.

1. At the 8va: Fourths become fifths, unisons become octaves, etc. While parallel 4ths sound fine, they do not invert contrapuntally, and double ctpt. at the octave avoids them.
2. At the 10th (8va+3rd): Parallel motion tends to be avoided altogether. This is because intervals that parallel acceptably in one texture (e.g. 3rds & 6ths) become unacceptable when inverted (8vas & 5ths).
3. At the 12th (8va+5th): With the exception of 3rds (which remain 3rds), acceptable parallels become unacceptable when inverted at the 12th. Accordingly, this type of contrapuntal inversion often employs many parallel 3rds.

How to Recognize Contrapuntal Inversions: If you suspect that two passages are contrapuntal inversions of each other, you should verify that two melodies have indeed exchanged registers, the higher voice becoming the lower and vice versa. If this is true, then contrapuntal inversion has indeed occurred. Calculate the interval of inversion as follows:

1. Determine the interval that the lower voice has been transposed UP
2. Determine the interval that the higher voice has been transposed DOWN.
3. If steps 1 and 2 are at the 8va or unisons, then the double counterpoint is at the 8va. Otherwise, add the results of steps 1 and 2, then subtract 1.

How to Write Contrapuntally Invertible Melodies: Obviously, contrapuntal inversion is an effective way to provide textural variation. What may not be obvious is that it requires careful planning to make melodies contrapuntally invertible without incurring errors in voice leading. For example, in double counterpoint at the 8va, parallel P4ths (acceptable) in the original texture become parallel P5ths (unacceptable) when inverted. If you want to write contrapuntally invertible melodies, you must avoid, between simultaneously sounding pitches of the original, otherwise consonant intervals that would invert as unprepared dissonance. Here is how to calculate the interval that would be produced in each type of contrapuntal inversion:

1. At the 8va: Subtract the interval between analogous pitches in the original melodies from NINE. This will tell you the interval that will be produced when the two melodies are contrapuntally inverted. For example: a 4th before inversion will become a 5th after inversion.
2. At the 10th: Subtract the interval between analogous pitches in the original melodies from ELEVEN.. For example: a 4th before inversion will become a 7th after inversion.
3. At the 12th: Subtract the interval between analogous pitches in the original melodies from THIRTEEN. For example: a 4th before inversion will become a 9th after inversion.

Next...